Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §15.6 Stokes’ Theorem and the Curl of F, #47 · Multivariable Calculus · Green, Stokes & divergence

(a) The second proof of Stokes' Theorem took … as the …

(a) The second proof of Stokes' Theorem took $M^{*} = M (x, y, f (x, y)) + P (x, y, f (x, y)) \partial f / \partial x$ as the $M$ in Green's Theorem. Compute $\partial M^{*} / \partial y$ from the chain rule and product rule (there are five terms).
(b) Similarly $N^{*} = N (x, y, f) + P (x, y, f) \partial f / \partial y$ has the $x$ derivative $N_{x} + N_{z} f_{x} + P_{x} f_{y} + P_{z} f_{x} f_{y} + P f_{y x}$ . Check that $N_{x}^{*} - M_{y}^{*}$ matches the right side of equation (8), as needed in the proof.

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §15.6 Stokes’ Theorem and the Curl of F, #47

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