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Study Guide for Stewart's Multivar...

8th Edition
Stewart + 1 other
Publisher: Brooks Cole
ISBN: 9781305271845
BuyFind

Study Guide for Stewart's Multivar...

8th Edition
Stewart + 1 other
Publisher: Brooks Cole
ISBN: 9781305271845

Solutions

Chapter
Section
Chapter 15.9, Problem 1PT
Textbook Problem

Find the Jacobian for x = u2v2, y = u2 + v2.

  1. a) 2uv3 − 2u3v
  2. b) 4u2v − 4uv2
  3. c) 2u2v − 2uv2
  4. d) 4uv3 − 4u3v

Expert Solution
To determine

The appropriate option for the Jacobian for x=u2v2,y=u2+v2, from the given options (a)-(d).

Answer to Problem 1PT

The appropriate option is (d)_.

Explanation of Solution

Formula used:

A transformation T is C1 means that all the component functions have continuous first partial derivatives.

For a C1 transformation T from uv-space to xy-space, the Jacobian of T is the determinant,

(x,y)(u,v)=|xuxvyuyv|=xuyvxvyu

Calculation:

Given transformation is x=u2v2,y=u2+v2.

The Jacobian of the transformation x=u2v2,y=u2+v2 is calculated as follows.

(x,y)(u,v)=|u(u2v2)v(u2v2)u(u2+v2)v(u2+v2)|=|2uv22u2v2u2v|=2uv2(2v)2u2v(2u)=4uv34u3v

Thus, (x,y)(u,v)=4uv34u3v.

Therefore, the appropriate option is (d)_.

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Study Guide for Stewart's Multivariable Calculus, 8th

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