# The Method of Characteristics > Ordinary differential equations you can solve; partial ones look like a different world. The method of characteristics is the bridge. Starting from the transport equation u\_t + c u\_x = 0, this lecture builds the idea out of the chain rule alone: walk through the x-t plane along the right curve, and the equation collapses into an ordinary one that says u never changes. That gives the general solution f(x - ct) in a single line. We then run the same two steps on a speed that varies with time, whose characteristics are parabolas, and finish with the nonlinear case, where each characteristic carries the value that sets its own slope, the fast ones catch the slow ones, and a shock is born. - Canonical watch page: [The Method of Characteristics](https://academa.ai/lectures/the-method-of-characteristics) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Mathematics - Published: 2026-08-28T22:51:44.000Z - Updated: 2026-08-28T22:51:44.000Z - Duration: PT778S (12 minutes 58 seconds) - Chapters: 5 - Views: 0 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M14TZPH69VE3VGWWFYD6E49T/0/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14TZPH69VE3VGWWFYD6E49T/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M14TZPH69VE3VGWWFYD6E49T/0/dark/poster.jpg) ## Description Walk along the right curves and a first-order PDE becomes an ODE: characteristics, the solution f(x - ct), and where shocks come from. ## Chapters - [00:00–04:42.685 · A First-Order PDE](https://academa.ai/lectures/the-method-of-characteristics?t=0) - [04:42.685–06:34.182 · Walking Along a Curve](https://academa.ai/lectures/the-method-of-characteristics?t=282.6851458333333) - [06:34.182–08:21.837 · The General Solution](https://academa.ai/lectures/the-method-of-characteristics?t=394.1817083333333) - [08:21.837–10:3.878 · A Speed That Changes](https://academa.ai/lectures/the-method-of-characteristics?t=501.8374583333333) - [10:3.878–12:58 · When Characteristics Collide](https://academa.ai/lectures/the-method-of-characteristics?t=603.8784791666667) ## Transcript ### [00:00 · A First-Order PDE](https://academa.ai/lectures/the-method-of-characteristics?t=0) The method of characteristics turns a partial differential equation into an ordinary differential equation along carefully chosen paths. We will build that idea from a moving shape, then follow it until those paths collide. Let's start with water. A long, narrow tank of it, seen from the side, and something has disturbed the surface: a swell, sitting here. Now pick a spot along the tank (any spot) and ask one very small question. How high is the water, right there? Down here it is barely raised. Walk toward the swell and it is higher. Keep going, up to the top, higher still, and past that it drops away again. So every position hands you back one number. That is all a function is, and this one has a name: u. Except water does not hold still, and that ruins it. Watch. And I owe you an explanation for what you are about to see, because the swell is going to keep its shape exactly. That is not a law of water: it is the simplest case, and the one to understand first. Picture the whole tank drifting downstream at a steady speed, carrying whatever sits on it the way a river carries a leaf. Nothing spreads, nothing dies. It just goes. Now stop moving, and let it come to you instead. Stand at this one place. The water climbs past you, and then it sinks away again, and you never took a step. So knowing where you are is no longer enough to know how high the water is. You have to say when as well. u of x and t. Two inputs means two ways to change it, so let us change them one at a time. First, freeze the clock. Nothing moves; this is one instant, held. Stand here, and walk this far along the tank. You changed x by that much, and look: u changed too, by that much. Same instant, different place, different height. Take a shorter walk and u changes less. A longer one, and it changes more. There is a rate hiding in there (height gained per step of x) and that rate is what u sub x means. Now the other one. Put x back where it was and leave it there: one place, this stake, and you do not move again. On the right I will keep a record of the height at that one stake, against time. Not against position; the position is fixed now. Against time. Watch both dots. The swell arrives, the water climbs the stake, and the record climbs with it. The crest passes, and the two of them come back down together. One number, drawn twice. You never moved, and u changed anyway. So there is a second rate (height gained per tick of the clock) and that one is u sub t. Two rates, then. And they are not independent: here is why. Here is the surface at one instant, and here it is a moment later. Nothing about the shape changed. Every part of it simply moved to the right, by the speed c times that moment. So the height at your feet now is not a new number at all. It is the height that used to sit a little way upstream (exactly c delta t upstream) and has since arrived. Now read that as a rate. The water rises under you at whatever rate the shape slopes downward, multiplied by how fast the shape is coming. Steeper slope, or faster water, and the height changes quicker. Tidy it up, and there is the transport equation. Every symbol in it is something we pointed at first. And now the reason anyone would want to solve it. This is a rule about rates, here and now. A solution is the height at every place at every time, including times that have not happened yet. Solving it means predicting the water. And there is more than one way to do that. You could guess that the shape merely travels and verify it; we have very nearly done that already. You could break the profile into sine waves and march each one forward. Or you could hunt for paths through the x t plane along which u is not allowed to change at all. That last one is the method of characteristics. It is one method among several, but it is the one that turns a partial differential equation into an ordinary one, and that is a trade worth understanding. ### [04:42.685 · Walking Along a Curve](https://academa.ai/lectures/the-method-of-characteristics?t=282.6851458333333) Let's try something that looks like a trick, and turns out to be the whole method. Instead of asking what u does everywhere at once, we are going to choose a curve with x prime of t equal to c, then walk across the x t plane and watch u as we go. Here is that curve, and here is where we stand on it at time t: at the position x of t, at height t. Now give the value of u at that moving point a name. Call it z of t. And z is a function of one variable. One variable means we can simply differentiate it, and the chain rule tells us exactly what comes out. u sub x times x prime of t, coming through the x slot, plus u sub t, coming through the t slot. Nothing clever has happened yet. But look hard at what is sitting there: u sub x, u sub t, and a factor x prime of t that nobody has told us how to choose. So let's choose it. Set x prime of t equal to c, the speed out of the equation. And now read the right hand side. u sub t plus c u sub x. That is precisely the left hand side of our equation, and our equation says it is zero. So z prime is zero. The value of u does not change at all as we walk along that curve. And which curve is it? x prime equals c says that x is x naught plus c t. A straight line of slope c in the x t plane. That line has a name: it is called a characteristic. Every equation of this kind has a whole family of them, one through each starting point. And the partial differential equation, which was a statement about two derivatives at once, has become a statement about one derivative along a line. ### [06:34.182 · The General Solution](https://academa.ai/lectures/the-method-of-characteristics?t=394.1817083333333) We already have the family of parallel characteristics. Keep the initial profile f in view on the right. Every point of the x t plane sits on exactly one of those paths, so one path is enough to recover its value. Now pick any point x, t, up here. Follow its characteristic back down to time zero. It lands on the horizontal axis at x minus c t. And u is constant all the way along that line. So the value up at x, t is the value down at the foot, which is nothing but the initial data f, evaluated at x minus c t. And there is our solution. That is the whole answer. No integrals, no infinite series, no separation of variables. Whatever shape you start with, you get that same shape back, only shifted. Watch what that means. Here is f, the blob sitting there at time zero. A moment later, the whole graph has picked itself up and moved to the right. Not one thing about its shape has changed. Later still, further along, and it is still exactly the same blob. That is the only thing this equation ever does. And here is one particle of dye. On the left it climbs its own characteristic, from that foot up to the point we asked about. On the right it rides the crest along at speed c. Solving this equation was never more than following the flow. One last remark, and it matters in a minute. The data is carried along untouched, so if f has a sharp corner in it, that corner rides along forever and never smooths out. Transport has no memory and no diffusion. It only translates. ### [08:21.837 · A Speed That Changes](https://academa.ai/lectures/the-method-of-characteristics?t=501.8374583333333) Now let's change exactly one thing. Instead of a constant speed c, let the speed depend on time. u sub t plus t u sub x equals zero. Look back at what the derivation actually used. Nothing in it needed c to be constant. Along a curve where u does not change, the chain rule asks x prime of t to equal whatever sits in front of u sub x. Here, that is t. So the characteristics solve x prime equals t. That is an ordinary differential equation, and an easy one. Integrate, and x of t is x naught plus t squared over two. Parabolas. Each one leaves the axis at its own x naught and then takes off. At time zero the slope is zero, so everything starts out standing still. Later they lean over more and more. The dye is accelerating. Now invert it. If I am standing at position x at time t, which parabola am I on? Solve for x naught, and it is x minus t squared over two. And u is constant along each parabola, so u of x, t is f of x minus t squared over two. Same recipe, two lines of work, and the answer is once again the starting profile with a rewritten argument. It is worth checking. Differentiate: u sub t is minus t times f prime, and u sub x is f prime. So u sub t plus t u sub x is minus t f prime, plus t f prime, which is zero. It works. ### [10:3.878 · When Characteristics Collide](https://academa.ai/lectures/the-method-of-characteristics?t=603.8784791666667) One last equation, and it is the one that made this method famous. u sub t plus u times u sub x equals zero. The coefficient in front of u sub x is now u itself: the value being carried sets its own speed. Along any curve, the chain rule gives u sub t plus x prime of t times u sub x. Choose x prime of t equal to u, and that derivative becomes the equation's left hand side, which is zero. So u stays constant along the characteristic. A characteristic leaving x naught therefore carries the fixed value f of x naught. Its slope x prime is that same fixed number, so each path is a straight line, but different starting values produce different slopes. Take decreasing initial data. The piece starting at x naught equals zero point five carries the large value one point seven five, so it travels fast. The piece starting at three point five carries only zero point two five, so it barely crawls. The two middle values lie between them. The quick characteristics chase the slow ones ahead. All four paths meet at x equals four, t equals two. They arrive carrying four different values of u, but u is supposed to be a function. At that point it cannot remain smooth and single-valued. The same collision appears in the wave profile. At first the front is smooth. As faster values catch slower ones, it steepens, and steepens again, until the front is nearly vertical. That breaking point is a shock. The smooth solution stops exactly when the first characteristics meet. The method is four steps long, and their order matters. First read the transport speed from the equation. First, write the equation with the two derivatives lined up, and read off the coefficient sitting in front of u sub x. Second, solve x prime of t equals that coefficient. Its solutions are the characteristics, one through each starting point x naught. Third, use the equation to show that u does not change along those curves. Thus u equals f of x naught all the way along. Fourth, invert. Express x naught in terms of x and t, then substitute it into f. That final expression is the solution. Each step is ordinary calculus: read the speed, find its curves, carry the initial value along them, and trace the requested point back to its start. The partial differential equation becomes ordinary along exactly those paths. