Question

Calculus 3 Problem

Answer 1

Step-by-step explanation:

Given:

$\vec{\textbf{F}}(x, y, z) = ye^z\hat{\textbf{i}} + xe^z\hat{\textbf{j}} + xye^z\hat{\textbf{k}}$

A vector field is conservative if

$\vec{\nabla}\textbf{×}\vec{\text{F}} = 0$

Looking at the components,

$\left(\vec{\nabla}\textbf{×}\vec{\text{F}}\right)_x = \left(\dfrac{\partial F_z}{\partial y} - \dfrac{\partial F_y}{\partial z}\right)_x$

$= xe^z - ye^z \neq 0$

Since the x- component is not equal to zero, then the field is not conservative so there is no scalar potential $\phi$.