Stokes_theorem_9-14

Stokes_theorem_9-14 - Stokes Theorem: Background Similar to...

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1 Stokes’ Theorem : Background j Q i P F r d F C ˆ ˆ , + = r r r ± R is flat (in the xy plane). Its normal direction is k . (Q x -P y ) is the k component of the curl. ± Green's Theorem uses only this component because the normal direction is always k . ± For Stokes' Theorem on a curved S, we need all three components of curl F . ∫∫ R dxdy y P x Q ± The surface integral in Green's Theorem is ± Similar to Green's Theorem: a line integral equals a surface integral. C R [] [ ] ∫∫ = + R y x C dxdy P Q Qdy Pdx ± The line integral is still the work around a curve:
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2 Stokes’ Theorem (Cont.) ∫∫ × = S C dS n F r d F ˆ ) ( r r r ± Stokes’ Theorem : a line integral equals a surface integral in 3D. ± The total flux of the curl of a vector field, F , through a surface equals to the line integral of the vector field, F , around the edge. n and C follow RHR
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Stokes_theorem_9-14 - Stokes Theorem: Background Similar to...

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