15_6_14 - cos v d v = Z x = 2 x = 1 d w k 2 w 2 = 1 k tan-1...

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14. The flux of F = m r | r | 3 out of the cube 1 x , y , z 2 is equal to three times the total flux out of the pair of opposite faces z = 1 and z = 2, which have outward normals - k and k respectively. This latter flux is 2 m I 2 - m I 1 , where I k = Z 2 1 dx Z 2 1 dy ( x 2 + y 2 + k 2 ) 3 / 2 Let y = x 2 + k 2 tan u dy = x 2 + k 2 sec 2 u du = Z 2 1 dx x 2 + k 2 Z y = 2 y = 1 cos u du = Z 2 1 dx x 2 + k 2 ( sin u ) ± ± ± ± y = 2 y = 1 = Z 2 1 dx x 2 + k 2 y p x 2 + y 2 + k 2 ± ± ± ± 2 1 ! = J k 2 - J k 1 , where J kn = n Z 2 1 dx ( x 2 + k 2 ) x 2 + n 2 + k 2 Let x = n 2 + k 2 tan v dx = n 2 + k 2 sec 2 v d v = n Z x = 2 x = 1 sec 2 v d v ² ( n 2 + k 2 ) tan 2 v + k 2 ³ sec v = n Z x = 2 x = 1 cos v d v ( n 2 + k 2 ) sin 2 v + k 2 cos 2 v = n Z x = 2 x = 1 cos v d v k 2 + n 2 sin 2 v Let w = n sin v d w = n
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Unformatted text preview: cos v d v = Z x = 2 x = 1 d w k 2 + w 2 = 1 k tan-1 w k ± ± ± ± x = 2 x = 1 = 1 k tan-1 n sin v k ± ± ± ± x = 2 x = 1 = 1 k tan-1 nx k √ x 2 + n 2 + k 2 ± ± ± ± 2 1 = 1 k ´ tan-1 2 n k √ 4 + n 2 + k 2-tan-1 n k √ 1 + n 2 + k 2 µ ....
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This note was uploaded on 03/08/2012 for the course MATH 120 taught by Professor Onurfen during the Spring '12 term at Middle East Technical University.

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