M2521806Extra

M2521806Extra - SECTIONS 18.6 AND 18.7: ADDITIONAL NOTES...

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SECTIONS 18.6 AND 18.7: ADDITIONAL NOTES AND REVISIONS SECTION 18.6: DIVERGENCE (GAUSS’S) THEOREM Instead of doing my Example in my notes (#8), I will do the following Example: Example Find the flux of F x , y , z ( ) = 2 x , x 2 z 3 ,5 z through any sphere S of radius 4. Solution Flux = F n dS S ∫∫ = div F ( ) dV Q ∫∫∫ , where div F = x 2 x ( ) + y x 2 z 3 ( ) + z 5 z ( ) = 2 + 0 + 5 = 7 and Q is the region bounded by S . Flux = 7 dV Q ∫∫∫ = 7 dV Q ∫∫∫ = 7 Volume of Q ( ) = 7 4 3 π 4 ( ) 3 since the volume of a sphere of radius r is 4 3 r 3 = 7 256 3 = 1792 3
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SECTION 18.7: STOKES’S THEOREM I will skip my Example (#6). I may show in class why Green’s Theorem is merely a special case of Stokes’s Theorem. We will make the usual assumptions for Stokes’s Theorem. According to the theorem, Work W = F T ds C = curl F ( ) n dS S ∫∫ If S is a region of the xy -plane, we can call it
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M2521806Extra - SECTIONS 18.6 AND 18.7: ADDITIONAL NOTES...

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