   Chapter 16.3, Problem 3E

Chapter
Section
Textbook Problem

Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇ f.3. F(x, y) = (xy + y2) i + (x2 + 2xy) j

To determine

Whether F is a conservative vector field and find corresponding function f such that F=f .

Explanation

Given data:

Vector field is F(x,y)=(xy+y2)i+(x2+2xy)j .

Formula used:

Consider a vector field as F(x,y)=P(x,y)i+Q(x,y)j . The condition for vector field F being a conservative field is,

Py=Qx (1)

Here,

Py is continuous first-order partial derivative of P, and

Qx is continuous first-order partial derivative of Q,

Compare the vector field F(x,y)=(xy+y2)i+(x2+2xy)j with F(x,y)=P(x,y)i+Q(x,y)j .

P=xy+y2 (2)

Q=x2+2xy (3)

Apply partial differentiation with respect to y on both sides of equation (2).

Py=y(xy+y2)=y(xy)+y(y2)=xy(y)+y(y2)=x(1)+2y

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