Test the vector field F to determine if it is conservative.
-$$\mathbf { F } = x ^ { 4 } y ^ { 4 } z ^ { 4 } i + x ^ { 4 } y ^ { 4 } z ^ { 4 } j + z ^ { 4 } k$$
A) Conservative
B) Not conservative

Question

Quizplus · Accepted Answer

The vector field is not conservative because its curl is non-zero. 
\begin{align*}
	ext{curl}\;\mathbf{F} &= \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \ 
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ 
x^4y^4z^4 & x^4y^4z^4 & z^4
\end{vmatrix} \
&= (4x^3y^4z^4 - 0)i + (4x^4y^3z^4 - 0)j + (0 - 0)k \
&= 4x^3y^4z^4i + 4x^4y^3z^4j
\end{align*}
Since the curl of $\mathbf{F}$ is non-zero, $\mathbf{F}$ is not conservative.

Test the Vector Field F to Determine If It Is F=x4y4z4i+x4y4z4j+z4k\mathbf { F } = x ^ { 4 } y ^ { 4 } z ^ { 4 } i + x ^ { 4 } y ^ { 4 } z ^ { 4 } j + z ^ { 4 } kF=x4y4z4i+x4y4z4j+z4k

Test the Vector Field F to Determine If It Is $\mathbf { F } = x ^ { 4 } y ^ { 4 } z ^ { 4 } i + x ^ { 4 } y ^ { 4 } z ^ { 4 } j + z ^ { 4 } k$