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Chapter 14: Q. 23 (page 1096)

In Exercises 17鈥�24, find a potential function for the given vector field.

G(x,y,z)=cosyi+(sinz鈭�xsiny)j+ycoszk

Short Answer

Expert verified

A potential function for the given vector field is g(x,y,z)=xcosy+ysinz.

Step by step solution

01

Step 1. Given Information

In given exercises we have to find a potential function for the given vector field.

G(x,y,z)=cosyi+(sinz鈭�xsiny)j+ycoszk

02

Step 2. Since G(x,y,z)=cosyi+(sinz−xsiny)j+ycoszk

G(x,y,z)=鈭�cosydx+B+CG(x,y,z)=cosy鈭�dx+B+CG(x,y,z)=cosy路x+B+伪+CG(x,y,z)=xcosy+伪+B+C

where 伪 is an arbitrary constant and B is the integral with respect to y of the terms in role="math" localid="1650474027433" G2(x,y,z) in which the factor x does not appear.

03

Step 3. In this case, that is all of G2(x,y,z), so

B=鈭�(sinz鈭�xsiny)dyB=鈭�sinzdy鈭�x鈭�sinydyB=sinz鈭�dy鈭�x(-cosy)+尾B=sinz路y+xcosy+尾B=ysinz+xcosy+尾

where尾 is an arbitrary constant.

04

Step 4. Now finding C=∫ycoszdz

C=y鈭�coszdzC=ysinz+纬

where纬 is an arbitrary constant.

Setting the constants equal to zero since they do not affect the gradient of g(x,y,z)

We have,

g(x,y,z)=xcosy+ysinz

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Most popular questions from this chapter

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