s11wk12su - Math 23 B Dodson Week 6 Homework 16.5 Curl and...

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Unformatted text preview: Math 23 B. Dodson Week 6 Homework: 16.5 Curl and Divergence 16.6 Surfaces and Surface Area 16.7 Surface integrals 16.8 Stokes’ Theorem [first half] 16.8 Stokes’ Theorem [second half] 16.9 Divergence Theorem [Syllabus covers just the “first half”] Problem 16.5.5: Find Div(F ) and Curl(F ) when F =< ex sin y, ex cos y, z > . Solution: For F =< P, Q, R >, Div(F ) = Px + Qy + Rz , so Div(F ) = ex sin y − ex sin y + 1 = 1. Using the cross product formula we have Curl(F ) =< Ry − Qz , Rx − Pz , Qx − Py >=< 0, 0, 0 > . Notice that if F =< fx , fy , fz > is the gradient of f, we always get Curl = 0 (why?) and on a region like R3 , a solid ball, or a rectangular solid we have the Theorem that Curl(F ) = 0 exactly when there is f so that F = Grad(f ). 2 Problem 16.9.7: Use the Divergence Theorem to calculate the surface integral S F · dS, where F (x, y, z ) = (ex sin y )i + (ex sin y )j + yz 2 k, and S is the surface of the box B bounded by planes x = 0, x = 1, y = 0, y = 1, z = 0, z = 2. Solution: The divergence div(F ) = = ex sin y + ex (− sin y ) + 2yz = 2yz. So S F · dS = 1 1 B 2yz dV 2 = 2yz dzdydx 0 = 2. 0 0 ...
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This note was uploaded on 11/24/2011 for the course MATH 23 taught by Professor Yukich during the Spring '06 term at Lehigh University .

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s11wk12su - Math 23 B Dodson Week 6 Homework 16.5 Curl and...

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