Both Green's theorem and Stokes' theorem are higher-dimensional versions of the fundamental theorem of calculus, see how!

Here we cover four different ways to extend the fundamental theorem of calculus to multiple dimensions. Green's theorem and the 2D divergence theorem do this for two dimensions, then …

Given a surface, boundary curve, and 3D vector field, convert between surface integrals and line integrals using Stokes’ theorem. (Or is Stokes’ theorem not applicable in this case?)

This seems to be a pretty straightforward argument (Proof of Stoke's Theorem): calculate both sides and show that they're equivalent; however, I was wondering if there was a good way to …

Explore Green's and Stokes' theorems in multivariable calculus with Khan Academy's comprehensive lessons and interactive exercises.

Oops. Something went wrong. Please try again. Uh oh, it looks like we ran into an error. You need to refresh. If this problem persists, tell us.

哎呀.出了错.请再试一试. 哦不, 看来我们遇到了一个错误. 请 刷新 该页面. 如果此问题仍然存在, 请 告诉我们.

Parameterizing the surface

Written in this form, it's easier to see that the 2D divergence theorem is secretly just saying the same thing as Green's theorem.

在这个视频中，我要来证明-- 其实，在这个视频和后面的几个视频中-- 我要证明斯托克斯定理的一些特例， 或者说，针对一个特殊情形的斯托克斯定理 我这样做是因为 这样对它的证明会简单 …