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Chapter 14: Q. 20 (page 1095)

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

G(x,y)=i鈭�j

Short Answer

Expert verified

A potential function for the given vector field is g(x,y)=x-y.

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)=i鈭�j

02

Step 2. Since G(x,y)=i−j

g(x,y)=鈭�dx+Bg(x,y)=x+伪+B

where 伪 is an arbitrary constant and B is the integral with respect to y of the terms in role="math" localid="1650472329576" G2(x,y) in which the factor x does not appear.

03

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

B=-鈭�dyB=-y+尾

where尾 is an arbitrary constant.

04

Step 4. Setting the constants equal to zero since they do not affect the gradient of g(x,y)

We have,

g(x,y)=x-y

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