Explanation Given: The disk u 2 + v 2 ≤ 1 The ellipse ( x a ) 2 + ( y b ) 2 ≤ 1 Formula used: Jacobian = [ ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v ] Calculation: The disk u 2 + v 2 ≤ 1 can be mapped onto the ellipse ( x a ) 2 + ( y b ) 2 ≤ 1 using the map: G ( u , v ) = ( u a , v b ) . That is, ⇒ u = x a ⇒ x = a u ⇒ v = y b ⇒ y = b v In order to convert the integral, let us first evaluate the Jacobian: Jacobian = [ ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v ] = [ ∂ ...

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 16.6, Problem 34E

To determine

To find:

A mapping G that maps the disk $u^{2} + v^{2} \leq 1$ onto the interior of the ellipse ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} \leq 1$ .

Use change of variables formula to prove that the area of the ellipse is $π a b$ .

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Chapter 16.6, Problem 33E

Chapter 16.6, Problem 35E

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Chapter 16 Solutions

CALCULUS W/SAPLING ACCESS >IC<

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 5PQ Ch. 16.1 - Prob. 6PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E

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A mapping G that maps the disk u 2 + v 2 ≤ 1 onto the interior of the ellipse ( x a ) 2 + ( y b ) 2 ≤ 1 . Use change of variables formula to prove that the area of the ellipse is π a b .

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To find: A mapping G that maps the disk u2+v2≤1onto the interior of the ellipse (xa)2+(yb)2≤1. Use change of variables formula to prove that the area of the ellipse is πab.

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Chapter 16 Solutions

To find:

A mapping G that maps the disk $u^{2} + v^{2} \leq 1$ onto the interior of the ellipse ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} \leq 1$ .

Use change of variables formula to prove that the area of the ellipse is $π a b$ .