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M14TZPH69VE3VGWWFYD6E49T/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [A First-Order PDE](https://academa.ai/lectures/the-method-of-characteristics?t=0) Span: 00:00–04:42.685 (0s–282.6851458333333s). #### Objects - after: a Point \[yellow\] labelled "upright("now")" drawn in axes\_5 (location=(6.0, 0.5401230931362038)) - axes: an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='upright("height")') - axes\_2: an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='u') - axes\_3: an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='u') - axes\_4: an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='u') - axes\_5: an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='u') - came\_from: a Point \[red\] drawn in axes\_5 (location=(4.72, 0.5401230931362038)) - card: a Title that says "Partial Differential Equations — The Method of Characteristics" - clock: a VariableNumber - derivation: a Derivation \[text\] that says "$u(x, thin t + Delta t) = u(x - c thin Delta t, thin t) \\ u\_t = -c thin u\_x \\ u\_t + c thin u\_x = 0$" - drop: a Line \[green\] drawn in axes (start=(\, 0.0), end=(\, ((0.95 \* ((((0.55 \* ((spot - 3.0)…, dashed=True) - fetch: a Line \[red\] labelled "c thin Delta t" drawn in axes\_5 (start=(4.72, 0.5401230931362038), end=(6.0, 0.5401230931362038)) - flowing: a Polygon \[blue\] drawn in axes\_2 (vertices=((0.0, ((0.95 \* ((((0.55 \* (-3.0 - (0.8 \* clock))) \* (-3.0 - (0…) - frozen: a FunctionPlot \[blue\] drawn in axes\_3 (function=\) - head\_dt: a Heading that says "Hold $x$. Let $t$ Run." - head\_dx: a Heading that says "Hold $t$. Change $x$." - head\_law: a Heading that says "Slide the Shape, and Read the Height" - head\_time: a Heading that says "It Will Not Hold Still" - head\_water: a Heading that says "Water in a Long Tank" - head\_ways: a Heading that says "Ways to Solve $u\_t + c thin u\_x = 0$" - held: a Point \[yellow\] labelled "u" drawn in axes\_2 (location=(6.8, ((0.95 \* ((((0.55 \* (3.8 - (0.8 \* clock))) \* (3.8 - (0.8 …) - history: an Axes (x\_range=(-0.5, 9.0), y\_range=(-0.35, 1.5), x\_label='t') - inputs: a Derivation \[text\] that says "$u = u(x) \\ u = u(x, t)$" - is\_now: a Polygon \[blue\] drawn in axes\_5 (vertices=((0.0, 0.023063338044952277), (0.08333333333333333, 0.024440700…) - mark: a Line \[green\] drawn in axes\_5 (start=(6.0, 0.0), end=(6.0, 1.45), dashed=True) - now\_dot: a Point \[yellow\] drawn in history (location=(\, ((0.95 \* ((((0.55 \* (3.8 - (0.8 \*…) - pool\_fill: an AreaUnder \[blue\] drawn in axes\_3 (x\_range=(0.0, 10.0), target='frozen') - post: a Line \[green\] drawn in axes\_2 (start=(6.8, 0.0), end=(6.8, 1.45), dashed=True) - reach: a VariableNumber (initial\_value=5.8) - reading: a Point \[yellow\] labelled "u" drawn in axes (location=(\, ((0.95 \* ((((0.55 \* ((spot - 3.0)…) - record: a FunctionPlot \[cyan\] drawn in history (function=\) - rider: a Point \[yellow\] drawn in axes\_4 (location=(6.8, ((0.95 \* ((((0.55 \* (3.8 - (0.8 \* tick))) \* (3.8 - (0.8 \*…) - spot: a VariableNumber (initial\_value=1.0) - stake: a Line \[green\] drawn in axes\_4 (start=(6.8, 0.0), end=(6.8, 1.45), dashed=True) - step\_x: a SecantLine \[yellow\] drawn in axes\_3 (target='frozen', x1=3.9, x2=\) - still\_water: a Polygon \[blue\] drawn in axes (vertices=((0.0, 0.20876303068025867), (0.08333333333333333, 0.2374972354…) - tick: a VariableNumber - tide: a Polygon \[blue\] drawn in axes\_4 (vertices=((0.0, ((0.95 \* ((((0.55 \* (-3.0 - (0.8 \* tick))) \* (-3.0 - (0.…) - was: a Polygon \[gray\] drawn in axes\_5 (vertices=((0.0, 0.07025740770405511), (0.08333333333333333, 0.0773264846…, filled=False) - way\_one: a Text \[text\] that says "1. Guess that the shape only travels. Check it." - way\_three: a Text \[text\] that says "3. Find paths along which $u$ cannot change." - way\_two: a Text \[text\] that says "2. Split the profile into sine waves. Advance each." #### Beats ##### [00:00](https://academa.ai/lectures/the-method-of-characteristics?t=0) Narration: The method of characteristics turns a partial differential equation into an ordinary differential equation along carefully chosen paths. We will build that idea from a moving shape, then follow it until those paths collide. Board: Empty. Actions: - [00:00](https://academa.ai/lectures/the-method-of-characteristics?t=0): card is shown on the screen, written out. - [00:1.5](https://academa.ai/lectures/the-method-of-characteristics?t=1.5): card: enter:write-left-to-right. - [00:13.607](https://academa.ai/lectures/the-method-of-characteristics?t=13.6065): card is hidden from the screen — left the board. ##### [00:14.806](https://academa.ai/lectures/the-method-of-characteristics?t=14.8065) Narration: Let's start with water. A long, narrow tank of it, seen from the side, and something has disturbed the surface: a swell, sitting here. Board: Unchanged from the preceding beat in this scene. Actions: - [00:14.806](https://academa.ai/lectures/the-method-of-characteristics?t=14.8065): head\_water is shown on the screen, written out. - [00:18.115](https://academa.ai/lectures/the-method-of-characteristics?t=18.115): axes is shown on the screen, written out. - [00:20.89](https://academa.ai/lectures/the-method-of-characteristics?t=20.889999999999997): still\_water is shown on the screen, written out. ##### [00:25.561](https://academa.ai/lectures/the-method-of-characteristics?t=25.5615) Narration: Now pick a spot along the tank (any spot) and ask one very small question. How high is the water, right there? Board: axes — an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='upright("height")'); head\_water — a Heading that says "Water in a Long Tank"; still\_water — a Polygon \[blue\] drawn in axes (vertices=((0.0, 0.20876303068025867), (0.08333333333333333, 0.2374972354…) Actions: - [00:26.618](https://academa.ai/lectures/the-method-of-characteristics?t=26.618): drop is shown on the screen, written out. - [00:32.365](https://academa.ai/lectures/the-method-of-characteristics?t=32.364999999999995): reading is shown on the screen, written out. ##### [00:34.88](https://academa.ai/lectures/the-method-of-characteristics?t=34.8805) Narration: Down here it is barely raised. Walk toward the swell and it is higher. Keep going, up to the top, higher still, and past that it drops away again. Board: axes — an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='upright("height")'); head\_water — a Heading that says "Water in a Long Tank"; still\_water — a Polygon \[blue\] drawn in axes (vertices=((0.0, 0.20876303068025867), (0.08333333333333333, 0.2374972354…); drop — a Line \[green\] drawn in axes (start=(\, 0.0), end=(\, ((0.95 \* ((((0.55 \* ((spot - 3.0)…, dashed=True); reading — a Point \[yellow\] labelled "u" drawn in axes (location=(\, ((0.95 \* ((((0.55 \* ((spot - 3.0)…) Actions: - [00:37.644](https://academa.ai/lectures/the-method-of-characteristics?t=37.64399999999999): drop is redrawn as the numbers it depends on change. - [00:37.644](https://academa.ai/lectures/the-method-of-characteristics?t=37.64399999999999): reading is redrawn as the numbers it depends on change. - [00:37.644](https://academa.ai/lectures/the-method-of-characteristics?t=37.64399999999999): spot ticks to 2.3. - [00:40.848](https://academa.ai/lectures/the-method-of-characteristics?t=40.848): drop is redrawn as the numbers it depends on change. - [00:40.848](https://academa.ai/lectures/the-method-of-characteristics?t=40.848): reading is redrawn as the numbers it depends on change. - [00:40.848](https://academa.ai/lectures/the-method-of-characteristics?t=40.848): spot ticks to 3.0. - [00:43.461](https://academa.ai/lectures/the-method-of-characteristics?t=43.461): drop is redrawn as the numbers it depends on change. - [00:43.461](https://academa.ai/lectures/the-method-of-characteristics?t=43.461): reading is redrawn as the numbers it depends on change. - [00:43.461](https://academa.ai/lectures/the-method-of-characteristics?t=43.461): spot ticks to 5.6. ##### [00:45.372](https://academa.ai/lectures/the-method-of-characteristics?t=45.372499999999995) Narration: So every position hands you back one number. That is all a function is, and this one has a name: u. Board: Unchanged from the preceding beat in this scene. Actions: - [00:45.988](https://academa.ai/lectures/the-method-of-characteristics?t=45.98799999999999): drop is redrawn as the numbers it depends on change. - [00:45.988](https://academa.ai/lectures/the-method-of-characteristics?t=45.98799999999999): reading is redrawn as the numbers it depends on change. - [00:45.988](https://academa.ai/lectures/the-method-of-characteristics?t=45.98799999999999): spot ticks to 0.5. - [00:52.832](https://academa.ai/lectures/the-method-of-characteristics?t=52.832499999999996): axes is hidden from the screen — left the board. - [00:52.832](https://academa.ai/lectures/the-method-of-characteristics?t=52.832499999999996): still\_water is hidden from the screen — axes left the board. - [00:52.832](https://academa.ai/lectures/the-method-of-characteristics?t=52.832499999999996): drop is hidden from the screen — axes left the board. - [00:52.832](https://academa.ai/lectures/the-method-of-characteristics?t=52.832499999999996): reading is hidden from the screen — axes left the board. - [00:52.832](https://academa.ai/lectures/the-method-of-characteristics?t=52.832499999999996): head\_water is hidden from the screen — left the board. ##### [00:53.432](https://academa.ai/lectures/the-method-of-characteristics?t=53.4325) Narration: Except water does not hold still, and that ruins it. Watch. Board: Empty. Actions: - [00:53.432](https://academa.ai/lectures/the-method-of-characteristics?t=53.4325): head\_time is shown on the screen, written out. - [00:53.432](https://academa.ai/lectures/the-method-of-characteristics?t=53.4325): axes\_2 is shown on the screen, written out. - [00:55.969](https://academa.ai/lectures/the-method-of-characteristics?t=55.969): axes\_2 moves to a new place on the board. - [00:55.969](https://academa.ai/lectures/the-method-of-characteristics?t=55.969): inputs is shown on the screen, written out. - [00:57.258](https://academa.ai/lectures/the-method-of-characteristics?t=57.257999999999996): flowing is shown on the screen, written out. ##### [00:58.578](https://academa.ai/lectures/the-method-of-characteristics?t=58.5775) Narration: And I owe you an explanation for what you are about to see, because the swell is going to keep its shape exactly. That is not a law of water: it is the simplest case, and the one to understand first. Picture the whole tank drifting downstream at a steady speed, carrying whatever sits on it the way a river carries a leaf. Nothing spreads, nothing dies. It just goes. Board: axes\_2 — an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='u'); head\_time — a Heading that says "It Will Not Hold Still"; flowing — a Polygon \[blue\] drawn in axes\_2 (vertices=((0.0, ((0.95 \* ((((0.55 \* (-3.0 - (0.8 \* clock))) \* (-3.0 - (0…) Actions: - [01:12.103](https://academa.ai/lectures/the-method-of-characteristics?t=72.103): flowing is redrawn as the numbers it depends on change. - [01:12.103](https://academa.ai/lectures/the-method-of-characteristics?t=72.103): held is redrawn as the numbers it depends on change. - [01:12.103](https://academa.ai/lectures/the-method-of-characteristics?t=72.103): clock ticks to 1.6. - [01:17.839](https://academa.ai/lectures/the-method-of-characteristics?t=77.839): flowing is redrawn as the numbers it depends on change. - [01:17.839](https://academa.ai/lectures/the-method-of-characteristics?t=77.839): held is redrawn as the numbers it depends on change. - [01:17.839](https://academa.ai/lectures/the-method-of-characteristics?t=77.839): clock ticks to 3.0. ##### [01:22.316](https://academa.ai/lectures/the-method-of-characteristics?t=82.316) Narration: Now stop moving, and let it come to you instead. Board: Unchanged from the preceding beat in this scene. Actions: - [01:23.048](https://academa.ai/lectures/the-method-of-characteristics?t=83.048): post is shown on the screen, written out. - [01:24.58](https://academa.ai/lectures/the-method-of-characteristics?t=84.58): held is shown on the screen, written out. ##### [01:26.319](https://academa.ai/lectures/the-method-of-characteristics?t=86.3185) Narration: Stand at this one place. The water climbs past you, and then it sinks away again, and you never took a step. Board: axes\_2 — an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='u'); head\_time — a Heading that says "It Will Not Hold Still"; flowing — a Polygon \[blue\] drawn in axes\_2 (vertices=((0.0, ((0.95 \* ((((0.55 \* (-3.0 - (0.8 \* clock))) \* (-3.0 - (0…); post — a Line \[green\] drawn in axes\_2 (start=(6.8, 0.0), end=(6.8, 1.45), dashed=True); held — a Point \[yellow\] labelled "u" drawn in axes\_2 (location=(6.8, ((0.95 \* ((((0.55 \* (3.8 - (0.8 \* clock))) \* (3.8 - (0.8 …) Actions: - [01:28.559](https://academa.ai/lectures/the-method-of-characteristics?t=88.559): flowing is redrawn as the numbers it depends on change. - [01:28.559](https://academa.ai/lectures/the-method-of-characteristics?t=88.559): held is redrawn as the numbers it depends on change. - [01:28.559](https://academa.ai/lectures/the-method-of-characteristics?t=88.559): clock ticks to 4.75. - [01:30.173](https://academa.ai/lectures/the-method-of-characteristics?t=90.173): flowing is redrawn as the numbers it depends on change. - [01:30.173](https://academa.ai/lectures/the-method-of-characteristics?t=90.173): held is redrawn as the numbers it depends on change. - [01:30.173](https://academa.ai/lectures/the-method-of-characteristics?t=90.173): clock ticks to 7.5. ##### [01:33.686](https://academa.ai/lectures/the-method-of-characteristics?t=93.6865) Narration: So knowing where you are is no longer enough to know how high the water is. You have to say when as well. u of x and t. Board: Unchanged from the preceding beat in this scene. Actions: - [01:38.563](https://academa.ai/lectures/the-method-of-characteristics?t=98.56299999999999): inputs is shown on the screen, written out. - [01:41.314](https://academa.ai/lectures/the-method-of-characteristics?t=101.3145): axes\_2 is hidden from the screen — left the board. - [01:41.314](https://academa.ai/lectures/the-method-of-characteristics?t=101.3145): flowing is hidden from the screen — axes\_2 left the board. - [01:41.314](https://academa.ai/lectures/the-method-of-characteristics?t=101.3145): post is hidden from the screen — axes\_2 left the board. - [01:41.314](https://academa.ai/lectures/the-method-of-characteristics?t=101.3145): held is hidden from the screen — axes\_2 left the board. - [01:41.314](https://academa.ai/lectures/the-method-of-characteristics?t=101.3145): head\_time is hidden from the screen — left the board. - [01:41.314](https://academa.ai/lectures/the-method-of-characteristics?t=101.3145): inputs is hidden from the screen — left the board. ##### [01:42.514](https://academa.ai/lectures/the-method-of-characteristics?t=102.5145) Narration: Two inputs means two ways to change it, so let us change them one at a time. First, freeze the clock. Nothing moves; this is one instant, held. Board: Empty. Actions: - [01:42.514](https://academa.ai/lectures/the-method-of-characteristics?t=102.5145): head\_dx is shown on the screen, written out. - [01:47.519](https://academa.ai/lectures/the-method-of-characteristics?t=107.51899999999999): axes\_3 is shown on the screen, written out. - [01:50.491](https://academa.ai/lectures/the-method-of-characteristics?t=110.491): frozen is shown on the screen, written out. - [01:51.106](https://academa.ai/lectures/the-method-of-characteristics?t=111.106): pool\_fill is shown on the screen, written out. ##### [01:52.438](https://academa.ai/lectures/the-method-of-characteristics?t=112.438) Narration: Stand here, and walk this far along the tank. You changed x by that much, and look: u changed too, by that much. Same instant, different place, different height. Board: axes\_3 — an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='u'); head\_dx — a Heading that says "Hold $t$. Change $x$."; frozen — a FunctionPlot \[blue\] drawn in axes\_3 (function=\); pool\_fill — an AreaUnder \[blue\] drawn in axes\_3 (x\_range=(0.0, 10.0), target='frozen') Actions: - [01:54.353](https://academa.ai/lectures/the-method-of-characteristics?t=114.353): step\_x is shown on the screen, written out. ##### [02:5.194](https://academa.ai/lectures/the-method-of-characteristics?t=125.1935) Narration: Take a shorter walk and u changes less. A longer one, and it changes more. There is a rate hiding in there (height gained per step of x) and that rate is what u sub x means. Board: axes\_3 — an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='u'); head\_dx — a Heading that says "Hold $t$. Change $x$."; frozen — a FunctionPlot \[blue\] drawn in axes\_3 (function=\); pool\_fill — an AreaUnder \[blue\] drawn in axes\_3 (x\_range=(0.0, 10.0), target='frozen'); step\_x — a SecantLine \[yellow\] drawn in axes\_3 (target='frozen', x1=3.9, x2=\) Actions: - [02:5.948](https://academa.ai/lectures/the-method-of-characteristics?t=125.948): step\_x is redrawn as the numbers it depends on change. - [02:5.948](https://academa.ai/lectures/the-method-of-characteristics?t=125.948): reach ticks to 4.4. - [02:8.793](https://academa.ai/lectures/the-method-of-characteristics?t=128.793): step\_x is redrawn as the numbers it depends on change. - [02:8.793](https://academa.ai/lectures/the-method-of-characteristics?t=128.793): reach ticks to 7.2. - [02:12.078](https://academa.ai/lectures/the-method-of-characteristics?t=132.078): step\_x is redrawn as the numbers it depends on change. - [02:12.078](https://academa.ai/lectures/the-method-of-characteristics?t=132.078): reach ticks to 5.2. - [02:18.731](https://academa.ai/lectures/the-method-of-characteristics?t=138.731): axes\_3 is hidden from the screen — left the board. - [02:18.731](https://academa.ai/lectures/the-method-of-characteristics?t=138.731): frozen is hidden from the screen — axes\_3 left the board. - [02:18.731](https://academa.ai/lectures/the-method-of-characteristics?t=138.731): pool\_fill is hidden from the screen — axes\_3 left the board. - [02:18.731](https://academa.ai/lectures/the-method-of-characteristics?t=138.731): step\_x is hidden from the screen — axes\_3 left the board. - [02:18.731](https://academa.ai/lectures/the-method-of-characteristics?t=138.731): head\_dx is hidden from the screen — left the board. ##### [02:19.331](https://academa.ai/lectures/the-method-of-characteristics?t=139.331) Narration: Now the other one. Put x back where it was and leave it there: one place, this stake, and you do not move again. Board: Empty. Actions: - [02:19.331](https://academa.ai/lectures/the-method-of-characteristics?t=139.331): head\_dt is shown on the screen, written out. - [02:19.331](https://academa.ai/lectures/the-method-of-characteristics?t=139.331): axes\_4 is shown on the screen, written out. - [02:19.679](https://academa.ai/lectures/the-method-of-characteristics?t=139.679): tide is shown on the screen, written out. - [02:25.38](https://academa.ai/lectures/the-method-of-characteristics?t=145.38): stake is shown on the screen, written out. - [02:27.191](https://academa.ai/lectures/the-method-of-characteristics?t=147.191): rider is shown on the screen, written out. ##### [02:28.603](https://academa.ai/lectures/the-method-of-characteristics?t=148.6035) Narration: On the right I will keep a record of the height at that one stake, against time. Not against position; the position is fixed now. Against time. Board: axes\_4 — an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='u'); head\_dt — a Heading that says "Hold $x$. Let $t$ Run."; tide — a Polygon \[blue\] drawn in axes\_4 (vertices=((0.0, ((0.95 \* ((((0.55 \* (-3.0 - (0.8 \* tick))) \* (-3.0 - (0.…); stake — a Line \[green\] drawn in axes\_4 (start=(6.8, 0.0), end=(6.8, 1.45), dashed=True); rider — a Point \[yellow\] drawn in axes\_4 (location=(6.8, ((0.95 \* ((((0.55 \* (3.8 - (0.8 \* tick))) \* (3.8 - (0.8 \*…) Actions: - [02:30.206](https://academa.ai/lectures/the-method-of-characteristics?t=150.20599999999996): axes\_4 moves to a new place on the board. - [02:30.206](https://academa.ai/lectures/the-method-of-characteristics?t=150.20599999999996): history is shown on the screen, written out. - [02:32.412](https://academa.ai/lectures/the-method-of-characteristics?t=152.41199999999998): record is shown on the screen, written out. - [02:35.813](https://academa.ai/lectures/the-method-of-characteristics?t=155.813): now\_dot is shown on the screen, written out. ##### [02:38.398](https://academa.ai/lectures/the-method-of-characteristics?t=158.39849999999998) Narration: Watch both dots. The swell arrives, the water climbs the stake, and the record climbs with it. The crest passes, and the two of them come back down together. One number, drawn twice. Board: axes\_4 — an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='u'); history — an Axes (x\_range=(-0.5, 9.0), y\_range=(-0.35, 1.5), x\_label='t'); head\_dt — a Heading that says "Hold $x$. Let $t$ Run."; tide — a Polygon \[blue\] drawn in axes\_4 (vertices=((0.0, ((0.95 \* ((((0.55 \* (-3.0 - (0.8 \* tick))) \* (-3.0 - (0.…); stake — a Line \[green\] drawn in axes\_4 (start=(6.8, 0.0), end=(6.8, 1.45), dashed=True); rider — a Point \[yellow\] drawn in axes\_4 (location=(6.8, ((0.95 \* ((((0.55 \* (3.8 - (0.8 \* tick))) \* (3.8 - (0.8 \*…); record — a FunctionPlot \[cyan\] drawn in history (function=\); now\_dot — a Point \[yellow\] drawn in history (location=(\, ((0.95 \* ((((0.55 \* (3.8 - (0.8 \*…) Actions: - [02:41.348](https://academa.ai/lectures/the-method-of-characteristics?t=161.34799999999996): tide is redrawn as the numbers it depends on change. - [02:41.348](https://academa.ai/lectures/the-method-of-characteristics?t=161.34799999999996): rider is redrawn as the numbers it depends on change. - [02:41.348](https://academa.ai/lectures/the-method-of-characteristics?t=161.34799999999996): now\_dot is redrawn as the numbers it depends on change. - [02:41.348](https://academa.ai/lectures/the-method-of-characteristics?t=161.34799999999996): tick ticks to 4.75. - [02:46.34](https://academa.ai/lectures/the-method-of-characteristics?t=166.33999999999997): tide is redrawn as the numbers it depends on change. - [02:46.34](https://academa.ai/lectures/the-method-of-characteristics?t=166.33999999999997): rider is redrawn as the numbers it depends on change. - [02:46.34](https://academa.ai/lectures/the-method-of-characteristics?t=166.33999999999997): now\_dot is redrawn as the numbers it depends on change. - [02:46.34](https://academa.ai/lectures/the-method-of-characteristics?t=166.33999999999997): tick ticks to 8.0. ##### [02:52.652](https://academa.ai/lectures/the-method-of-characteristics?t=172.65249999999997) Narration: You never moved, and u changed anyway. So there is a second rate (height gained per tick of the clock) and that one is u sub t. Board: Unchanged from the preceding beat in this scene. Actions: - [03:2.16](https://academa.ai/lectures/the-method-of-characteristics?t=182.16049999999998): axes\_4 is hidden from the screen — left the board. - [03:2.16](https://academa.ai/lectures/the-method-of-characteristics?t=182.16049999999998): tide is hidden from the screen — axes\_4 left the board. - [03:2.16](https://academa.ai/lectures/the-method-of-characteristics?t=182.16049999999998): stake is hidden from the screen — axes\_4 left the board. - [03:2.16](https://academa.ai/lectures/the-method-of-characteristics?t=182.16049999999998): rider is hidden from the screen — axes\_4 left the board. - [03:2.16](https://academa.ai/lectures/the-method-of-characteristics?t=182.16049999999998): head\_dt is hidden from the screen — left the board. - [03:2.16](https://academa.ai/lectures/the-method-of-characteristics?t=182.16049999999998): history is hidden from the screen — left the board. - [03:2.16](https://academa.ai/lectures/the-method-of-characteristics?t=182.16049999999998): record is hidden from the screen — history left the board. - [03:2.16](https://academa.ai/lectures/the-method-of-characteristics?t=182.16049999999998): now\_dot is hidden from the screen — history left the board. ##### [03:2.76](https://academa.ai/lectures/the-method-of-characteristics?t=182.76049999999998) Narration: Two rates, then. And they are not independent: here is why. Here is the surface at one instant, and here it is a moment later. Board: Empty. Actions: - [03:2.76](https://academa.ai/lectures/the-method-of-characteristics?t=182.76049999999998): head\_law is shown on the screen, written out. - [03:2.76](https://academa.ai/lectures/the-method-of-characteristics?t=182.76049999999998): axes\_5 is shown on the screen, written out. - [03:9.402](https://academa.ai/lectures/the-method-of-characteristics?t=189.40199999999996): was is shown on the screen, written out. - [03:11.004](https://academa.ai/lectures/the-method-of-characteristics?t=191.00399999999993): is\_now is shown on the screen, written out. ##### [03:12.614](https://academa.ai/lectures/the-method-of-characteristics?t=192.61399999999998) Narration: Nothing about the shape changed. Every part of it