Unit 10: Stokes' Theorem, the Divergence Theorem, and the Big Picture
Boundary versus interior, argued one dimension higher, until every theorem in the course turns out to be the same sentence.
8 topics · 28 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
10.1 Stokes' Theorem
- How it worksHow does Stokes' theorem generalize Green's theorem by relating a surface integral of curl to a line integral around the surface's boundary curve?
- Worked exampleFind the circulation of F(x, y, z) = <-y, x, 0> around the boundary of the hemisphere x2 + y2 + z2 = 1, z ≥ 0, using Stokes' theorem.
- Common mistakeWhy doesn't the specific shape of the surface spanning a boundary curve change the value of a Stokes' theorem circulation?
Your turn1 of 2
Find the circulation of F(x, y, z) = <-4·y, 4·x, 0> around the boundary of the hemisphere x2 + y2 + z2 = 7, z ≥ 0, using Stokes' theorem.10.2 The Divergence Theorem
- How it worksHow does the divergence theorem convert a flux integral over a closed surface into a triple integral of divergence over the solid it encloses?
- Worked exampleFind the flux of F(x, y, z) = <x, y, z> through the sphere x2 + y2 + z2 = 9 using the divergence theorem.
- Going furtherWhat happens to the divergence theorem's triple integral when the vector field has zero divergence everywhere inside the solid?
Your turn1 of 2
Find the flux of F(x, y, z) = <4·x, 4·y, 4·z> through the sphere of radius 4, using the divergence theorem.10.3 Applications of the Divergence Theorem
- How it worksHow does the divergence theorem let you replace a difficult flux integral over a complicated surface with a simpler triple integral of divergence?
- Common mistakeWhy doesn't the divergence theorem require the vector field's divergence to stay constant throughout the solid region?
- Worked exampleFind the flux of F(x, y, z) = <x3, y3, z3> through the surface of the cube 0 ≤ x, y, z ≤ 1 using the divergence theorem.
Your turn1 of 2
Find the flux of F(x, y, z) = <4·x3, 4·y3, 4·z3> through the surface of the cube 0 ≤ x, y, z ≤ 2, using the divergence theorem.10.4 Choosing the Right Theorem
- How it worksHow do you decide among Green's theorem, Stokes' theorem, and the divergence theorem based on whether you are handed a closed curve or a closed surface?
- Worked exampleShow which theorem applies to converting a line integral around the boundary triangle of a tilted plane into a surface integral, and set up that conversion.
- Ties togetherHow do Green's theorem, Stokes' theorem, and the divergence theorem all express the exact same idea at different dimensions?
Your turn1 of 2
Find the flux of F(x, y, z) = <4·x, 4·y, 6·z> through the sphere of radius 2, deciding first that the divergence theorem applies because the surface is closed.10.5 The Fundamental Theorem of Calculus in Higher Dimensions
- How it worksHow does evaluating an antiderivative at two endpoints generalize step by step into Green's, Stokes', and the divergence theorem?
- Going furtherWhat happens to this unifying boundary-versus-interior pattern when it gets pushed all the way back down to the single-variable fundamental theorem of calculus itself?
- Ties togetherHow does this single unifying pattern connect the divergence theorem, Stokes' theorem, and Green's theorem into one family of results?
Your turn1 of 2
As an instance of the higher-dimensional pattern, find the flux of F(x, y, z) = <4·x, 4·y, 4·z> through the surface of the cube 0 ≤ x, y, z ≤ 3.10.6 Physical Interpretations of Curl and Divergence
- How it worksHow does divergence measure a fluid's local rate of expansion while curl measures its local rate of rotation, as two independent physical quantities?
- Worked exampleFind the curl and divergence of the vector field F(x, y, z) = <-y, x, z>, and describe the corresponding fluid flow's rotation and expansion.
- Common mistakeWhy doesn't a fluid flow with zero divergence everywhere mean the fluid has stopped moving?
- Ties togetherHow do curl and divergence, brought together by Stokes' theorem and the divergence theorem, complete the picture of a vector field this whole course has been building?
Your turn1 of 2
For F(x, y, z) = <-a·y, a·x, b·z>, find the z-component of the curl, as a function of a.10.7 Stokes' Theorem and Orientation-Compatible Boundaries
- How it worksHow does the right-hand rule link a surface's chosen orientation to the direction its boundary curve must be traversed for Stokes' theorem to hold?
- Worked exampleFind the correctly oriented direction of travel around the boundary circle of the hemisphere x2 + y2 + z2 = 1, z ≥ 0, oriented with an outward-pointing normal.
- Common mistakeWhy doesn't Stokes' theorem give the same answer if the boundary curve gets traversed in the wrong direction for the surface's chosen orientation?
- Going furtherWhat happens to the orientation rule when the same boundary curve happens to bound two different surfaces on opposite sides?
Your turn1 of 2
The correctly oriented boundary circle of the hemisphere x2+y2+z2=r2, z≥0 (outward normal) is traversed counterclockwise when viewed from above, with unit tangent direction (-sin(theta), cos(theta), 0) at angle theta. Find its y-component, as a function of theta.10.8 Computing a Flux Two Ways: Directly and via the Divergence Theorem
- How it worksHow do you set up a flux integral directly over a closed surface using its outward normal vector, piece by piece?
- Worked exampleFind the flux of F(x, y, z) = <x, y, z> through the surface of the cube 0 ≤ x, y, z ≤ 1 by direct computation over all six faces.
- Worked exampleFind the same flux from the previous step using the divergence theorem, and confirm the two answers agree.
- Common mistakeWhy doesn't the divergence theorem shortcut require checking every face of a closed surface separately the way the direct computation does?
- Going furtherWhat happens to the direct, face-by-face computation when one face of the closed surface is curved instead of flat?
Your turn1 of 2
Find the flux of F(x, y, z) = <4·x, 4·y, 4·z> through the surface of the cube 0 ≤ x, y, z ≤ 3, by direct computation over all six faces.