# Stokes' Theorem in Practice > One exam question, worked from the first reading to the final check. The line integral of F = -yz i + (4y + 1) j + xy k around a circle of radius 3 standing in the plane y = 4, walked clockwise as seen from the positive y-axis. Stokes' theorem trades that walk around the rim for a flux through the flat disc the rim bounds, so we compute the curl one slot at a time, settle which way the unit normal points by reading the right-hand rule off the wording of the problem rather than off a preference, and integrate a constant through a disc of known area. The answer is verified twice. - Canonical watch page: [Stokes' Theorem in Practice](https://academa.ai/lectures/stokes-theorem-in-practice) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Mathematics - Published: 2026-08-28T22:51:42.000Z - Updated: 2026-08-28T22:51:42.000Z - Duration: PT281S (4 minutes 41 seconds) - Chapters: 1 - Views: 0 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M14TZG80Q03WZ8E386YW13DE/0/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14TZG80Q03WZ8E386YW13DE/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M14TZG80Q03WZ8E386YW13DE/0/dark/poster.jpg) ## Description Stokes' theorem on one exam question: swap the walk around the circle for the flux through the disc, and read the normal off the wording. ## Chapters - [00:00–04:41 · Stokes' Theorem in Practice](https://academa.ai/lectures/stokes-theorem-in-practice?t=0) ## Transcript ### [00:00 · Stokes' Theorem in Practice](https://academa.ai/lectures/stokes-theorem-in-practice?t=0) We want the circulation of this vector field around one circle. Begin by reading every geometric fact in the question against that circle. Its radius is three, it lies in the plane y equals four, and its centre is the point zero, four, zero. That plane is perpendicular to the y axis, so the circle stands like a coin in a translated copy of the x z plane. Viewed from the positive y axis while looking toward the origin, the red arrows run clockwise. This direction will determine the sign of the answer. Stokes' theorem replaces circulation around the boundary with flux of the curl through a spanning surface. The boundary direction and the surface normal must be compatible. The surface may be any surface with this rim. A bowl can dip behind it, and a dome can rise in front of it. Both have exactly the same boundary. The flat disc is the useful choice. It lies in y equals four, its unit normal is constant, and its area is pi times three squared, or nine pi. Write the field as three components: minus y z, four y plus one, and x y. These are the three entries the curl differentiates. Each curl slot uses the other two components in cyclic order. The determinant mnemonic records the same pattern with i, j, k on top, the partial derivatives in the middle, and the field along the bottom. For the first slot, differentiate x y with respect to y, then subtract the z derivative of four y plus one. Those terms are x and zero, so the first slot is x. For the second slot, the z derivative of minus y z is minus y. Subtract the x derivative of x y, which is y. Minus y minus y is minus two y. For the third slot, the x derivative of four y plus one is zero. Subtract the y derivative of minus y z, which is minus z, and the result is z. Together the three slots give x, minus two y, z. Notice that four y plus one contributed nothing because both derivatives taken from it were in directions it does not depend on. Keep that term marked for the final check. Now the right-hand rule fixes the normal. From the side the normal points toward, the positive walk around the boundary must look counterclockwise. Move to the positive y viewpoint. The given red walk looks clockwise from there, so the compatible normal points away from that viewer, back toward the origin. Therefore n hat is minus j