simply moved to the right, by the speed c times that moment. Board: axes\_5 — an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='u'); head\_law — a Heading that says "Slide the Shape, and Read the Height"; was — a Polygon \[gray\] drawn in axes\_5 (vertices=((0.0, 0.07025740770405511), (0.08333333333333333, 0.0773264846…, filled=False); is\_now — a Polygon \[blue\] drawn in axes\_5 (vertices=((0.0, 0.023063338044952277), (0.08333333333333333, 0.024440700…) Actions: - [03:17.118](https://academa.ai/lectures/the-method-of-characteristics?t=197.11799999999994): mark is shown on the screen, written out. - [03:18.628](https://academa.ai/lectures/the-method-of-characteristics?t=198.62799999999993): after is shown on the screen, written out. ##### [03:20.772](https://academa.ai/lectures/the-method-of-characteristics?t=200.77199999999996) Narration: So the height at your feet now is not a new number at all. It is the height that used to sit a little way upstream (exactly c delta t upstream) and has since arrived. Board: axes\_5 — an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='u'); head\_law — a Heading that says "Slide the Shape, and Read the Height"; was — a Polygon \[gray\] drawn in axes\_5 (vertices=((0.0, 0.07025740770405511), (0.08333333333333333, 0.0773264846…, filled=False); is\_now — a Polygon \[blue\] drawn in axes\_5 (vertices=((0.0, 0.023063338044952277), (0.08333333333333333, 0.024440700…); mark — a Line \[green\] drawn in axes\_5 (start=(6.0, 0.0), end=(6.0, 1.45), dashed=True); after — a Point \[yellow\] labelled "upright("now")" drawn in axes\_5 (location=(6.0, 0.5401230931362038)) Actions: - [03:26.786](https://academa.ai/lectures/the-method-of-characteristics?t=206.78599999999994): came\_from is shown on the screen, written out. - [03:28.736](https://academa.ai/lectures/the-method-of-characteristics?t=208.73599999999993): axes\_5 moves to a new place on the board. - [03:28.736](https://academa.ai/lectures/the-method-of-characteristics?t=208.73599999999993): derivation is shown on the screen, written out. - [03:30.791](https://academa.ai/lectures/the-method-of-characteristics?t=210.79099999999994): fetch is shown on the screen, written out. ##### [03:32.366](https://academa.ai/lectures/the-method-of-characteristics?t=212.36649999999997) Narration: Now read that as a rate. The water rises under you at whatever rate the shape slopes downward, multiplied by how fast the shape is coming. Steeper slope, or faster water, and the height changes quicker. Board: axes\_5 — an Axes (x\_range=(0.0, 10.0), y\_range=(-0.35, 1.5), y\_label='u'); head\_law — a Heading that says "Slide the Shape, and Read the Height"; was — a Polygon \[gray\] drawn in axes\_5 (vertices=((0.0, 0.07025740770405511), (0.08333333333333333, 0.0773264846…, filled=False); is\_now — a Polygon \[blue\] drawn in axes\_5 (vertices=((0.0, 0.023063338044952277), (0.08333333333333333, 0.024440700…); mark — a Line \[green\] drawn in axes\_5 (start=(6.0, 0.0), end=(6.0, 1.45), dashed=True); after — a Point \[yellow\] labelled "upright("now")" drawn in axes\_5 (location=(6.0, 0.5401230931362038)); came\_from — a Point \[red\] drawn in axes\_5 (location=(4.72, 0.5401230931362038)); fetch — a Line \[red\] labelled "c thin Delta t" drawn in axes\_5 (start=(4.72, 0.5401230931362038), end=(6.0, 0.5401230931362038)) Actions: - [03:33.818](https://academa.ai/lectures/the-method-of-characteristics?t=213.81799999999996): derivation is shown on the screen, written out. ##### [03:46.155](https://academa.ai/lectures/the-method-of-characteristics?t=226.15549999999996) Narration: Tidy it up, and there is the transport equation. Every symbol in it is something we pointed at first. Board: Unchanged from the preceding beat in this scene. Actions: - [03:48.222](https://academa.ai/lectures/the-method-of-characteristics?t=228.22199999999995): derivation is shown on the screen, written out. ##### [03:53.907](https://academa.ai/lectures/the-method-of-characteristics?t=233.90699999999995) Narration: And now the reason anyone would want to solve it. This is a rule about rates, here and now. A solution is the height at every place at every time, including times that have not happened yet. Solving it means predicting the water. Board: Unchanged from the preceding beat in this scene. Actions: - [04:8.814](https://academa.ai/lectures/the-method-of-characteristics?t=248.81449999999995): axes\_5 is hidden from the screen — left the board. - [04:8.814](https://academa.ai/lectures/the-method-of-characteristics?t=248.81449999999995): was is hidden from the screen — axes\_5 left the board. - [04:8.814](https://academa.ai/lectures/the-method-of-characteristics?t=248.81449999999995): is\_now is hidden from the screen — axes\_5 left the board. - [04:8.814](https://academa.ai/lectures/the-method-of-characteristics?t=248.81449999999995): mark is hidden from the screen — axes\_5 left the board. - [04:8.814](https://academa.ai/lectures/the-method-of-characteristics?t=248.81449999999995): after is hidden from the screen — axes\_5 left the board. - [04:8.814](https://academa.ai/lectures/the-method-of-characteristics?t=248.81449999999995): came\_from is hidden from the screen — axes\_5 left the board. - [04:8.814](https://academa.ai/lectures/the-method-of-characteristics?t=248.81449999999995): fetch is hidden from the screen — axes\_5 left the board. - [04:8.814](https://academa.ai/lectures/the-method-of-characteristics?t=248.81449999999995): derivation is hidden from the screen — left the board. - [04:8.814](https://academa.ai/lectures/the-method-of-characteristics?t=248.81449999999995): head\_law is hidden from the screen — left the board. ##### [04:10.014](https://academa.ai/lectures/the-method-of-characteristics?t=250.01449999999994) Narration: And there is more than one way to do that. Board: Empty. Actions: - [04:10.014](https://academa.ai/lectures/the-method-of-characteristics?t=250.01449999999994): head\_ways is shown on the screen, written out. ##### [04:13.238](https://academa.ai/lectures/the-method-of-characteristics?t=253.23799999999994) Narration: You could guess that the shape merely travels and verify it; we have very nearly done that already. You could break the profile into sine waves and march each one forward. Or you could hunt for paths through the x t plane along which u is not allowed to change at all. Board: head\_ways — a Heading that says "Ways to Solve $u\_t + c thin u\_x = 0$" Actions: - [04:13.993](https://academa.ai/lectures/the-method-of-characteristics?t=253.99299999999997): way\_one is shown on the screen, written out. - [04:19.519](https://academa.ai/lectures/the-method-of-characteristics?t=259.51899999999995): way\_two is shown on the screen, written out. - [04:23.803](https://academa.ai/lectures/the-method-of-characteristics?t=263.803): way\_three is shown on the screen, written out. ##### [04:29.07](https://academa.ai/lectures/the-method-of-characteristics?t=269.06999999999994) Narration: That last one is the method of characteristics. It is one method among several, but it is the one that turns a partial differential equation into an ordinary one, and that is a trade worth understanding. Board: way\_one — a Text \[text\] that says "1. Guess that the shape only travels. Check it."; way\_two — a Text \[text\] that says "2. Split the profile into sine waves. Advance each."; way\_three — a Text \[text\] that says "3. Find paths along which $u$ cannot change."; head\_ways — a Heading that says "Ways to Solve $u\_t + c thin u\_x = 0$" Actions: - [04:41.643](https://academa.ai/lectures/the-method-of-characteristics?t=281.6434791666666): head\_ways is hidden from the screen — left the board. - [04:41.643](https://academa.ai/lectures/the-method-of-characteristics?t=281.6434791666666): way\_one is hidden from the screen — left the board. - [04:41.643](https://academa.ai/lectures/the-method-of-characteristics?t=281.6434791666666): way\_three is hidden from the screen — left the board. - [04:41.643](https://academa.ai/lectures/the-method-of-characteristics?t=281.6434791666666): way\_two is hidden from the screen — left the board. ### Scene 2: [Walking Along a Curve](https://academa.ai/lectures/the-method-of-characteristics?t=282.6851458333333) Span: 04:42.685–06:34.182 (282.6851458333333s–394.1817083333333s). #### Objects - choice: a Text \[text\] that says "Choose the curve so that $x'(t) = c$." - family: a Line \[blue\] drawn in xt (start=(0.2, 0.0), end=(3.2, 3.6)) - family\_2: a Line \[blue\] drawn in xt (start=(1.8, 0.0), end=(4.8, 3.6)) - family\_3: a Line \[blue\] drawn in xt (start=(2.7, 0.0), end=(5.7, 3.6)) - heading: a Heading that says "Walking Along a Curve" - rider: a Point \[green\] labelled "(x(t), t)" drawn in xt (location=(2.698796779428898, 1.8), label\_font\_size=30) - straight: a Line \[blue\] labelled "x = x\_0 + c t" drawn in xt (start=(1.0, 0.0), end=(4.0, 3.6)) - verdict: a Math \[text\] that says "$u(x(t), t) = upright("constant")$" - wander: a ParametricCurve \[gray\] drawn in xt (function=\, t\_range=(0.0, 3.6)) - work: a Derivation \[text\] that says "$z(t) &= u(x(t), t) \\ z'(t) &= u\_x thin x'(t) + u\_t \\ &= u\_t + c thin u\_x \\ &= 0$" - xt: an Axes (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), y\_label='t') #### Beats ##### [04:42.685](https://academa.ai/lectures/the-method-of-characteristics?t=282.6851458333333) Narration: Let's try something that looks like a trick, and turns out to be the whole method. Instead of asking what u does everywhere at once, we are going to choose a curve with x prime of t equal to c, then walk across the x t plane and watch u as we go. Board: Empty. Actions: - [04:42.685](https://academa.ai/lectures/the-method-of-characteristics?t=282.6851458333333): heading is shown on the screen, written out. - [04:42.685](https://academa.ai/lectures/the-method-of-characteristics?t=282.6851458333333): choice is shown on the screen, written out. - [04:42.685](https://academa.ai/lectures/the-method-of-characteristics?t=282.6851458333333): xt is shown on the screen, written out. - [04:50.754](https://academa.ai/lectures/the-method-of-characteristics?t=290.7541458333333): wander is shown on the screen, drawn. - [04:56.059](https://academa.ai/lectures/the-method-of-characteristics?t=296.05914583333333): rider is shown on the screen, written out. ##### [04:58.297](https://academa.ai/lectures/the-method-of-characteristics?t=298.2966458333333) Narration: Here is that curve, and here is where we stand on it at time t: at the position x of t, at height t. Now give the value of u at that moving point a name. Call it z of t. Board: choice — a Text \[text\] that says "Choose the curve so that $x'(t) = c$."; xt — an Axes (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), y\_label='t'); heading — a Heading that says "Walking Along a Curve"; wander — a ParametricCurve \[gray\] drawn in xt (function=\, t\_range=(0.0, 3.6)); rider — a Point \[green\] labelled "(x(t), t)" drawn in xt (location=(2.698796779428898, 1.8), label\_font\_size=30) Actions: - [05:9.129](https://academa.ai/lectures/the-method-of-characteristics?t=309.1291458333333): work is shown on the screen, written out. ##### [05:11.25](https://academa.ai/lectures/the-method-of-characteristics?t=311.2496458333333) Narration: And z is a function of one variable. One variable means we can simply differentiate it, and the chain rule tells us exactly what comes out. Board: Unchanged from the preceding beat in this scene. Actions: - [05:17.786](https://academa.ai/lectures/the-method-of-characteristics?t=317.7861458333333): work is shown on the screen, written out. ##### [05:21.033](https://academa.ai/lectures/the-method-of-characteristics?t=321.0331458333333) Narration: u sub x times x prime of t, coming through the x slot, plus u sub t, coming through the t slot. Nothing clever has happened yet. But look hard at what is sitting there: u sub x, u sub t, and a factor x prime of t that nobody has told us how to choose. Board: Unchanged from the preceding beat in this scene. Actions: - [05:21.439](https://academa.ai/lectures/the-method-of-characteristics?t=321.4391458333333): work (the "u\_x" part) is emphasized. - [05:22.809](https://academa.ai/lectures/the-method-of-characteristics?t=322.80914583333333): work (the "u\_x" part) is no longer emphasized. - [05:22.809](https://academa.ai/lectures/the-method-of-characteristics?t=322.80914583333333): work (the "x'(t)" part) is emphasized. - [05:26.223](https://academa.ai/lectures/the-method-of-characteristics?t=326.2231458333333): work (the "u\_t" part) is emphasized. - [05:26.223](https://academa.ai/lectures/the-method-of-characteristics?t=326.2231458333333): work (the "x'(t)" part) is no longer emphasized. - [05:36.312](https://academa.ai/lectures/the-method-of-characteristics?t=336.3121458333333): work (the "u\_t" part) is no longer emphasized. - [05:36.312](https://academa.ai/lectures/the-method-of-characteristics?t=336.3121458333333): work (the "x'(t)" part) is emphasized. - [05:40.109](https://academa.ai/lectures/the-method-of-characteristics?t=340.1086458333333): work (the "x'(t)" part) is no longer emphasized. ##### [05:40.709](https://academa.ai/lectures/the-method-of-characteristics?t=340.7086458333333) Narration: So let's choose it. Set x prime of t equal to c, the speed out of the equation. Board: Unchanged from the preceding beat in this scene. Actions: - [05:41.544](https://academa.ai/lectures/the-method-of-characteristics?t=341.5441458333333): choice is indicated — a transient flash. - [05:42.74](https://academa.ai/lectures/the-method-of-characteristics?t=342.7401458333333): work is shown on the screen, written out. ##### [05:47.555](https://academa.ai/lectures/the-method-of-characteristics?t=347.5546458333333) Narration: And now read the right hand side. u sub t plus c u sub x. That is precisely the left hand side of our equation, and our equation says it is zero. Board: Unchanged from the preceding beat in this scene. Actions: - [05:58.049](https://academa.ai/lectures/the-method-of-characteristics?t=358.04914583333334): work is shown on the screen, written out. ##### [05:59.526](https://academa.ai/lectures/the-method-of-characteristics?t=359.5261458333333) Narration: So z prime is zero. The value of u does not change at all as we walk along that curve. And which curve is it? x prime equals c says that x is x naught plus c t. A straight line of slope c in the x t plane. Board: Unchanged from the preceding beat in this scene. Actions: - [06:2.992](https://academa.ai/lectures/the-method-of-characteristics?t=362.9921458333333): verdict is shown on the screen, written out. - [06:10.062](https://academa.ai/lectures/the-method-of-characteristics?t=370.0621458333333): straight is shown on the screen, drawn. - [06:10.062](https://academa.ai/lectures/the-method-of-characteristics?t=370.0621458333333): wander is hidden from the screen. - [06:10.062](https://academa.ai/lectures/the-method-of-characteristics?t=370.0621458333333): rider is hidden from the screen. ##### [06:16.038](https://academa.ai/lectures/the-method-of-characteristics?t=376.03764583333333) Narration: That line has a name: it is called a characteristic. Every equation of this kind has a whole family of them, one through each starting point. And the partial differential equation, which was a statement about two derivatives at once, has become a statement about one derivative along a line. Board: choice — a Text \[text\] that says "Choose the curve so that $x'(t) = c$."; verdict — a Math \[text\] that says "$u(x(t), t) = upright("constant")$"; xt — an Axes (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), y\_label='t'); heading — a Heading that says "Walking Along a Curve"; straight — a Line \[blue\] labelled "x = x\_0 + c t" drawn in xt (start=(1.0, 0.0), end=(4.0, 3.6)) Actions: - [06:18.441](https://academa.ai/lectures/the-method-of-characteristics?t=378.4411458333333): straight is indicated — a transient flash. - [06:22.051](https://academa.ai/lectures/the-method-of-characteristics?t=382.0511458333333): family is shown on the screen, drawn. - [06:22.201](https://academa.ai/lectures/the-method-of-characteristics?t=382.2011458333333): family\_2 is shown on the screen, drawn. - [06:22.351](https://academa.ai/lectures/the-method-of-characteristics?t=382.3511458333333): family\_3 is shown on the screen, drawn. - [06:33.14](https://academa.ai/lectures/the-method-of-characteristics?t=393.14004166666666): choice is hidden from the screen — left the board. - [06:33.14](https://academa.ai/lectures/the-method-of-characteristics?t=393.14004166666666): heading is hidden from the screen — left the board. - [06:33.14](https://academa.ai/lectures/the-method-of-characteristics?t=393.14004166666666): verdict is hidden from the screen — left the board. - [06:33.14](https://academa.ai/lectures/the-method-of-characteristics?t=393.14004166666666): work is hidden from the screen — left the board. - [06:33.14](https://academa.ai/lectures/the-method-of-characteristics?t=393.14004166666666): xt is hidden from the screen — left the board. - [06:33.14](https://academa.ai/lectures/the-method-of-characteristics?t=393.14004166666666): straight is hidden from the screen — xt left the board. - [06:33.14](https://academa.ai/lectures/the-method-of-characteristics?t=393.14004166666666): family is hidden from the screen — xt left the board. - [06:33.14](https://academa.ai/lectures/the-method-of-characteristics?t=393.14004166666666): family\_2 is hidden from the screen — xt left the board. - [06:33.14](https://academa.ai/lectures/the-method-of-characteristics?t=393.14004166666666): family\_3 is hidden from the screen — xt left the board. ### Scene 3: [The General Solution](https://academa.ai/lectures/the-method-of-characteristics?t=394.1817083333333) Span: 06:34.182–08:21.837 (394.1817083333333s–501.8374583333333s). #### Objects - clock: a VariableNumber - corner: a FunctionPlot \[red\] labelled "sharp corner" drawn in profile (function=\) - crest: a Point \[green\] drawn in profile (location=((clock + 1.5), 1.0)) - fan: an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 3.6), y\_label='t') - foot: a Point \[yellow\] labelled "x - c t" drawn in fan (location=(1.5, 0.0)) - heading: a Heading that says "The General Solution" - here: a Point \[yellow\] labelled "(x, t)" drawn in fan (location=(4.5, 3.0)) - live\_profile: a Polygon \[blue\] drawn in profile (vertices=((-1.0, ((((-2.5 - clock) \* (-2.5 - clock)) + 1.0) \*\* -1.0)), (…, fill\_opacity=0.15) - profile: an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 1.5), y\_label='u') - snap\_0: a FunctionPlot \[blue\] labelled "f(x)" drawn in profile (function=\) - snap\_1: a FunctionPlot \[blue\] drawn in profile (function=\) - snap\_2: a FunctionPlot \[blue\] drawn in profile (function=\) - solution: a Math \[text\] that says "$u(x, t) = f(x - c t)$" - trace: a Line \[blue\] drawn in fan (start=(1.5, 0.0), end=(4.5, 3.0)) - traveller: a Point \[green\] drawn in fan (location=((clock + 1.5), \)) #### Beats ##### [06:34.182](https://academa.ai/lectures/the-method-of-characteristics?t=394.1817083333333) Narration: We already have the family of parallel characteristics. Keep the initial profile f in view on the right. Every point of the x t plane sits on exactly one of those paths, so one path is enough to recover its value. Board: Empty. Actions: - [06:34.182](https://academa.ai/lectures/the-method-of-characteristics?t=394.1817083333333): heading is shown on the screen, written out. - [06:34.182](https://academa.ai/lectures/the-method-of-characteristics?t=394.1817083333333): profile is shown on the screen, written out. - [06:34.182](https://academa.ai/lectures/the-method-of-characteristics?t=394.1817083333333): snap\_0 is shown on the screen, written out. - [06:41.578](https://academa.ai/lectures/the-method-of-characteristics?t=401.5777083333333): profile moves to a new place on the board. - [06:41.578](https://academa.ai/lectures/the-method-of-characteristics?t=401.5777083333333): fan is shown on the screen, written out. ##### [06:47.715](https://academa.ai/lectures/the-method-of-characteristics?t=407.7152083333333) Narration: Now pick any point x, t, up here. Follow its characteristic back down to time zero. It lands on the horizontal axis at x minus c t. Board: fan — an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 3.6), y\_label='t'); profile — an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 1.5), y\_label='u'); heading — a Heading that says "The General Solution"; snap\_0 — a FunctionPlot \[blue\] labelled "f(x)" drawn in profile (function=\) Actions: - [06:48.342](https://academa.ai/lectures/the-method-of-characteristics?t=408.3417083333333): here is shown on the screen, written out. - [06:51.048](https://academa.ai/lectures/the-method-of-characteristics?t=411.0477083333333): trace is shown on the screen, drawn. - [06:54.74](https://academa.ai/lectures/the-method-of-characteristics?t=414.7397083333333): foot is shown on the screen, written out. ##### [06:58.869](https://academa.ai/lectures/the-method-of-characteristics?t=418.8687083333333) Narration: And u is constant all the way along that line. So the value up at x, t is the value down at the foot, which is nothing but the initial data f, evaluated at x minus c t. And there is our solution. Board: fan — an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 3.6), y\_label='t'); profile — an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 1.5), y\_label='u'); heading — a Heading that says "The General Solution"; snap\_0 — a FunctionPlot \[blue\] labelled "f(x)" drawn in profile (function=\); here — a Point \[yellow\] labelled "(x, t)" drawn in fan (location=(4.5, 3.0)); trace — a Line \[blue\] drawn in fan (start=(1.5, 0.0), end=(4.5, 3.0)); foot — a Point \[yellow\] labelled "x - c t" drawn in fan (location=(1.5, 0.0)) Actions: - [07:12.22](https://academa.ai/lectures/the-method-of-characteristics?t=432.2197083333333): solution is shown on the screen, written out. - [07:12.22](https://academa.ai/lectures/the-method-of-characteristics?t=432.2197083333333): trace is hidden from the screen. ##### [07:13.842](https://academa.ai/lectures/the-method-of-characteristics?t=433.84170833333326) Narration: That is the whole answer. No integrals, no infinite series, no separation of variables. Whatever shape you start with, you get that same shape back, only shifted. Board: fan — an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 3.6), y\_label='t'); solution — a Math \[text\] that says "$u(x, t) = f(x - c t)$"; profile — an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 1.5), y\_label='u'); heading — a Heading that says "The General Solution"; snap\_0 — a FunctionPlot \[blue\] labelled "f(x)" drawn in profile (function=\); here — a Point \[yellow\] labelled "(x, t)" drawn in fan (location=(4.5, 3.0)); foot — a Point \[yellow\] labelled "x - c t" drawn in fan (location=(1.5, 0.0)) Actions: - [07:21.505](https://academa.ai/lectures/the-method-of-characteristics?t=441.50470833333327): snap\_0 is indicated — a transient flash. ##### [07:26.226](https://academa.ai/lectures/the-method-of-characteristics?t=446.2257083333333) Narration: Watch what that means. Here is f, the blob sitting there at time zero. Board: Unchanged from the preceding beat in this scene. Actions: - [07:28.49](https://academa.ai/lectures/the-method-of-characteristics?t=448.4897083333333): snap\_0 is indicated — a transient flash. ##### [07:32.747](https://academa.ai/lectures/the-method-of-characteristics?t=452.7472083333333) Narration: A moment later, the whole graph has picked itself up and moved to the right. Not one thing about its shape has changed. Board: Unchanged from the preceding beat in this scene. Actions: - [07:35.685](https://academa.ai/lectures/the-method-of-characteristics?t=455.6847083333333): snap\_1 is shown on the screen, written out. - [07:35.685](https://academa.ai/lectures/the-method-of-characteristics?t=455.6847083333333): snap\_0 is hidden from the screen. ##### [07:40.302](https://academa.ai/lectures/the-method-of-characteristics?t=460.3017083333333) Narration: Later still, further along, and it is still exactly the same blob. That is the only thing this equation ever does. Board: fan — an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 3.6), y\_label='t'); solution — a Math \[text\] that says "$u(x, t) = f(x - c t)$"; profile — an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 1.5), y\_label='u'); heading — a Heading that says "The General Solution"; here — a Point \[yellow\] labelled "(x, t)" drawn in fan (location=(4.5, 3.0)); foot — a Point \[yellow\] labelled "x - c t" drawn in fan (location=(1.5, 0.0)); snap\_1 — a FunctionPlot \[blue\] drawn in profile (function=\) Actions: - [07:40.604](https://academa.ai/lectures/the-method-of-characteristics?t=460.6037083333333): snap\_2 is shown on the screen, written out. - [07:40.604](https://academa.ai/lectures/the-method-of-characteristics?t=460.6037083333333): snap\_1 is hidden from the screen. ##### [07:48.622](https://academa.ai/lectures/the-method-of-characteristics?t=468.6222083333333) Narration: And here is one particle of dye. On the left it climbs its own characteristic, from that foot up to the point we asked about. On the right it rides the crest along at speed c. Solving this equation was never more than following the flow. Board: fan — an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 3.6), y\_label='t'); solution — a Math \[text\] that says "$u(x, t) = f(x - c t)$"; profile — an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 