   Chapter 15.9, Problem 5E

Chapter
Section
Textbook Problem

Find the Jacobian of the transformation.5. x = uv, y = vw, z = wu

To determine

To find: Jacobian transformation of x=uv, y=vw and z=wu.

Explanation

Definition used: Jacobian transformation

The Jacobian of the transformation T given by x=g(u,v), y=h(u,v) and z=k(u,v) is,

(x,y,z)(u,v,w)=|xuxvxwyuyvywzuzvzw|=xu(yvzwywzv)xv(yuzwywzu)+xw(yuzvyvzu)

Calculation:

Given that, x=uv

Take partial derivative of x with respect to u to obtain xu.

xu=(1)v=v

Take partial derivative of x with respect to v to obtain xv.

xv=u(1)=u

Take partial derivative of x with respect to w to obtain xw.

xw=0

Given that, y=vw

Take partial derivative of y with respect to u to obtain yu.

yu=0

Take partial derivative of y with respect to v to obtain yv

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