hat: zero, minus one, zero. A free vector keeps that direction when its base is moved away from the y-axis line. The orientation pair moves together. Reverse the boundary walk and the compatible normal flips to plus j. Restore the given clockwise walk and the normal returns to minus j. Because the disc is flat, that same normal belongs at every point. Slide it across the surface and only its base point changes; its direction remains minus j everywhere. Dot the curl x, minus two y, z with the normal zero, minus one, zero. The x term is multiplied by zero, and the z term is multiplied by zero. They vanish. The middle product is minus two y times minus one, leaving plus two y. Every point of this disc has y equal to four. As a point crosses the disc, its reading stays at y equals four: two y is eight at every point, even while x and z change. So the surface integral is eight times the area of a radius-three disc. The area is nine pi, and eight times nine pi is seventy-two pi. Therefore the requested circulation is seventy-two pi. The unused middle component has a structural explanation. Four y plus one in the j direction is the gradient of two y squared plus y, and every gradient has zero curl. A direct parametrization gives r of t equal to three cosine t, four, three sine t. Its dot product with the field simplifies to thirty-six d t. Integrating thirty-six from zero to two pi gives seventy-two pi again. The boundary calculation and the flux calculation agree. ## 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/01M14TZG80Q03WZ8E386YW13DE/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: [Stokes' Theorem in Practice](https://academa.ai/lectures/stokes-theorem-in-practice?t=0) Span: 00:00–04:41.159 (0s–281.1592083333333s). #### Objects - answer: a Math \[text\] that says "$integral\_C arrow(F) dot dif arrow(r) = 72 pi$" - bowl: a Surface \[green\] drawn in frame (function=\, u\_range=(-3.0, 3.0), v\_range=(-3.0, 3.0)) - centre: a Point \[text\] labelled "(0, 4, 0)" drawn in frame (location=(0.0, 4.0, 0.0)) - checks\_heading: a Heading that says "Two Independent Checks" - components: a Math \[text\] that says "$arrow(F) = chevron.l -y z, thin 4 y + 1, thin x y chevron.r$" - curl\_gradient: a Math \[text\] that says "$nabla times nabla(2 y^2 + y) = arrow(0)$" - curl\_heading: a Heading that says "The Curl, One Slot at a Time" - curl\_result: a Math \[text\] that says "$nabla times arrow(F) = chevron.l x, thin -2 y, thin z chevron.r$" - curl\_work: a Derivation \[text\] that says "$(nabla times arrow(F))\_1 &= (partial F\_3)/(partial y) - (partial F\_2)/(partial z) = x - 0 = x \\ (nabla times arrow(F))\_2 &= (partial F\_1)/(partial z) - (partial F\_3)/(partial x) = -y - y = -2 y \\ (nabla times arrow(F))\_3 &= (partial F\_2)/(…$" - direct\_check: a Derivation \[text\] that says "$arrow(r)(t) = chevron.l 3 cos t, thin 4, thin 3 sin t chevron.r \\ arrow(F) dot dif arrow(r) = 36 thin dif t \\ integral\_0^(2 pi) 36 thin dif t = 72 pi$" - disc: a Circle \[blue\] drawn in frame (center=(0.0, 4.0, 0.0), radius=3.0, normal\_vector=(0.0, 1.0, 0.0)) - dome: a Surface \[magenta\] drawn in frame (function=\, u\_range=(-3.0, 3.0), v\_range=(-3.0, 3.0)) - flux\_heading: a Heading that says "The Flux Through the Disc" - flux\_work: a Derivation \[text\] that says "$nabla times arrow(F) &= chevron.l x, thin -2 y, thin z chevron.r \\ hat(n) &= chevron.l 0, thin -1, thin 0 chevron.r \\ (nabla times arrow(F)) dot hat(n) &= x dot 0 + (-2 y)(-1) + z dot 0 \\ &= 2 y = 2(4) = 8 \\ integral\_C arrow(F) dot dif arr…$" - frame: an Axes3D (x\_range=(-4.0, 4.0), y\_range=(-2.0, 6.0), z\_range=(-4.0, 4.0)) - gradient\_check: a Math \[text\] that says "$(4 y + 1) hat(j) = nabla(2 y^2 + y)$" - mnemonic: a Math \[text\] that says "$nabla times arrow(F) = op("det") mat(hat(i), hat(j), hat(k); partial\_x, partial\_y, partial\_z; -y