1.5), y\_label='u'); heading — a Heading that says "The General Solution"; here — a Point \[yellow\] labelled "(x, t)" drawn in fan (location=(4.5, 3.0)); foot — a Point \[yellow\] labelled "x - c t" drawn in fan (location=(1.5, 0.0)); snap\_2 — a FunctionPlot \[blue\] drawn in profile (function=\) Actions: - [07:49.911](https://academa.ai/lectures/the-method-of-characteristics?t=469.9107083333333): traveller is shown on the screen, written out. - [07:49.911](https://academa.ai/lectures/the-method-of-characteristics?t=469.9107083333333): crest is shown on the screen, written out. - [07:49.911](https://academa.ai/lectures/the-method-of-characteristics?t=469.9107083333333): snap\_2 is hidden from the screen. - [07:49.911](https://academa.ai/lectures/the-method-of-characteristics?t=469.9107083333333): live\_profile is shown on the screen, written out. - [07:52.431](https://academa.ai/lectures/the-method-of-characteristics?t=472.4307083333333): traveller is redrawn as the numbers it depends on change. - [07:52.431](https://academa.ai/lectures/the-method-of-characteristics?t=472.4307083333333): crest is redrawn as the numbers it depends on change. - [07:52.431](https://academa.ai/lectures/the-method-of-characteristics?t=472.4307083333333): live\_profile is redrawn as the numbers it depends on change. - [07:52.431](https://academa.ai/lectures/the-method-of-characteristics?t=472.4307083333333): clock ticks to 3.0. ##### [08:5.14](https://academa.ai/lectures/the-method-of-characteristics?t=485.13970833333326) Narration: One last remark, and it matters in a minute. The data is carried along untouched, so if f has a sharp corner in it, that corner rides along forever and never smooths out. Transport has no memory and no diffusion. It only translates. Board: fan — an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 3.6), y\_label='t'); solution — a Math \[text\] that says "$u(x, t) = f(x - c t)$"; profile — an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 1.5), y\_label='u'); heading — a Heading that says "The General Solution"; here — a Point \[yellow\] labelled "(x, t)" drawn in fan (location=(4.5, 3.0)); foot — a Point \[yellow\] labelled "x - c t" drawn in fan (location=(1.5, 0.0)); traveller — a Point \[green\] drawn in fan (location=((clock + 1.5), \)); crest — a Point \[green\] drawn in profile (location=((clock + 1.5), 1.0)); live\_profile — a Polygon \[blue\] drawn in profile (vertices=((-1.0, ((((-2.5 - clock) \* (-2.5 - clock)) + 1.0) \*\* -1.0)), (…, fill\_opacity=0.15) Actions: - [08:11.85](https://academa.ai/lectures/the-method-of-characteristics?t=491.8497083333333): live\_profile is hidden from the screen. - [08:11.85](https://academa.ai/lectures/the-method-of-characteristics?t=491.8497083333333): traveller is hidden from the screen. - [08:11.85](https://academa.ai/lectures/the-method-of-characteristics?t=491.8497083333333): crest is hidden from the screen. - [08:11.85](https://academa.ai/lectures/the-method-of-characteristics?t=491.8497083333333): corner is shown on the screen, drawn. - [08:20.796](https://academa.ai/lectures/the-method-of-characteristics?t=500.7957916666666): fan is hidden from the screen — left the board. - [08:20.796](https://academa.ai/lectures/the-method-of-characteristics?t=500.7957916666666): here is hidden from the screen — fan left the board. - [08:20.796](https://academa.ai/lectures/the-method-of-characteristics?t=500.7957916666666): foot is hidden from the screen — fan left the board. - [08:20.796](https://academa.ai/lectures/the-method-of-characteristics?t=500.7957916666666): heading is hidden from the screen — left the board. - [08:20.796](https://academa.ai/lectures/the-method-of-characteristics?t=500.7957916666666): profile is hidden from the screen — left the board. - [08:20.796](https://academa.ai/lectures/the-method-of-characteristics?t=500.7957916666666): corner is hidden from the screen — profile left the board. - [08:20.796](https://academa.ai/lectures/the-method-of-characteristics?t=500.7957916666666): solution is hidden from the screen — left the board. ### Scene 4: [A Speed That Changes](https://academa.ai/lectures/the-method-of-characteristics?t=501.8374583333333) Span: 08:21.837–10:3.878 (501.8374583333333s–603.8784791666667s). #### Objects - answer: a Math \[text\] that says "$u(x, t) = f(x - frac(t^2, 2))$" - heading\_intro: a Heading that says "A Speed That Changes With Time" - insight: a Math \[text\] that says "$x'(t) = t$" - paths: a ParametricCurve \[blue\] drawn in xt (function=\, t\_range=(0.0, 2.8)) - paths\_2: a ParametricCurve \[blue\] drawn in xt (function=\, t\_range=(0.0, 2.8)) - paths\_3: a ParametricCurve \[blue\] drawn in xt (function=\, t\_range=(0.0, 2.8)) - paths\_4: a ParametricCurve \[blue\] drawn in xt (function=\, t\_range=(0.0, 2.8)) - point: a Point \[yellow\] drawn in xt (location=(-0.5, 0.0)) - point\_2: a Point \[yellow\] drawn in xt (location=(0.5, 0.0)) - point\_3: a Point \[yellow\] drawn in xt (location=(1.5, 0.0)) - point\_4: a Point \[yellow\] drawn in xt (location=(2.5, 0.0)) - problem: a Tex \[text\] that says "Example. Solve $u\_t + t thin u\_x = 0$ with $u(x, 0) = f(x)$." - problem\_intro: a Tex \[text\] that says "Solve $u\_t + t thin u\_x = 0$ with $u(x, 0) = f(x)$." - verification: a Derivation \[text\] that says "$u\_t &= -t thin f'(x - frac(t^2, 2)) \\ u\_x &= f'(x - frac(t^2, 2)) \\ u\_t + t thin u\_x &= -t thin f'(x - frac(t^2, 2)) + t thin f'(x - frac(t^2, 2)) = 0$" - verify\_answer: a Math \[text\] that says "$u(x, t) = f(x - frac(t^2, 2))$" - verify\_heading: a Heading that says "Verify the Solution" - work: a Derivation \[text\] that says "$x'(t) &= t \\ x(t) &= x\_0 + frac(t^2, 2) \\ x\_0 &= x - frac(t^2, 2)$" - xt: an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 3.0), y\_label='t') #### Beats ##### [08:21.837](https://academa.ai/lectures/the-method-of-characteristics?t=501.8374583333333) Narration: Now let's change exactly one thing. Instead of a constant speed c, let the speed depend on time. u sub t plus t u sub x equals zero. Board: Empty. Actions: - [08:21.837](https://academa.ai/lectures/the-method-of-characteristics?t=501.8374583333333): heading\_intro is shown on the screen, written out. - [08:21.837](https://academa.ai/lectures/the-method-of-characteristics?t=501.8374583333333): problem\_intro is shown on the screen, written out. ##### [08:33.293](https://academa.ai/lectures/the-method-of-characteristics?t=513.2929583333333) Narration: Look back at what the derivation actually used. Nothing in it needed c to be constant. Along a curve where u does not change, the chain rule asks x prime of t to equal whatever sits in front of u sub x. Here, that is t. Board: problem\_intro — a Tex \[text\] that says "Solve $u\_t + t thin u\_x = 0$ with $u(x, 0) = f(x)$."; heading\_intro — a Heading that says "A Speed That Changes With Time" Actions: - [08:43.764](https://academa.ai/lectures/the-method-of-characteristics?t=523.7644583333333): insight is shown on the screen, written out. - [08:49.918](https://academa.ai/lectures/the-method-of-characteristics?t=529.9184583333333): heading\_intro is hidden from the screen — left the board. - [08:49.918](https://academa.ai/lectures/the-method-of-characteristics?t=529.9184583333333): insight is hidden from the screen — left the board. - [08:49.918](https://academa.ai/lectures/the-method-of-characteristics?t=529.9184583333333): problem\_intro is hidden from the screen — left the board. ##### [08:50.518](https://academa.ai/lectures/the-method-of-characteristics?t=530.5184583333333) Narration: So the characteristics solve x prime equals t. That is an ordinary differential equation, and an easy one. Integrate, and x of t is x naught plus t squared over two. Parabolas. Board: Empty. Actions: - [08:50.518](https://academa.ai/lectures/the-method-of-characteristics?t=530.5184583333333): problem is shown on the screen, written out. - [08:52.352](https://academa.ai/lectures/the-method-of-characteristics?t=532.3524583333333): work is shown on the screen, written out. - [08:55.092](https://academa.ai/lectures/the-method-of-characteristics?t=535.0924583333333): xt is shown on the screen, written out. - [08:58.981](https://academa.ai/lectures/the-method-of-characteristics?t=538.9814583333333): work is shown on the screen, written out. - [09:3.695](https://academa.ai/lectures/the-method-of-characteristics?t=543.6954583333332): paths is shown on the screen, drawn. - [09:3.995](https://academa.ai/lectures/the-method-of-characteristics?t=543.9954583333333): paths\_2 is shown on the screen, drawn. - [09:4.295](https://academa.ai/lectures/the-method-of-characteristics?t=544.2954583333333): paths\_3 is shown on the screen, drawn. - [09:4.595](https://academa.ai/lectures/the-method-of-characteristics?t=544.5954583333333): paths\_4 is shown on the screen, drawn. ##### [09:5.561](https://academa.ai/lectures/the-method-of-characteristics?t=545.5609583333332) Narration: Each one leaves the axis at its own x naught and then takes off. At time zero the slope is zero, so everything starts out standing still. Later they lean over more and more. The dye is accelerating. Board: xt — an Axes (x\_range=(-1.0, 7.0), y\_range=(0.0, 3.0), y\_label='t'); problem — a Tex \[text\] that says "Example. Solve $u\_t + t thin u\_x = 0$ with $u(x, 0) = f(x)$."; paths — a ParametricCurve \[blue\] drawn in xt (function=\, t\_range=(0.0, 2.8)); paths\_2 — a ParametricCurve \[blue\] drawn in xt (function=\, t\_range=(0.0, 2.8)); paths\_3 — a ParametricCurve \[blue\] drawn in xt (function=\, t\_range=(0.0, 2.8)); paths\_4 — a ParametricCurve \[blue\] drawn in xt (function=\, t\_range=(0.0, 2.8)) Actions: - [09:7.499](https://academa.ai/lectures/the-method-of-characteristics?t=547.4994583333333): point is shown on the screen, grown. - [09:7.499](https://academa.ai/lectures/the-method-of-characteristics?t=547.4994583333333): point\_2 is shown on the screen, grown. - [09:7.499](https://academa.ai/lectures/the-method-of-characteristics?t=547.4994583333333): point\_3 is shown on the screen, grown. - [09:7.499](https://academa.ai/lectures/the-method-of-characteristics?t=547.4994583333333): point\_4 is shown on the screen, grown. - [09:9.499](https://academa.ai/lectures/the-method-of-characteristics?t=549.4994583333333): point is hidden from the screen. - [09:9.499](https://academa.ai/lectures/the-method-of-characteristics?t=549.4994583333333): point\_2 is hidden from the screen. - [09:9.499](https://academa.ai/lectures/the-method-of-characteristics?t=549.4994583333333): point\_3 is hidden from the screen. - [09:9.499](https://academa.ai/lectures/the-method-of-characteristics?t=549.4994583333333): point\_4 is hidden from the screen. - [09:11.261](https://academa.ai/lectures/the-method-of-characteristics?t=551.2614583333333): paths is indicated — a transient flash. - [09:15.383](https://academa.ai/lectures/the-method-of-characteristics?t=555.3834583333334): paths\_4 is indicated — a transient flash. ##### [09:19.187](https://academa.ai/lectures/the-method-of-characteristics?t=559.1869583333333) Narration: Now invert it. If I am standing at position x at time t, which parabola am I on? Solve for x naught, and it is x minus t squared over two. Board: Unchanged from the preceding beat in this scene. Actions: - [09:25.932](https://academa.ai/lectures/the-method-of-characteristics?t=565.9324583333333): work is shown on the screen, written out. ##### [09:30.909](https://academa.ai/lectures/the-method-of-characteristics?t=570.9094583333333) Narration: And u is constant along each parabola, so u of x, t is f of x minus t squared over two. Same recipe, two lines of work, and the answer is once again the starting profile with a rewritten argument. Board: Unchanged from the preceding beat in this scene. Actions: - [09:31.861](https://academa.ai/lectures/the-method-of-characteristics?t=571.8614583333333): answer is shown on the screen, written out. - [09:38.885](https://academa.ai/lectures/the-method-of-characteristics?t=578.8854583333333): A box is drawn around answer. - [09:44.574](https://academa.ai/lectures/the-method-of-characteristics?t=584.5739583333333): answer is hidden from the screen — left the board. - [09:44.574](https://academa.ai/lectures/the-method-of-characteristics?t=584.5739583333333): problem is hidden from the screen — left the board. - [09:44.574](https://academa.ai/lectures/the-method-of-characteristics?t=584.5739583333333): work is hidden from the screen — left the board. - [09:44.574](https://academa.ai/lectures/the-method-of-characteristics?t=584.5739583333333): xt is hidden from the screen — left the board. - [09:44.574](https://academa.ai/lectures/the-method-of-characteristics?t=584.5739583333333): paths is hidden from the screen — xt left the