SME
Calculus III (Multivariable) · Problem type
Which theorem fits?
Use this for “Which theorem turns the flux of F out of a closed surface into a triple integral?”
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Look at what is integrated and over what: a path, a closed plane curve, a loop in space or a closed surface.
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Assumptions
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Each theorem trades an integral over a boundary for one over what the boundary encloses.
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A line integral of a gradient ∇φ needs only φ at the two ends, so any closed loop gives 0.
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Green’s theorem is the flat case of Stokes’: a curve in the xy-plane with n = ⟨0, 0, 1⟩.
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Stokes’ theorem round a circle
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Vector-valued functions, arc length and curvature
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Related lessons
Green's, Stokes' and Divergence theorems
Main lesson
The Divergence theorem on a box
Use this for “Use the Divergence theorem to find the flux of F = ⟨x², y², z²⟩ out of the box 0 ≤ x ≤ 2, 0 ≤ y ≤ 1, 0 ≤ z ≤ 3.”
Stokes’ theorem round a circle
Use this for “Use Stokes’ theorem to find the circulation of F = ⟨−2y, 2x, 0⟩ round the circle x² + y² = 4, z = 0.”