z, 4 y + 1, x y)$" - normal: a Vector \[green\] labelled "hat(n)" drawn in frame (start=(\, 4.0, \, \, \, t\_range=(0.0, 6.283185307179586)) - reverse\_walk\_2: an Orientation \[red\] drawn in frame (path=((3.0, 4.0, -7.347880794884119e-16), (2.9855541800165906, 4.0, …, closed=True, arrows=4) - right\_hand: a Panel that says "From the side the normal points toward, the positive boundary direction looks counterclockwise." - rim: a ParametricCurve \[yellow\] drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)) - statement: a Math \[text\] that says "$integral\_C arrow(F) dot dif arrow(r) = integral.double\_S (nabla times arrow(F)) dot hat(n) thin dif S$" - theorem: a Panel that says "$S$ may be any oriented surface with boundary $C$, provided its normal agrees with the direction around $C$." - theorem\_heading: a Heading that says "Stokes' Theorem" - walk: an Orientation \[red\] drawn in frame (path=((3.0, 4.0, 0.0), (2.9855541800165906, 4.0, 0.2940514209886818)…, closed=True, arrows=4) #### Beats ##### [00:00](https://academa.ai/lectures/stokes-theorem-in-practice?t=0) Narration: We want the circulation of this vector field around one circle. Begin by reading every geometric fact in the question against that circle. Board: Empty. Actions: - [00:00](https://academa.ai/lectures/stokes-theorem-in-practice?t=0): question is shown on the screen, written out. - [00:00](https://academa.ai/lectures/stokes-theorem-in-practice?t=0): frame is shown on the screen, written out. - [00:2.682](https://academa.ai/lectures/stokes-theorem-in-practice?t=2.682): rim is shown on the screen, drawn. ##### [00:8.204](https://academa.ai/lectures/stokes-theorem-in-practice?t=8.2045) Narration: Its radius is three, it lies in the plane y equals four, and its centre is the point zero, four, zero. That plane is perpendicular to the y axis, so the circle stands like a coin in a translated copy of the x z plane. Board: frame — an Axes3D (x\_range=(-4.0, 4.0), y\_range=(-2.0, 6.0), z\_range=(-4.0, 4.0)); question — a Panel that says "Evaluate $integral\_C arrow(F) dot dif arrow(r)$ for $arrow(F) = -y z thin hat(i) + (4 y + 1) thin hat(j) + x y thin hat(k)$ $C$: radius $3$, plane $y=4$, clockwise from the positive $y$-axis"; rim — a ParametricCurve \[yellow\] drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)) Actions: - [00:9.389](https://academa.ai/lectures/stokes-theorem-in-practice?t=9.389): point is shown on the screen, grown. - [00:10.589](https://academa.ai/lectures/stokes-theorem-in-practice?t=10.588999999999999): point is hidden from the screen. - [00:12.105](https://academa.ai/lectures/stokes-theorem-in-practice?t=12.105): point\_2 is shown on the screen, grown. - [00:13.174](https://academa.ai/lectures/stokes-theorem-in-practice?t=13.174): centre is shown on the screen, written out. - [00:13.305](https://academa.ai/lectures/stokes-theorem-in-practice?t=13.305): point\_2 is hidden from the screen. - [00:14.079](https://academa.ai/lectures/stokes-theorem-in-practice?t=14.078999999999999): point\_3 is shown on the screen, grown. - [00:15.579](https://academa.ai/lectures/stokes-theorem-in-practice?t=15.578999999999999): point\_3 is hidden from the screen. ##### [00:24.094](https://academa.ai/lectures/stokes-theorem-in-practice?t=24.094499999999996) Narration: Viewed from the positive y axis while looking toward the origin, the red arrows run clockwise. This direction will determine the sign of the answer. Board: frame — an Axes3D (x\_range=(-4.0, 4.0), y\_range=(-2.0, 6.0), z\_range=(-4.0, 4.0)); question — a Panel that says "Evaluate $integral\_C arrow(F) dot dif arrow(r)$ for $arrow(F) = -y z thin hat(i) + (4 y + 1) thin hat(j) + x y thin hat(k)$ $C$: radius $3$, plane $y=4$, clockwise from the positive $y$-axis"; rim — a ParametricCurve \[yellow\] drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)); centre — a