Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §15.3 Green’s Theorem, #46 · Multivariable Calculus · Green, Stokes & divergence

4

A region $R$ is “simply connected” when every closed curve inside $R$ can be squeezed to a point without leaving $R$ . Test these regions:

1. $x y$ plane without $(0, 0)$

2. $x y z$ space without $(0, 0, 0)$

3. sphere $x^{2} + y^{2} + z^{2} = 1$

4. a torus (or doughnut)
Q. Why do all paths give the same answer?

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §15.3 Green’s Theorem, #46

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