Furthermore, the theorem has applications in fluid mechanics and electromagnetism. We use Stokes’ theorem to derive Faraday’s law, an important result involving electric fields. Stokes’
Let C be the closed curve illustrated below. For F ( x, y, z) = ( y, z, x), compute ∫ C F ⋅ d s using Stokes' Theorem. Solution : Since we are given a line integral and told to use Stokes' theorem, we need to compute a surface integral ∬ S curl F ⋅ d S
Stokes' theorem connects to the standard gradient, curl, and divergence theorems by the following relations. If is a function on , (2) where (the dual space) is the duality
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is easy to calculate, Stokes’ theorem allows us to calculate the easier surface integral. In Example 16.7.2, we could have calculated ∬Scurl ⇀ F ⋅ d ⇀ S by calculating ∬S curl
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