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Assignment 7 solutions

# Assignment 7 solutions - MIT OpenCourseWare...

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MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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18.085 - Mathematical Methods for Engineers I Prof. Gilbert Strang Solutions - Problem Set 7 3.4, Problem 4. Show that u = r cos + r 1 cos solves Laplace’s equation (13), and express u in terms of x and y . Find v = ( u x , u y ) and verify that v n = 0 on the circle x 2 + y 2 = 1. This is the velocity of ﬂow · past a circle in Figure 3.18. Show u = r cos + r 1 cos solves Laplace’s equation: Laplace’s equation in r, ∂ is: 2 u 1 �u 1 2 u + + = 0 . (13) �r 2 r �r r 2 �∂ 2 Now, if u = r cos + r 1 cos , then �u �u = cos r 2 cos = r sin r 1 sin �r �∂ 2 u 3 2 u 1 �r 2 = 2 r cos �∂ 2 = r cos r cos ∂. Plugging this into the left-hand side of (13), we get 2 r 3 cos + 1 (cos r 2 cos ) + 1 ( r cos r 1 cos ) = (2 1 1) r 3 cos + (1 1) 1 cos = 0 , 2 r r r as desired. U in ( x, y ): y Now, since x = r cos ∂, y = r sin r = x 2 + y 2 , ∂ = tan 1 . So x x x (1 + x 2 + y 2 ) u = r cos + r 1 cos = x + = . x 2 + y 2 x 2 + y 2 So �u ( x 2 + y 2 ) 1 x (2 x ) y 2 x 2 = 1 + · = 1 + . �x ( x 2 + y 2 ) 2 ( x 2 + y 2 ) 2 �u = 0 + x ( 1 ( x 2 + y 2 ) 2 2 y ) = 2 xy .
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