11.09 - Section 12.8: Change of variables in multiple...

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Unformatted text preview: Section 12.8: Change of variables in multiple integrals Why the Jacobian? Lets look at it in 3-space. Consider the rectangular parallelepiped [ u , u + u ] [ v , v + v ] [ w , w + w ], with corner ( u , v , w ) and sides given by the vectors u ,0,0 , 0, v ,0 , and 0,0, w . When we map forward by the transformation T ( u , v , w ) = ( f ( u , v , w ), g ( u , v , w ), h ( u , v , w )), we have vec( T ( u , v , w ), T ( u + u , v , w )) = f ( u + u , v , w ) f ( u , v , w ), g ( u + u , v , w ) g ( u , v , w ), h ( u + u , v , w ) h ( u , v , w ) which is approximately equal to (1) f u ( u , v , w ) u , g u ( u , v , w ) u , h u ( u , v , w ) u Likewise vec( T ( u , v , w ) T ( u , v + v , w )) (2) f v ( u , v , w ) v , g v ( u , v , w ) v , h v ( u , v , w ) v and vec( T ( u , v , w ) T ( u , v , w + w )) (3) f w ( u , v , w...
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This note was uploaded on 02/13/2012 for the course MATH 241 taught by Professor Staff during the Fall '11 term at UMass Lowell.

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11.09 - Section 12.8: Change of variables in multiple...

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