Examples7.6

Examples7.6 - 1 Example 1 Evaluate the surface integral of...

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Unformatted text preview: 1 Example 1 Evaluate the surface integral of the vector field F = 3 x 2 i- 2 yx j + 8 k over the surface S that is the graph of z = 2 x- y over the rectangle [0 , 2] [0 , 2] . Solution. Use the formula for a surface integral over a graph z = g ( x,y ) : ZZ S F d S = ZZ D F - g x i- g y j + k dxdy. In our case we get Z 2 Z 2 (3 x 2 ,- 2 yx, 8) (- 2 , 1 , 1) dxdy = Z 2 Z 2 (- 6 x 2- 2 yx + 8) dxdy = Z 2- 2 x 3- yx 2 + 8 x 2 x =0 dy = Z 2- 4 y dy =- 2 y 2 | 2 =- 8 . 2 Example 2 Let S be the triangle with vertices (1 , , 0) , (0 , 2 , 0) , and (0 , 1 , 1) , and let F = xyz ( i + j ) . Calculate the surface integral ZZ S F d S , if the triangle is oriented by the downward normal. Solution. Since S lies in a plane (see the right hand part of the Figure), it is part of the graph of a linear function z = ax + by + c. v = 2 - 2 u v = 1 - u (0, 2) (0, 1) (1, 0) z y x (0,1,1) (0,2,0) D S v u Substituting the vertices of the triangle for ( x,y,z ) , we get the equation...
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This note was uploaded on 07/19/2010 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.

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Examples7.6 - 1 Example 1 Evaluate the surface integral of...

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