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Multivariable calculus

Course: Multivariable calculus>Unit 5

Lesson 9: Proof of Stokes' theorem

Stokes' theorem proof part 3

Writing our surface integral as a double integral over the domain of our parameters. Created by Sal Khan.

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• In previous videos the dS was defined as the magnitude of || rx cross ry|| . Why doesn't he take the magnitude of the cross product in this video?
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• Note the vector notation over the dS. He's using the vector dS, which is a vector that points normal to the surface, and whose magnitude I believe is the infinitessimal area dS (or the magnitude of the cross-product of r sub-x and r sub-y, times dxdy).

In this video, he takes the surface integral of the dot product of curl(F) and vector dS instead of the dot product of curl(F) and the unit normal vector, times scalar dS. He explains in-depth how these are equivalent in this video: https://www.khanacademy.org/math/multivariable-calculus/surface-integrals/3d_flux/v/vector-representation-of-a-surface-integral