board. - [09:44.574](https://academa.ai/lectures/the-method-of-characteristics?t=584.5739583333333): paths\_2 is hidden from the screen — xt left the board. - [09:44.574](https://academa.ai/lectures/the-method-of-characteristics?t=584.5739583333333): paths\_3 is hidden from the screen — xt left the board. - [09:44.574](https://academa.ai/lectures/the-method-of-characteristics?t=584.5739583333333): paths\_4 is hidden from the screen — xt left the board. ##### [09:45.174](https://academa.ai/lectures/the-method-of-characteristics?t=585.1739583333333) Narration: It is worth checking. Differentiate: u sub t is minus t times f prime, and u sub x is f prime. So u sub t plus t u sub x is minus t f prime, plus t f prime, which is zero. It works. Board: Empty. Actions: - [09:45.174](https://academa.ai/lectures/the-method-of-characteristics?t=585.1739583333333): verify\_heading is shown on the screen, written out. - [09:45.174](https://academa.ai/lectures/the-method-of-characteristics?t=585.1739583333333): verify\_answer is shown on the screen, written out. - [09:48.715](https://academa.ai/lectures/the-method-of-characteristics?t=588.7154583333333): verification is shown on the screen, written out. - [09:49.818](https://academa.ai/lectures/the-method-of-characteristics?t=589.8184583333333): verification (the "-t thin f'(x - frac(t^2, 2))" part) is emphasized. - [09:52.117](https://academa.ai/lectures/the-method-of-characteristics?t=592.1174583333333): verification is shown on the screen, written out. - [09:55.553](https://academa.ai/lectures/the-method-of-characteristics?t=595.5534583333333): verification is shown on the screen, written out. - [09:59.338](https://academa.ai/lectures/the-method-of-characteristics?t=599.3384583333333): verification (the "+ t thin f'(x - frac(t^2, 2))" part) is emphasized. - [09:59.338](https://academa.ai/lectures/the-method-of-characteristics?t=599.3384583333333): verification (the "-t thin f'(x - frac(t^2, 2))" part) is no longer emphasized. - [10:1.068](https://academa.ai/lectures/the-method-of-characteristics?t=601.0684583333333): verification (the "+ t thin f'(x - frac(t^2, 2))" part) is no longer emphasized. - [10:1.068](https://academa.ai/lectures/the-method-of-characteristics?t=601.0684583333333): verification (the "= 0" part) is emphasized. - [10:2.587](https://academa.ai/lectures/the-method-of-characteristics?t=602.5868125): verification (the "= 0" part) is no longer emphasized. - [10:2.837](https://academa.ai/lectures/the-method-of-characteristics?t=602.8368125): verification is hidden from the screen — left the board. - [10:2.837](https://academa.ai/lectures/the-method-of-characteristics?t=602.8368125): verify\_answer is hidden from the screen — left the board. - [10:2.837](https://academa.ai/lectures/the-method-of-characteristics?t=602.8368125): verify\_heading is hidden from the screen — left the board. ### Scene 5: [When Characteristics Collide](https://academa.ai/lectures/the-method-of-characteristics?t=603.8784791666667) Span: 10:3.878–12:58.431 (603.8784791666667s–778.4305625000001s). #### Objects - breaking\_heading: a Heading that says "The Profile Steepens Into a Shock" - collide\_heading: a Heading that says "Faster Values Catch Slower Values" - crossing: a Point \[red\] labelled "upright("shock")" drawn in xt (location=(4.0, 2.0)) - early\_axes: an Axes (x\_range=(0.0, 6.0), y\_range=(0.0, 2.0), y\_label='u') - early\_caption: a Text \[text\] that says "Earlier" - early\_curve: a FunctionPlot \[blue\] drawn in early\_axes (function=\) - equation: a Math \[text\] that says "$u\_t + u thin u\_x = 0$" - heading: a Heading that says "When Characteristics Collide" - initial: an Axes (x\_range=(0.0, 4.0), y\_range=(0.0, 2.1), x\_label='x\_0') - initial\_curve: a FunctionPlot \[blue\] drawn in initial (function=\, x\_range=(0.2, 3.8)) - late\_axes: an Axes (x\_range=(0.0, 6.0), y\_range=(0.0, 2.0), y\_label='u') - late\_caption: a Text \[text\] that says "At the shock" - late\_curve: a FunctionPlot \[red\] drawn in late\_axes (function=\) - middle\_axes: an Axes (x\_range=(0.0, 6.0), y\_range=(0.0, 2.0), y\_label='u') - middle\_caption: a Text \[text\] that says "Steeper" - middle\_curve: a FunctionPlot \[blue\] drawn in middle\_axes (function=\) - nonlinear\_work: a Derivation \[text\] that says "$frac(dif, dif t) u(x(t), t) &= u\_t + x'(t) thin u\_x \\ x'(t) &= u \\ frac(dif u, dif t) &= u\_t + u thin u\_x = 0 \\ u &= f(x\_0) quad arrow.r quad x'(t) = f(x\_0)$" - note: a Panel that says "The coefficient of $u\_x$ is $u$ itself, so every characteristic carries the value that fixes its own slope." - point: a Point \[yellow\] drawn in initial (location=(0.5, 1.75)) - point\_2: a Point \[yellow\] drawn in initial (location=(3.5, 0.25)) - point\_3: a Point \[yellow\] drawn in xt (location=(4.0, 2.0)) - profile\_points: a Point \[red\] labelled "1.75" drawn in initial (location=(0.5, 1.75), label\_font\_size=24) - profile\_points\_2: a Point \[blue\] labelled "0.25" drawn in initial (location=(3.5, 0.25), label\_font\_size=24) - profile\_points\_3: a Point \[yellow\] labelled "1.25" drawn in initial (location=(1.5, 1.25), label\_font\_size=24) - profile\_points\_4: a Point \[green\] labelled "0.75" drawn in initial (location=(2.5, 0.75), label\_font\_size=24) - ray\_origins: a Point \[red\] drawn in xt (location=(0.5, 0.0)) - ray\_origins\_2: a Point \[blue\] drawn in xt (location=(3.5, 0.0)) - ray\_origins\_3: a Point \[yellow\] drawn in xt (location=(1.5, 0.0)) - ray\_origins\_4: a Point \[green\] drawn in xt (location=(2.5, 0.0)) - rays: a Line \[red\] drawn in xt (start=(0.5, 0.0), end=(4.0, 2.0)) - rays\_2: a Line \[blue\] drawn in xt (start=(3.5, 0.0), end=(4.0, 2.0)) - rays\_3: a Line \[yellow\] drawn in xt (start=(1.5, 0.0), end=(4.0, 2.0)) - rays\_4: a Line \[green\] drawn in xt (start=(2.5, 0.0), end=(4.0, 2.0)) - recipe\_heading: a Heading that says "The Method, In Four Steps" - steps: a Block \[text\] that says "Write $u\_t + a thin u\_x = 0$ and read off $a$. Solve $x'(t) = a$ from each starting point $x\_0$. Use constancy along each characteristic: $u = f(x\_0)$. Invert for $x\_0$ in terms of $x$ and $t$, then substitute." - xt: an Axes (x\_range=(0.0, 7.0), y\_range=(0.0, 3.0), y\_label='t') #### Beats ##### [10:3.878](https://academa.ai/lectures/the-method-of-characteristics?t=603.8784791666667) Narration: One last equation, and it is the one that made this method famous. u sub t plus u times u sub x equals zero. The coefficient in front of u sub x is now u itself: the value being carried sets its own speed. Board: Empty. Actions: - [10:3.878](https://academa.ai/lectures/the-method-of-characteristics?t=603.8784791666667): heading is shown on the screen, written out. - [10:3.878](https://academa.ai/lectures/the-method-of-characteristics?t=603.8784791666667): equation is shown on the screen, written out. ##### [10:20.227](https://academa.ai/lectures/the-method-of-characteristics?t=620.2274791666667) Narration: Along any curve, the chain rule gives u sub t plus x prime of t times u sub x. Choose x prime of t equal to u, and that derivative becomes the equation's left hand side, which is zero. So u stays constant along the characteristic. Board: equation — a Math \[text\] that says "$u\_t + u thin u\_x = 0$"; heading — a Heading that says "When Characteristics Collide" Actions: - [10:21.789](https://academa.ai/lectures/the-method-of-characteristics?t=621.7894791666668): nonlinear\_work is shown on the screen, written out. - [10:22.625](https://academa.ai/lectures/the-method-of-characteristics?t=622.6254791666667): nonlinear\_work (the "u\_t" part) is emphasized. - [10:23.716](https://academa.ai/lectures/the-method-of-characteristics?t=623.7164791666667): nonlinear\_work (the "u\_t" part) is no longer emphasized. - [10:23.716](https://academa.ai/lectures/the-method-of-characteristics?t=623.7164791666667): nonlinear\_work (the "x'(t)" part) is emphasized. - [10:25.307](https://academa.ai/lectures/the-method-of-characteristics?t=625.3074791666667): nonlinear\_work (the "u\_x" part) is emphasized. - [10:25.307](https://academa.ai/lectures/the-method-of-characteristics?t=625.3074791666667): nonlinear\_work (the "x'(t)" part) is no longer emphasized. - [10:26.944](https://academa.ai/lectures/the-method-of-characteristics?t=626.9444791666667): nonlinear\_work is shown on the screen, written out. - [10:26.944](https://academa.ai/lectures/the-method-of-characteristics?t=626.9444791666667): nonlinear\_work (the "u\_x" part) is no longer emphasized. - [10:33.19](https://academa.ai/lectures/the-method-of-characteristics?t=633.1904791666667): nonlinear\_work is shown on the screen, written out. ##### [10:38.121](https://academa.ai/lectures/the-method-of-characteristics?t=638.1209791666668) Narration: A characteristic leaving x naught therefore carries the fixed value f of x naught. Its slope x prime is that same fixed number, so each path is a straight line, but different starting values produce different slopes. Board: Unchanged from the preceding beat in this scene. Actions: - [10:41.824](https://academa.ai/lectures/the-method-of-characteristics?t=641.8244791666667): nonlinear\_work is shown on the screen, written out. - [10:46.7](https://academa.ai/lectures/the-method-of-characteristics?t=646.7004791666667): note is shown on the screen, written out. - [10:53.539](https://academa.ai/lectures/the-method-of-characteristics?t=653.5389791666666): equation is hidden from the screen — left the board. - [10:53.539](https://academa.ai/lectures/the-method-of-characteristics?t=653.5389791666666): heading is hidden from the screen — left the board. - [10:53.539](https://academa.ai/lectures/the-method-of-characteristics?t=653.5389791666666): nonlinear\_work is hidden from the screen — left the board. - [10:53.539](https://academa.ai/lectures/the-method-of-characteristics?t=653.5389791666666): note is hidden from the screen — left the board. ##### [10:54.139](https://academa.ai/lectures/the-method-of-characteristics?t=654.1389791666667) Narration: Take decreasing initial data. The piece starting at x naught equals zero point five carries the large value one point seven five, so it travels fast. The piece starting at three point five carries only zero point two five, so it barely crawls. The two middle values lie between them. The quick characteristics chase the slow ones ahead. Board: Empty. Actions: - [10:54.139](https://academa.ai/lectures/the-method-of-characteristics?t=654.1389791666667): collide\_heading is shown on the screen, written out. - [10:54.139](https://academa.ai/lectures/the-method-of-characteristics?t=654.1389791666667): initial is shown on the screen, written out. - [10:54.139](https://academa.ai/lectures/the-method-of-characteristics?t=654.1389791666667): xt is shown on the screen, written out. - [10:54.835](https://academa.ai/lectures/the-method-of-characteristics?t=654.8354791666667): initial\_curve is shown on the screen, written out. - [10:59.398](https://academa.ai/lectures/the-method-of-characteristics?t=659.3984791666667): point is shown on the screen, grown. - [11:1.398](https://academa.ai/lectures/the-method-of-characteristics?t=661.3984791666667): point is hidden from the screen. - [11:2.01](https://academa.ai/lectures/the-method-of-characteristics?t=662.0104791666668): profile\_points is shown on the screen, written out. - [11:2.01](https://academa.ai/lectures/the-method-of-characteristics?t=662.0104791666668): ray\_origins is shown on the screen, written out. - [11:4.193](https://academa.ai/lectures/the-method-of-characteristics?t=664.1934791666667): rays is shown on the screen, drawn. - [11:6.282](https://academa.ai/lectures/the-method-of-characteristics?t=666.2824791666667): point\_2 is shown on the screen, grown. - [11:8.233](https://academa.ai/lectures/the-method-of-characteristics?t=668.2334791666667): profile\_points\_2 is shown on the screen, written out. - [11:8.233](https://academa.ai/lectures/the-method-of-characteristics?t=668.2334791666667): ray\_origins\_2 is shown on the screen, written out. - [11:8.282](https://academa.ai/lectures/the-method-of-characteristics?t=668.2824791666667): point\_2 is hidden from the screen. - [11:10.613](https://academa.ai/lectures/the-method-of-characteristics?t=670.6134791666667): rays\_2 is shown on the screen, drawn. - [11:12.517](https://academa.ai/lectures/the-method-of-characteristics?t=672.5174791666667): profile\_points\_3 is shown on the screen, written out. - [11:12.667](https://academa.ai/lectures/the-method-of-characteristics?t=672.6674791666667): ray\_origins\_3 is shown on the screen, written out. - [11:12.817](https://academa.ai/lectures/the-method-of-characteristics?t=672.8174791666667): rays\_3 is shown on the