Point \[text\] labelled "(0, 4, 0)" drawn in frame (location=(0.0, 4.0, 0.0)) Actions: - [00:28.309](https://academa.ai/lectures/stokes-theorem-in-practice?t=28.308999999999997): walk is shown on the screen, written out. - [00:33.243](https://academa.ai/lectures/stokes-theorem-in-practice?t=33.2435): frame moves to a new place on the board. - [00:33.243](https://academa.ai/lectures/stokes-theorem-in-practice?t=33.2435): question is hidden from the screen — left the board. ##### [00:33.843](https://academa.ai/lectures/stokes-theorem-in-practice?t=33.84349999999999) Narration: Stokes' theorem replaces circulation around the boundary with flux of the curl through a spanning surface. The boundary direction and the surface normal must be compatible. Board: frame — an Axes3D (x\_range=(-4.0, 4.0), y\_range=(-2.0, 6.0), z\_range=(-4.0, 4.0)); rim — a ParametricCurve \[yellow\] drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)); centre — a Point \[text\] labelled "(0, 4, 0)" drawn in frame (location=(0.0, 4.0, 0.0)); walk — an Orientation \[red\] drawn in frame (path=((3.0, 4.0, 0.0), (2.9855541800165906, 4.0, 0.2940514209886818)…, closed=True, arrows=4) Actions: - [00:33.843](https://academa.ai/lectures/stokes-theorem-in-practice?t=33.84349999999999): theorem\_heading is shown on the screen, written out. - [00:34.83](https://academa.ai/lectures/stokes-theorem-in-practice?t=34.82999999999999): statement is shown on the screen, written out. - [00:35.364](https://academa.ai/lectures/stokes-theorem-in-practice?t=35.36399999999999): statement (the "integral\_C arrow(F) dot dif arrow(r)" part) is emphasized. - [00:36.409](https://academa.ai/lectures/stokes-theorem-in-practice?t=36.40899999999999): theorem is shown on the screen, written out. - [00:37.35](https://academa.ai/lectures/stokes-theorem-in-practice?t=37.34999999999999): statement (the "integral.double\_S (nabla times arrow(F)) dot hat(n) thin dif S" part) is emphasized. - [00:37.35](https://academa.ai/lectures/stokes-theorem-in-practice?t=37.34999999999999): statement (the "integral\_C arrow(F) dot dif arrow(r)" part) is no longer emphasized. ##### [00:44.915](https://academa.ai/lectures/stokes-theorem-in-practice?t=44.915499999999994) Narration: The surface may be any surface with this rim. A bowl can dip behind it, and a dome can rise in front of it. Both have exactly the same boundary. Board: frame — an Axes3D (x\_range=(-4.0, 4.0), y\_range=(-2.0, 6.0), z\_range=(-4.0, 4.0)); rim — a ParametricCurve \[yellow\] drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)); centre — a Point \[text\] labelled "(0, 4, 0)" drawn in frame (location=(0.0, 4.0, 0.0)); walk — an Orientation \[red\] drawn in frame (path=((3.0, 4.0, 0.0), (2.9855541800165906, 4.0, 0.2940514209886818)…, closed=True, arrows=4); theorem — a Panel that says "$S$ may be any oriented surface with boundary $C$, provided its normal agrees with the direction around $C$."; statement — a Math \[text\] that says "$integral\_C arrow(F) dot dif arrow(r) = integral.double\_S (nabla times arrow(F)) dot hat(n) thin dif S$"; theorem\_heading — a Heading that says "Stokes' Theorem" Actions: - [00:48.828](https://academa.ai/lectures/stokes-theorem-in-practice?t=48.827999999999996): bowl is shown on the screen, faded in. - [00:50.593](https://academa.ai/lectures/stokes-theorem-in-practice?t=50.59299999999999): bowl is hidden from the screen. - [00:50.593](https://academa.ai/lectures/stokes-theorem-in-practice?t=50.59299999999999): dome is shown on the screen, faded in. - [00:52.787](https://academa.ai/lectures/stokes-theorem-in-practice?t=52.78699999999999): dome is hidden from the screen. ##### [00:55.674](https://academa.ai/lectures/stokes-theorem-in-practice?t=55.67399999999999) Narration: The flat disc is the useful choice. It lies in y equals four, its unit normal is constant, and its area