Question

Calculus 3 Problem

7. Determine if the field F(x, y, z) = ye^z i + xe^z j + xy e^z k is conservative. If it is, find a potential function.​

Answer 1

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{\\abla}\textbf{×}\vec{\text{F}} = 0`

Looking at the components,

`\left(\vec{\\abla}\textbf{×}\vec{\text{F}}\right)_x = \left((\partial F_z)/(\partial y) - (\partial F_y)/(\partial z)\right)_x`

`= xe^z - ye^z \\eq 0`

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