surface interal - 30 Surface integrals Suppose we are given...

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± ² ² ² ² ² ² ² ² ² ² ² ² 30. Surface integrals Suppose we are given a smooth 2-manifold M R 3 . Let g : U −→ M W, be a diffeomorphism, where U R 2 , with coordinates s and t . We can deFne two tangent vectors, which span the tangent plane to M at P = g ( s 0 ,t 0 ): ∂g T s ( s 0 0 ) = ( s 0 0 ) ∂s T t ( s 0 0 ) = ( s 0 0 ) . ∂t get an element of area on M , d S = T s × T t d s d t. Using this we can deFne the area of M W to be area( M W ) = d S = T s × T t d s d t. M W U Example 30.1. We can parametrise the torus, M = { ( x,y,z ) | ( a x 2 + y 2 ) 2 + z 2 = b 2 } , as follows. Let U = (0 , 2 π ) × (0 , 2 π ) , and W = R 3 \ { ( ) | x 0 and y = 0 , or x 2 + y 2 a 2 and z = 0 } . Let g : U M W, be the function g ( s,t ) = (( a + b cos t ) cos s, ( a + b cos t ) sin s,b sin t ) . Let’s calculate the tangent vectors, T s = = ( ( a + b cos t ) sin s, ( a + b cos t ) cos s, 0) , T t = = ( b sin t cos s, b sin t sin cos t ) . So T s × T t = ˆ ı j ˆ k ˆ ( a + b cos t ) sin s ( a + b cos t ) cos s 0 b sin t cos s b sin t sin s b cos t = ( a + b cos t ) b cos s cos t ˆ ı + ( a + b cos t ) b sin s cos t ˆ k. j + ( a + b cos t ) b sin t
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surface interal - 30 Surface integrals Suppose we are given...

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