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lecture20-09

# lecture20-09 - 18.02 Multivariable Calculus(Spring 2009...

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18.02 Multivariable Calculus (Spring 2009): Lecture 20 General Change of Variables March 20 Reading Material: From Simmons : 20.3 and 20.4. From Course Notes : CV. Last time: Polar coordinates. Today: General Change of Variables. Consider the ellipse S defined by S = { ( x, y ) / 2 x 2 + 9 y 2 8 } . What is the area A of S ? Clearly one way to compute A is to compute A = S 1 dA and if we use the standard rectangular coordinates we get A = 2 - 2 8 - 2 x 2 / 3 - 8 - 2 x 2 / 3 1 dy dx and this is messy ! The idea here is to change into a more suitable coordinate system (u,v) in which the ellipse gets transformed into a circle of radius 1. Rewrite S = { ( x, y ) / 1 4 x 2 + 9 8 y 2 1 } . 1

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We can write the change of variable T ( x, y ) = ( u, v ) as follows: 1 2 x = u 9 8 y = v and as a consequence T - 1 ( u, v ) = ( x, y ) as x = 2 u (1.1) y = 8 9 v (1.2) It is now clear that T ( S ) = D , the disk centered at the origin of redius 1 in the u and v coordinate system: As for the polar coordinates we need to know how dA can be expressed in terms of dudv , that is
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lecture20-09 - 18.02 Multivariable Calculus(Spring 2009...

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