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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

If z =f(u, v), where u = xy, v = y/x, and f has continuous second partial derivatives, show that x 2 2 z x 2 y 2 2 z y 2 = 4 u v 2 z u   v + 2 v z v

To determine

To show: The equation x22zx2y22zy2=4uv2zuv+2vzv if z=f(u,v),u=xyandv=yx also f hascontinuous second order partial derivatives.

Explanation

Proof:

Given:

The functions are, z=f(u,v),u=xyandv=yx .

Calculation:

Compute the value of zx by using Chain Rule,

zx=zuux+zvvx=zux(xy)+zvx(yx)=zuy+zv(yx2)

Compute the value of zy by using Chain Rule,

zy=zuuy+zvvy=zuy(xy)+zvy(yx)=zux+zv(1x)

Take the partial derivative of zx with respect to x ,

2zx2=x(zuy+zv(yx2))=yx(zu)+x(yx2)zv+(yx2)x(zv)=y[2zu2y+2zvu(yx2)]+2yx3zvyx2[2zuvy+2zv2(yx2)]

Simplify further,

2zx2=y[2zu2y+2zvu(yx2)]+2yx3zvyx2[2zuvy+2zv2(yx2)]=2yx3zvyx2[2zuvy+2zv2(yx2)]+y[2zu2y+2zvu(yx2)]=2yx3zvy2x22zuv+y2x42zv2+y22zu2y2x22zvu=2yx3zv2y2x22zuv+y2x42zv2+y22zu2(2z

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Chapter 14 Solutions

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