Slide_14 - MA1104 Multivariable Calculus Lecture 14 Dr KU...

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MA1104 Multivariable Calculus Lecture 14 Dr KU Cheng Yeaw Thursday March 12, 2009
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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
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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
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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 0 ( 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 .
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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.
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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 -plane that corresponds to the region R in the xy -plane. This change simplifies the computation when S has simpler description on the -plane than that of R on the xy -plane.
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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 -plane to the xy -plane via the equations: x = r cos θ, y = r sin θ.
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Generally, we consider a change of variables that is given by a transformation T from the uv -plane to the xy -plane: 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 first-order partial derivatives . We say that T is a C 1 transformation .
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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 ) . If no two points have the same image, T is called one-to-one .
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The figure shows the effect of a transformation T on a region S in the uv -plane.
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