Chapter 6, Problem 427RE

True or False? Justify your answer with a proof or a counterexample.

427. Vector field $\text{F}(x,y)=x^{2}y\text{i}+y^{2}x\text{j}$ is conservative.

To determine

To calculate:

To check whether the given statement is true or false.

Vector field $F=x^{2}yi+y^{2}xj$ is conservative.

Answer to Problem 427RE

True.

Explanation of Solution

Given information:

Vector field $F=x^{2}yi+y^{2}xj$

Calculation:

Here the vector field is two dimensional of the form

F = Pi + Qj

F is said to be conservative if $\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}:$

Here $\frac{\partial P}{\partial y}=2xy:\frac{\partial Q}{\partial x}=2yx=2xy:$

Since the condition required is satisfied F is conservative.

The given statement is true.