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Unformatted text preview: MA1104 Multivariable Calculus Lecture 14 Dr KU Cheng Yeaw Thursday March 12, 2009 Recap Last Lecture ... • we evaluate triple integrals via Cylindrical and Spherical Coordinates • prior to that, we have learned to evaluate double integrals via Polar Coordinates • we have seen how change of coordinates simplify computation of multiple integrals Overview In this lecture, • we observe that evaluation of multiple integrals via change of coordinates can be seen as a generalization of integration by substitution in Single Variable Calculus • we put this method of integration in a more general framework. To achieve this, we introduce the notion of transformation and its Jacobian Change of Variables In Single Variable Calculus, we often use a change of variable (substitution rule) to simplify an integral: Z b a f ( x ) dx = Z d c f ( g ( u )) g ( u ) du where x = g ( u ) and a = g ( c ) , b = g ( d ) . Sometimes, we write this as follows: Z b a f ( x ) dx = Z d c f ( x ( u ) ) dx du du . A change of variables can also be useful in double integrals. We have already seen one example of this: conversion to polar coordinates. The new variables r and θ are related to the old variables x and y by: x = r cos θ, y = r sin θ. The change of variables formula can be written as: ZZ R f ( x,y ) dA = ZZ S f ( r cos θ , r sin θ ) r dr dθ where S is the region in the rθplane that corresponds to the region R in the xyplane. This change simplifies the computation when S has simpler description on the rθplane than that of R on the xyplane. To this end, we shall deduce that the above conversion to polar coordinates is just a special case of a more general phenomenon. What is implicit in this conversion to polar coordinates is a transformation from the rθplane to the xyplane via the equations: x = r cos θ, y = r sin θ. Generally, we consider a change of variables that is given by a transformation T from the uvplane to the xyplane: T : uv plane → xy plane T : ( u,v ) 7→ ( x,y ) via the equations x = g ( u,v ) , y = h ( u,v ) where g and h have continuous firstorder partial derivatives . We say that T is a C 1 transformation . A transformation T is really just a function whose domain and range are both subsets of R 2 . If T ( u,v ) = ( x,y ) , then the point ( x,y ) is called the image of the point ( u,v ) ....
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This note was uploaded on 03/19/2012 for the course MATH 1104 taught by Professor Kuchengyeaw during the Spring '08 term at National University of Singapore.
 Spring '08
 Kuchengyeaw
 Integrals, Multivariable Calculus, Polar Coordinates

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