is pi times three squared, or nine pi. Board: Unchanged from the preceding beat in this scene. Actions: - [00:56.15](https://academa.ai/lectures/stokes-theorem-in-practice?t=56.15): disc is shown on the screen, faded in. - [01:0.248](https://academa.ai/lectures/stokes-theorem-in-practice?t=60.248): point\_4 is shown on the screen, grown. - [01:1.448](https://academa.ai/lectures/stokes-theorem-in-practice?t=61.448): point\_4 is hidden from the screen. - [01:6.785](https://academa.ai/lectures/stokes-theorem-in-practice?t=66.785): statement is hidden from the screen — left the board. - [01:6.785](https://academa.ai/lectures/stokes-theorem-in-practice?t=66.785): theorem is hidden from the screen — left the board. - [01:6.785](https://academa.ai/lectures/stokes-theorem-in-practice?t=66.785): theorem\_heading is hidden from the screen — left the board. ##### [01:7.385](https://academa.ai/lectures/stokes-theorem-in-practice?t=67.38499999999999) Narration: Write the field as three components: minus y z, four y plus one, and x y. These are the three entries the curl differentiates. Board: frame — an Axes3D (x\_range=(-4.0, 4.0), y\_range=(-2.0, 6.0), z\_range=(-4.0, 4.0)); rim — a ParametricCurve \[yellow\] drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)); centre — a Point \[text\] labelled "(0, 4, 0)" drawn in frame (location=(0.0, 4.0, 0.0)); walk — an Orientation \[red\] drawn in frame (path=((3.0, 4.0, 0.0), (2.9855541800165906, 4.0, 0.2940514209886818)…, closed=True, arrows=4); disc — a Circle \[blue\] drawn in frame (center=(0.0, 4.0, 0.0), radius=3.0, normal\_vector=(0.0, 1.0, 0.0)) Actions: - [01:7.385](https://academa.ai/lectures/stokes-theorem-in-practice?t=67.38499999999999): curl\_heading is shown on the screen, written out. - [01:8.929](https://academa.ai/lectures/stokes-theorem-in-practice?t=68.929): components is shown on the screen, written out. - [01:10.02](https://academa.ai/lectures/stokes-theorem-in-practice?t=70.02): components (the "-y z" part) is emphasized. - [01:11.518](https://academa.ai/lectures/stokes-theorem-in-practice?t=71.518): components (the "-y z" part) is no longer emphasized. - [01:11.518](https://academa.ai/lectures/stokes-theorem-in-practice?t=71.518): components (the "4 y + 1" part) is emphasized. - [01:13.306](https://academa.ai/lectures/stokes-theorem-in-practice?t=73.306): components (the "4 y + 1" part) is no longer emphasized. - [01:13.306](https://academa.ai/lectures/stokes-theorem-in-practice?t=73.306): components (the "x y" part) is emphasized. ##### [01:18.109](https://academa.ai/lectures/stokes-theorem-in-practice?t=78.109) Narration: Each curl slot uses the other two components in cyclic order. The determinant mnemonic records the same pattern with i, j, k on top, the partial derivatives in the middle, and the field along the bottom. Board: frame — an Axes3D (x\_range=(-4.0, 4.0), y\_range=(-2.0, 6.0), z\_range=(-4.0, 4.0)); rim — a ParametricCurve \[yellow\] drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)); centre — a Point \[text\] labelled "(0, 4, 0)" drawn in frame (location=(0.0, 4.0, 0.0)); walk — an Orientation \[red\] drawn in frame (path=((3.0, 4.0, 0.0), (2.9855541800165906, 4.0, 0.2940514209886818)…, closed=True, arrows=4); disc — a Circle \[blue\] drawn in frame (center=(0.0, 4.0, 0.0), radius=3.0, normal\_vector=(0.0, 1.0, 0.0)); components — a Math \[text\] that says "$arrow(F) = chevron.l -y z, thin 4 y + 1, thin x y chevron.r$"; curl\_heading — a Heading that says "The Curl, One Slot at a Time" Actions: - [01:18.109](https://academa.ai/lectures/stokes-theorem-in-practice?t=78.109): components (the "x y" part) is no longer emphasized. - [01:21.081](https://academa.ai/lectures/stokes-theorem-in-practice?t=81.08099999999999): pattern is shown on the screen, written out. - [01:22.904](https://academa.ai/lectures/stokes-theorem-in-practice?t=82.90399999999998): mnemonic