   Chapter 16, Problem 11RE

Chapter
Section
Textbook Problem

Show that F is a conservative vector field. Then find a function f such that F = ∇f.11. F(x, y) = (1 + xy)exy i + (ey + x2exy) j

To determine

To show : F is a conservative vector field and function f such that F=f .

Explanation

Given data:

F(x,y)=(1+xy)exyi+(ey+x2exy)j (1)

Formula used:

Consider the standard equation of an curl F for F=Pi+Qj+Rk

curlF=|ijkxyzPQR| (2)

Find the value of curlF .

Substitute (1+xy)exy for P , (ey+x2exy) for Q and 0 for R in equation (2).

curlF=|ijkxyz(1+xy)exy(ey+x2exy)0|={[y(0)z(ey+x2exy)]i[x(0)z(1+xy)exy]j+[x(ey+x2exy)y(1+xy)exy]k}=ij+[0+2xexy+yx2exyxexyx2yexyxexy]k=ij+[0+0]k

Simplify expression as follows.

curlF=0

Since curlF=0 , the F is a conservative vector and the domain of F is 2

Thus, F is a conservative vector field is shown.

Consider f=fx(x,y)i+fy(x,y)j

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