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Chapter 14: Q. 18 (page 1095)

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

F(x,y)=(eysec2x,eytanx)

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given Information

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

F(x,y)=(eysec2x,eytanx)

02

Step 2. Since F(x, y)=(eysec2x, eytanx)

f(x,y)=鈭�(eysec2x)dx+Bf(x,y)=ey鈭�sec2xdx+Bf(x,y)=eytanx+伪+B

where 伪 is an arbitrary constant and B is the integral with respect to y of the terms in F2(x,y) in which the factor x does not appear.

03

Step 3. In this case, that is all of F2(x,y), so

B=鈭�(eytanx)dyB=tanx鈭�eydyB=tanx路ey+尾B=eytanx+尾

where尾 is an arbitrary constant.

04

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

We have,

f(x,y)=eytanx

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