is shown on the screen, written out. - [01:22.904](https://academa.ai/lectures/stokes-theorem-in-practice?t=82.90399999999998): mnemonic is emphasized. - [01:31.53](https://academa.ai/lectures/stokes-theorem-in-practice?t=91.53049999999999): components moves to a new place on the board. - [01:31.53](https://academa.ai/lectures/stokes-theorem-in-practice?t=91.53049999999999): mnemonic is hidden from the screen — left the board. - [01:31.53](https://academa.ai/lectures/stokes-theorem-in-practice?t=91.53049999999999): pattern is hidden from the screen — left the board. ##### [01:32.13](https://academa.ai/lectures/stokes-theorem-in-practice?t=92.1305) Narration: For the first slot, differentiate x y with respect to y, then subtract the z derivative of four y plus one. Those terms are x and zero, so the first slot is x. Board: Unchanged from the preceding beat in this scene. Actions: - [01:32.13](https://academa.ai/lectures/stokes-theorem-in-practice?t=92.1305): curl\_work is shown on the screen, written out. - [01:34.801](https://academa.ai/lectures/stokes-theorem-in-practice?t=94.80099999999999): curl\_work (the "(partial F\_3)/(partial y)" part) is emphasized. - [01:37.958](https://academa.ai/lectures/stokes-theorem-in-practice?t=97.958): curl\_work (the "(partial F\_2)/(partial z)" part) is emphasized. - [01:37.958](https://academa.ai/lectures/stokes-theorem-in-practice?t=97.958): curl\_work (the "(partial F\_3)/(partial y)" part) is no longer emphasized. - [01:41.65](https://academa.ai/lectures/stokes-theorem-in-practice?t=101.64999999999999): curl\_work (the "(partial F\_2)/(partial z)" part) is no longer emphasized. - [01:41.65](https://academa.ai/lectures/stokes-theorem-in-practice?t=101.64999999999999): curl\_work (the "x#2" part) is emphasized. - [01:41.952](https://academa.ai/lectures/stokes-theorem-in-practice?t=101.952): curl\_work (the "0" part) is emphasized. - [01:41.952](https://academa.ai/lectures/stokes-theorem-in-practice?t=101.952): curl\_work (the "x#2" part) is no longer emphasized. - [01:43.136](https://academa.ai/lectures/stokes-theorem-in-practice?t=103.136): curl\_work (the "0" part) is no longer emphasized. - [01:43.136](https://academa.ai/lectures/stokes-theorem-in-practice?t=103.136): curl\_work (the "= x" part) is emphasized. ##### [01:45.007](https://academa.ai/lectures/stokes-theorem-in-practice?t=105.007) Narration: For the second slot, the z derivative of minus y z is minus y. Subtract the x derivative of x y, which is y. Minus y minus y is minus two y. Board: Unchanged from the preceding beat in this scene. Actions: - [01:45.605](https://academa.ai/lectures/stokes-theorem-in-practice?t=105.605): curl\_work is shown on the screen, written out. - [01:46.65](https://academa.ai/lectures/stokes-theorem-in-practice?t=106.65): curl\_work (the "= x" part) is no longer emphasized. - [01:46.65](https://academa.ai/lectures/stokes-theorem-in-practice?t=106.65): curl\_work (the "(partial F\_1)/(partial z)" part) is emphasized. - [01:51.085](https://academa.ai/lectures/stokes-theorem-in-practice?t=111.08500000000001): curl\_work (the "(partial F\_1)/(partial z)" part) is no longer emphasized. - [01:51.085](https://academa.ai/lectures/stokes-theorem-in-practice?t=111.08500000000001): curl\_work (the "(partial F\_3)/(partial x)" part) is emphasized. - [01:54.069](https://academa.ai/lectures/stokes-theorem-in-practice?t=114.069): curl\_work (the "(partial F\_3)/(partial x)" part) is no longer emphasized. - [01:54.069](https://academa.ai/lectures/stokes-theorem-in-practice?t=114.069): curl\_work (the "-y - y" part) is emphasized. - [01:56.089](https://academa.ai/lectures/stokes-theorem-in-practice?t=116.089): curl\_work (the "-2 y" part) is emphasized. - [01:56.089](https://academa.ai/lectures/stokes-theorem-in-practice?t=116.089): curl\_work (the "-y - y" part) is no longer emphasized. ##### [01:57.624](https://academa.ai/lectures/stokes-theorem-in-practice?t=117.6235) Narration: For the third slot, the x derivative of four y plus one is zero. Subtract the y derivative of minus y z, which is minus z, and the result is z. Board: Unchanged from the preceding beat in this scene. Actions: - [01:58.315](https://academa.ai/lectures/stokes-theorem-in-practice?t=118.315): curl\_work is shown on the screen, written out. - [01:59.557](https://academa.ai/lectures/stokes-theorem-in-practice?t=119.55699999999999): curl\_work (the "-2 y" part) is no longer emphasized. - [01:59.557](https://academa.ai/lectures/stokes-theorem-in-practice?t=119.55699999999999): curl\_work (the "(partial F\_2)/(partial x)" part) is emphasized. - [02:1.775](https://academa.ai/lectures/stokes-theorem-in-practice?t=121.77499999999999): curl\_work (the "(partial F\_2)/(partial x)" part) is no longer emphasized. - [02:1.775](https://academa.ai/lectures/stokes-theorem-in-practice?t=121.77499999999999): curl\_work (the "0" part) is emphasized. - [02:3.725](https://academa.ai/lectures/stokes-theorem-in-practice?t=123.725): curl\_work (the "(partial F\_1)/(partial y)" part) is emphasized. - [02:3.725](https://academa.ai/lectures/stokes-theorem-in-practice?t=123.725): curl\_work (the "0" part) is no longer emphasized. - [02:6.337](https://academa.ai/lectures/stokes-theorem-in-practice?t=126.33699999999999): curl\_work (the "(-z)" part) is emphasized. - [02:6.337](https://academa.ai/lectures/stokes-theorem-in-practice?t=126.33699999999999): curl\_work (the "(partial F\_1)/(partial y)" part) is no longer emphasized. - [02:7.765](https://academa.ai/lectures/stokes-theorem-in-practice?t=127.76499999999999): curl\_work (the "(-z)" part) is no longer emphasized. - [02:7.765](https://academa.ai/lectures/stokes-theorem-in-practice?t=127.76499999999999): curl\_work (the "= z" part) is emphasized. ##### [02:9.736](https://academa.ai/lectures/stokes-theorem-in-practice?t=129.7355) Narration: Together the three slots give x, minus two y, z. Notice that four y plus one contributed nothing because both derivatives taken from it were in directions it does not depend on. Keep that term marked for the final check. Board: Unchanged from the preceding beat in this scene. Actions: - [02:10.084](https://academa.ai/lectures/stokes-theorem-in-practice?t=130.084): curl\_result is shown on the screen, written out. - [02:11.883](https://academa.ai/lectures/stokes-theorem-in-practice?t=131.88299999999998): curl\_result (the "x" part) is emphasized. - [02:12.243](https://academa.ai/lectures/stokes-theorem-in-practice?t=132.243): curl\_result (the "-2 y" part) is emphasized. - [02:12.243](https://academa.ai/lectures/stokes-theorem-in-practice?t=132.243): curl\_result (the "x" part) is no longer emphasized. - [02:13.218](https://academa.ai/lectures/stokes-theorem-in-practice?t=133.21800000000002): curl\_result (the "-2 y" part) is no longer emphasized. - [02:13.218](https://academa.ai/lectures/stokes-theorem-in-practice?t=133.21800000000002): curl\_result (the "z" part) is emphasized. - [02:14.844](https://academa.ai/lectures/stokes-theorem-in-practice?t=134.844): components (the "4 y + 1" part) is emphasized. - [02:24.549](https://academa.ai/lectures/stokes-theorem-in-practice?t=144.5495): components is hidden from the screen — left the board. - [02:24.549](https://academa.ai/lectures/stokes-theorem-in-practice?t=144.5495): curl\_heading is hidden from the screen — left the board. - [02:24.549](https://academa.ai/lectures/stokes-theorem-in-practice?t=144.5495): curl\_result is hidden from the screen — left the board. - [02:24.549](https://academa.ai/lectures/stokes-theorem-in-practice?t=144.5495): curl\_work is hidden from the screen — left the board. ##### [02:25.149](https://academa.ai/lectures/stokes-theorem-in-practice?t=145.1495) Narration: