Stokes' Theorem and the Divergence Theorem
Both theorems say that integrating a derivative over a region equals integrating the original object over its boundary. With Green's theorem and the fundamental theorem, they are one statement in four settings.
What a learner can do afterwards
- State each theorem and identify the region and its boundary
- Convert a surface flux integral into a triple integral of divergence
- Explain how all four theorems in the course share one form
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Each theorem says that integrating a derivative over a region equals integrating the original object over the region's boundary. The fundamental theorem of calculus is the line case: the integral of f prime over [a, b] equals f(b) minus f(a), and the boundary is the two endpoints. Green's theorem is the flat case: a double integral of a curl or divergence over a plane region equals a line integral around its edge. The divergence theorem and Stokes' theorem are the cases in space.
The divergence theorem pairs a solid with its closed outer surface. The triple integral of the divergence over the solid equals the outward flux through the surface. With the outward normal, positive total flux means net outflow, negative means net inflow, and zero means the flow balances.
Take the field (3x, y, z) leaving a cube of side 3. Its divergence is 3 plus 1 plus 1, the constant 5. The cube volume is 3 cubed, which is 27. With constant divergence the flux is divergence times volume: 5 times 27 is 135. When a surface is closed, try the inside integral before parametrising six faces.
Stokes' theorem pairs a surface with its rim curve: the flux of curl F through the cap equals the circulation of F around the rim. Circulation adds up F dotted with the direction of travel around the rim. Only the rim matters, so you can swap a hard cap for an easier one with the same rim. Orientation follows the right hand rule: fingers curl along the rim direction and the thumb points along the cap normal. Flipping the normal flips the sign of the answer.
Take F = (-y, x, 0) around the circle of radius 2 in the plane z = 0, walked anticlockwise seen from above. Its curl is (0, 0, 2). Use the flat disk as the cap, with normal (0, 0, 1). The curl dotted with the normal is 2 everywhere, and the disk has area 4 pi, so the circulation is 8 pi.
Match each problem to its pairing before computing. A closed skin asks for the divergence trade, and a rim curve asks for the Stokes trade. Watch orientation closely: one flipped normal changes the sign of the whole answer.
Sum the derivative inside and you get the boundary total for free, so pick the easier side to compute.
The lesson and quiz items were written for this dataset. The lesson was later revised.
Extra items x1 to x6 were written for this dataset. The rewritten readings were written for this dataset. Other reading entries summarise the Khan Academy articles Stokes' theorem examples and Flux in three dimensions. Videos are from Khan Academy.
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