screen, drawn. - [11:12.967](https://academa.ai/lectures/the-method-of-characteristics?t=672.9674791666667): profile\_points\_4 is shown on the screen, written out. - [11:13.117](https://academa.ai/lectures/the-method-of-characteristics?t=673.1174791666667): ray\_origins\_4 is shown on the screen, written out. - [11:13.267](https://academa.ai/lectures/the-method-of-characteristics?t=673.2674791666667): rays\_4 is shown on the screen, drawn. ##### [11:18.551](https://academa.ai/lectures/the-method-of-characteristics?t=678.5509791666667) Narration: All four paths meet at x equals four, t equals two. They arrive carrying four different values of u, but u is supposed to be a function. At that point it cannot remain smooth and single-valued. Board: initial — an Axes (x\_range=(0.0, 4.0), y\_range=(0.0, 2.1), x\_label='x\_0'); xt — an Axes (x\_range=(0.0, 7.0), y\_range=(0.0, 3.0), y\_label='t'); collide\_heading — a Heading that says "Faster Values Catch Slower Values"; initial\_curve — a FunctionPlot \[blue\] drawn in initial (function=\, x\_range=(0.2, 3.8)); profile\_points — a Point \[red\] labelled "1.75" drawn in initial (location=(0.5, 1.75), label\_font\_size=24); ray\_origins — a Point \[red\] drawn in xt (location=(0.5, 0.0)); rays — a Line \[red\] drawn in xt (start=(0.5, 0.0), end=(4.0, 2.0)); profile\_points\_2 — a Point \[blue\] labelled "0.25" drawn in initial (location=(3.5, 0.25), label\_font\_size=24); ray\_origins\_2 — a Point \[blue\] drawn in xt (location=(3.5, 0.0)); rays\_2 — a Line \[blue\] drawn in xt (start=(3.5, 0.0), end=(4.0, 2.0)); profile\_points\_3 — a Point \[yellow\] labelled "1.25" drawn in initial (location=(1.5, 1.25), label\_font\_size=24); ray\_origins\_3 — a Point \[yellow\] drawn in xt (location=(1.5, 0.0)); rays\_3 — a Line \[yellow\] drawn in xt (start=(1.5, 0.0), end=(4.0, 2.0)); profile\_points\_4 — a Point \[green\] labelled "0.75" drawn in initial (location=(2.5, 0.75), label\_font\_size=24); ray\_origins\_4 — a Point \[green\] drawn in xt (location=(2.5, 0.0)); rays\_4 — a Line \[green\] drawn in xt (start=(2.5, 0.0), end=(4.0, 2.0)) Actions: - [11:19.909](https://academa.ai/lectures/the-method-of-characteristics?t=679.9094791666668): crossing is shown on the screen, written out. - [11:20.455](https://academa.ai/lectures/the-method-of-characteristics?t=680.4554791666667): point\_3 is shown on the screen, grown. - [11:22.455](https://academa.ai/lectures/the-method-of-characteristics?t=682.4554791666667): point\_3 is hidden from the screen. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): collide\_heading is hidden from the screen — left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): initial is hidden from the screen — left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): initial\_curve is hidden from the screen — initial left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): profile\_points is hidden from the screen — initial left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): profile\_points\_2 is hidden from the screen — initial left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): profile\_points\_3 is hidden from the screen — initial left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): profile\_points\_4 is hidden from the screen — initial left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): xt is hidden from the screen — left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): ray\_origins is hidden from the screen — xt left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): rays is hidden from the screen — xt left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): ray\_origins\_2 is hidden from the screen — xt left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): rays\_2 is hidden from the screen — xt left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): ray\_origins\_3 is hidden from the screen — xt left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): rays\_3 is hidden from the screen — xt left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): ray\_origins\_4 is hidden from the screen — xt left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): rays\_4 is hidden from the screen — xt left the board. - [11:31.925](https://academa.ai/lectures/the-method-of-characteristics?t=691.9254791666667): crossing is hidden from the screen — xt left the board. ##### [11:32.525](https://academa.ai/lectures/the-method-of-characteristics?t=692.5254791666667) Narration: The same collision appears in the wave profile. At first the front is smooth. As faster values catch slower ones, it steepens, and steepens again, until the front is nearly vertical. That breaking point is a shock. The smooth solution stops exactly when the first characteristics meet. Board: Empty. Actions: - [11:32.525](https://academa.ai/lectures/the-method-of-characteristics?t=692.5254791666667): breaking\_heading is shown on the screen, written out. - [11:32.525](https://academa.ai/lectures/the-method-of-characteristics?t=692.5254791666667): early\_axes is shown on the screen, written out. - [11:32.525](https://academa.ai/lectures/the-method-of-characteristics?t=692.5254791666667): middle\_axes is shown on the screen, written out. - [11:32.525](https://academa.ai/lectures/the-method-of-characteristics?t=692.5254791666667): late\_axes is shown on the screen, written out. - [11:32.525](https://academa.ai/lectures/the-method-of-characteristics?t=692.5254791666667): early\_caption is shown on the screen, written out. - [11:32.525](https://academa.ai/lectures/the-method-of-characteristics?t=692.5254791666667): middle\_caption is shown on the screen, written out. - [11:32.525](https://academa.ai/lectures/the-method-of-characteristics?t=692.5254791666667): late\_caption is shown on the screen, written out. - [11:37.134](https://academa.ai/lectures/the-method-of-characteristics?t=697.1344791666668): early\_curve is shown on the screen, drawn. - [11:41.024](https://academa.ai/lectures/the-method-of-characteristics?t=701.0244791666668): middle\_curve is shown on the screen, drawn. - [11:41.964](https://academa.ai/lectures/the-method-of-characteristics?t=701.9644791666667): late\_curve is shown on the screen, drawn. - [11:46.817](https://academa.ai/lectures/the-method-of-characteristics?t=706.8174791666668): late\_curve is indicated — a transient flash. - [11:52.007](https://academa.ai/lectures/the-method-of-characteristics?t=712.0074791666667): breaking\_heading is hidden from the screen — left the board. - [11:52.007](https://academa.ai/lectures/the-method-of-characteristics?t=712.0074791666667): early\_axes is hidden from the screen — left the board. - [11:52.007](https://academa.ai/lectures/the-method-of-characteristics?t=712.0074791666667): early\_curve is hidden from the screen — early\_axes left the board. - [11:52.007](https://academa.ai/lectures/the-method-of-characteristics?t=712.0074791666667): early\_caption is hidden from the screen — left the board. - [11:52.007](https://academa.ai/lectures/the-method-of-characteristics?t=712.0074791666667): late\_axes is hidden from the screen — left the board. - [11:52.007](https://academa.ai/lectures/the-method-of-characteristics?t=712.0074791666667): late\_curve is hidden from the screen — late\_axes left the board. - [11:52.007](https://academa.ai/lectures/the-method-of-characteristics?t=712.0074791666667): late\_caption is hidden from the screen — left the board. - [11:52.007](https://academa.ai/lectures/the-method-of-characteristics?t=712.0074791666667): middle\_axes is hidden from the screen — left the board. - [11:52.007](https://academa.ai/lectures/the-method-of-characteristics?t=712.0074791666667): middle\_curve is hidden from the screen — middle\_axes left the board. - [11:52.007](https://academa.ai/lectures/the-method-of-characteristics?t=712.0074791666667): middle\_caption is hidden from the screen — left the board. ##### [11:53.207](https://academa.ai/lectures/the-method-of-characteristics?t=713.2074791666668) Narration: The method is four steps long, and their order matters. First read the transport speed from the equation. Board: Unchanged from the preceding beat in this scene. Actions: - [11:53.207](https://academa.ai/lectures/the-method-of-characteristics?t=713.2074791666668): recipe\_heading is shown on the screen, written out. - [11:53.207](https://academa.ai/lectures/the-method-of-characteristics?t=713.2074791666668): steps is shown on the screen, written out. - [11:57.329](https://academa.ai/lectures/the-method-of-characteristics?t=717.3294791666667): steps (the "Write" part) is emphasized. ##### [12:0.785](https://academa.ai/lectures/the-method-of-characteristics?t=720.7849791666667) Narration: First, write the equation with the two derivatives lined up, and read off the coefficient sitting in front of u sub x. Board: steps — a Block \[text\] that says "Write $u\_t + a thin u\_x = 0$ and read off $a$. Solve $x'(t) = a$ from each starting point $x\_0$. Use constancy along each characteristic: $u = f(x\_0)$. Invert for $x\_0$ in terms of $x$ and $t$, then substitute."; recipe\_heading — a Heading that says "The Method, In Four Steps" Actions: - None. ##### [12:8.873](https://academa.ai/lectures/the-method-of-characteristics?t=728.8734791666667) Narration: Second, solve x prime of t equals that coefficient. Its solutions are the characteristics, one through each starting point x naught. Board: Unchanged from the preceding beat in this scene. Actions: - [12:9.349](https://academa.ai/lectures/the-method-of-characteristics?t=729.3494791666667): steps (the "Solve" part) is emphasized. - [12:9.349](https://academa.ai/lectures/the-method-of-characteristics?t=729.3494791666667): steps (the "Write" part) is no longer emphasized. ##### [12:20.015](https://academa.ai/lectures/the-method-of-characteristics?t=740.0154791666668) Narration: Third, use the equation to show that u does not change along those curves. Thus u equals f of x naught all the way along. Board: Unchanged from the preceding beat in this scene. Actions: - [12:20.363](https://academa.ai/lectures/the-method-of-characteristics?t=740.3634791666667): steps (the "Solve" part) is no longer emphasized. - [12:20.363](https://academa.ai/lectures/the-method-of-characteristics?t=740.3634791666667): steps (the "Use constancy" part) is emphasized. ##### [12:29.52](https://academa.ai/lectures/the-method-of-characteristics?t=749.5204791666667) Narration: Fourth, invert. Express x naught in terms of x and t, then substitute it into f. That final expression is the solution. Board: Unchanged from the preceding beat in this scene. Actions: - [12:29.868](https://academa.ai/lectures/the-method-of-characteristics?t=749.8684791666667): steps (the "Invert" part) is emphasized. - [12:29.868](https://academa.ai/lectures/the-method-of-characteristics?t=749.8684791666667): steps (the "Use constancy" part) is no longer emphasized. ##### [12:40.779](https://academa.ai/lectures/the-method-of-characteristics?t=760.7789791666667) Narration: Each step is ordinary calculus: read the speed, find its curves, carry the initial value along them, and trace the requested point back to its start. The partial differential equation becomes ordinary along exactly those paths. Board: Unchanged from the preceding beat in this scene. Actions: - [12:43.843](https://academa.ai/lectures/the-method-of-characteristics?t=763.8434791666667): steps (the "Invert" part) is no longer emphasized. - [12:43.843](https://academa.ai/lectures/the-method-of-characteristics?t=763.8434791666667): steps (the "Write" part) is emphasized. - [12:44.923](https://academa.ai/lectures/the-method-of-characteristics?t=764.9234791666668): steps (the "Solve" part) is emphasized. - [12:44.923](https://academa.ai/lectures/the-method-of-characteristics?t=764.9234791666668): steps (the "Write" part) is no longer emphasized. - [12:46.363](https://academa.ai/lectures/the-method-of-characteristics?t=766.3634791666667): steps (the "Solve" part) is no longer emphasized. - [12:46.363](https://academa.ai/lectures/the-method-of-characteristics?t=766.3634791666667): steps (the "Use constancy" part) is emphasized. - [12:48.928](https://academa.ai/lectures/the-method-of-characteristics?t=768.9284791666668): steps (the "Invert" part) is emphasized. - [12:48.928](https://academa.ai/lectures/the-method-of-characteristics?t=768.9284791666668): steps (the "Use constancy" part) is no longer emphasized. - [12:57.139](https://academa.ai/lectures/the-method-of-characteristics?t=777.1388958333334): steps (the "Invert" part) is no longer emphasized. - [12:57.389](https://academa.ai/lectures/the-method-of-characteristics?t=777.3888958333334): recipe\_heading is hidden from the screen — left the board. - [12:57.389](https://academa.ai/lectures/the-method-of-characteristics?t=777.3888958333334): steps is hidden from the screen — left the board.