Explanation Given: LetDbe the image of R = [ 1 , 4 ] × [ 1 , 4 ] under the map G ( u , v ) = ( u 2 v , v 2 u ) . Formula Used: Jac ( G ) = | ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v | ... ( 1 ) If G : D 0 → D and if f is continuous then, ∬ D f ( x , y ) d x d y = ∬ D 0 f ( x ( u , v ) , y ( u , v ) ) | ∂ ( x , y ) ∂ ( u , v ) | d u d v .... ( 2 ) Calculation: Given mapping is: G ( u , v ) = ( u 2 v , v 2 u ) a. To calculate the area of G ( R ) , first we need to find the Jacobian by using the formula(1): x = u 2 v ∂ x ∂ u = 2 u v , ∂ x ∂ v = − u 2 v 2 And y = v 2 u ∂ x ∂ u = − v 2 u 2 , ∂ x ∂ v = 2 u v Now, Jac ( G ) = | ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v | = | 2 u v − u 2 v 2 − v 2 u 2 2 v u | = ( 2 u v × 2 v u ) − ( − u 2 v 2 × − v 2 u 2 ) = 4 − 1 = 3 So, Jac ( G ) = 3 b...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 16.6, Problem 30E

To determine

To calculate:

a. $Jac (G)$ .

b. SketchD.

c. Use the Change of Variables Formula to compute $Area (D)$ and

$\iint_{D} f (x, y) d x d y$ ,where $f (x, y) = x + y$ .

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Chapter 16.6, Problem 29E

Chapter 16.6, Problem 31E

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Chapter 16 Solutions

CALCULUS (CLOTH)

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a. Jac ( G ) . b. SketchD. c. Use the Change of Variables Formula to compute Area ( D ) and ∬ D f ( x , y ) d x d y ,where f ( x , y ) = x + y .

Solution Summary: The author explains how to calculate the area of G ( R) by using the Change of Variables Formula.

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To calculate:

a. $Jac (G)$ .

b. SketchD.

c. Use the Change of Variables Formula to compute $Area (D)$ and

$\iint_{D} f (x, y) d x d y$ ,where $f (x, y) = x + y$ .

Want to see the full answer?

Chapter 16 Solutions

a. Jac ( G ) . b. SketchD. c. Use the Change of Variables Formula to compute Area ( D ) and ∬ D f ( x , y ) d x d y ,where f ( x , y ) = x + y .

Solution Summary: The author explains how to calculate the area of G ( R) by using the Change of Variables Formula.

Videos

To calculate: a. Jac(G). b. SketchD. c. Use the Change of Variables Formula to compute Area(D)and ∬Df(x,y)dxdy,where f(x,y)=x+y.

Want to see the full answer?

Chapter 16 Solutions

To calculate:

a. $Jac (G)$ .

b. SketchD.

c. Use the Change of Variables Formula to compute $Area (D)$ and

$\iint_{D} f (x, y) d x d y$ ,where $f (x, y) = x + y$ .