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Home Vector Calculus Theorems of Stokes and Gauss Stokes' Theorem  
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Stokes' Theorem
Stokes' Theorem relates a surface integral over S to a line integral over the boundary of S. It corresponds to Green's Theorem in the form Let S be an oriented surface over a region D. The boundary of S, ∂S, is the simple closed space curve whose direction depends on the orientation of S as shown in Figure 13.6.1. The notation denotes the line integral around ∂S in the direction determined by the orientation of S. Figure 13.6.1 The Boundary of S STOKES' THEOREM Given a vector field F(x, y, z) on an oriented surface S, (See Figure 13.6.2.) Figure 13.6.2 To put this equation in scalar form, let F = Pi + Qj + Rk, curl F = Hi + Lj + Mk. Then F · T ds = P dx + Q dy + R dz, and if S is oriented with the top side positive, Thus Stokes' Theorem has the scalar form Stokes' Theorem has two corollaries which are analogous to the Path Independence Theorem.


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