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Unformatted text preview: Math 23
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 > .
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.
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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