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surface interal

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

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�� �� 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 define two tangent vectors, which span the tangent plane to M at P = �g ( s 0 , t 0 ): ∂�g T s ( s 0 , t 0 ) = ( s 0 , t 0 ) ∂s T t ( s 0 , t 0 ) = ∂�g ( s 0 , t 0 ) . ∂t We get an element of area on M , d S = T s × T t d s d t. Using this we can define 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, y, z ) | 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 = ∂�g = ( ( a + b cos t ) sin s, ( a + b cos t ) cos s, 0) , ∂s T t = ∂�g = ( b sin t cos s, b sin t sin s, b cos t ) .

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