Now the right-hand rule fixes the normal. From the side the normal points toward, the positive walk around the boundary must look counterclockwise. Board: frame — an Axes3D (x\_range=(-4.0, 4.0), y\_range=(-2.0, 6.0), z\_range=(-4.0, 4.0)); rim — a ParametricCurve \[yellow\] drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)); centre — a Point \[text\] labelled "(0, 4, 0)" drawn in frame (location=(0.0, 4.0, 0.0)); walk — an Orientation \[red\] drawn in frame (path=((3.0, 4.0, 0.0), (2.9855541800165906, 4.0, 0.2940514209886818)…, closed=True, arrows=4); disc — a Circle \[blue\] drawn in frame (center=(0.0, 4.0, 0.0), radius=3.0, normal\_vector=(0.0, 1.0, 0.0)) Actions: - [02:25.149](https://academa.ai/lectures/stokes-theorem-in-practice?t=145.1495): normal\_heading is shown on the screen, written out. - [02:25.823](https://academa.ai/lectures/stokes-theorem-in-practice?t=145.82299999999998): right\_hand is shown on the screen, written out. ##### [02:35.269](https://academa.ai/lectures/stokes-theorem-in-practice?t=155.2695) Narration: Move to the positive y viewpoint. The given red walk looks clockwise from there, so the compatible normal points away from that viewer, back toward the origin. Board: frame — an Axes3D (x\_range=(-4.0, 4.0), y\_range=(-2.0, 6.0), z\_range=(-4.0, 4.0)); rim — a ParametricCurve \[yellow\] drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)); centre — a Point \[text\] labelled "(0, 4, 0)" drawn in frame (location=(0.0, 4.0, 0.0)); walk — an Orientation \[red\] drawn in frame (path=((3.0, 4.0, 0.0), (2.9855541800165906, 4.0, 0.2940514209886818)…, closed=True, arrows=4); disc — a Circle \[blue\] drawn in frame (center=(0.0, 4.0, 0.0), radius=3.0, normal\_vector=(0.0, 1.0, 0.0)); right\_hand — a Panel that says "From the side the normal points toward, the positive boundary direction looks counterclockwise."; normal\_heading — a Heading that says "The Normal Follows the Walk" Actions: - [02:36.082](https://academa.ai/lectures/stokes-theorem-in-practice?t=156.08199999999997): frame turns in its own slot. ##### [02:45.969](https://academa.ai/lectures/stokes-theorem-in-practice?t=165.96949999999998) Narration: Therefore n hat is minus j hat: zero, minus one, zero. A free vector keeps that direction when its base is moved away from the y-axis line. Board: Unchanged from the preceding beat in this scene. Actions: - [02:47.897](https://academa.ai/lectures/stokes-theorem-in-practice?t=167.89699999999996): normal is shown on the screen, written out. - [02:49.557](https://academa.ai/lectures/stokes-theorem-in-practice?t=169.55699999999996): normal\_result is shown on the screen, written out. ##### [02:57.843](https://academa.ai/lectures/stokes-theorem-in-practice?t=177.843) Narration: The orientation pair moves together. Reverse the boundary walk and the compatible normal flips to plus j. Restore the given clockwise walk and the normal returns to minus j. Board: frame — an Axes3D (x\_range=(-4.0, 4.0), y\_range=(-2.0, 6.0), z\_range=(-4.0, 4.0)); rim — a ParametricCurve \[yellow\] drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)); centre — a Point \[text\] labelled "(0, 4, 0)" drawn in frame (location=(0.0, 4.0, 0.0)); walk — an Orientation \[red\] drawn in frame (path=((3.0, 4.0, 0.0), (2.9855541800165906, 4.0, 0.2940514209886818)…, closed=True, arrows=4); disc — a Circle \[blue\] drawn in frame (center=(0.0, 4.0, 0.0), radius=3.0, normal\_vector=(0.0, 1.0, 0.0)); right\_hand — a Panel that says "From the side the normal points toward, the positive boundary direction looks counterclockwise."; normal\_result — a Math \[text\] that says "$hat(n) = -hat(j) = chevron.l 0, thin -1, thin 0 chevron.r$"; normal\_heading — a Heading that says "The Normal Follows the Walk"; normal — a Vector \[green\] labelled "hat(n)" drawn in frame (start=(\, 4.0, \, \