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Chapter 14: Q. 19 (page 1095)

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

G(x,y)=(5x4+y,x鈭�12y3)

Short Answer

Expert verified

A potential function for the given vector field is g(x,y)=x5+xy鈭�3y4.

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)=(5x4+y,x鈭�12y3)

02

Step 2. Since G(x,y)=(5x4+y,x−12y3)

G(x,y)=鈭�(5x4+y)dx+Bg(x,y)=5鈭�x4dx+y鈭�dx+Bg(x,y)=5x4+14+1+yx+伪+Bg(x,y)=5x55+xy+伪+Bg(x,y)=x5+xy+伪+B

where 伪 is an arbitrary constant and B is the integral with respect to y of the terms in localid="1650472400181" 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�(x鈭�12y3)dyB=x鈭�dy鈭�12鈭�y3dyB=xy鈭�12y3+13+1+尾B=xy鈭�12y44+尾B=xy鈭�3y4+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)=x5+xy鈭�3y4

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