Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §15.6 Stokes’ Theorem and the Curl of F, #15 · Multivariable Calculus · Green, Stokes & divergence

(b) Second proof: The difference between those integrals is

(important) Suppose two surfaces $S$ and $T$ have the same boundary $C$ , and the direction around $C$ is the same.

(a) Prove $\iint_{S} curl F \cdot n d S = \iint_{T} curl F \cdot n d S$ .

(b) Second proof: The difference between those integrals is $∭_{V} div (curl F) d V$ . By what Theorem? What region is $V$ ? Why is this integral zero?

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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, #15

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