Use the curl test to determine whether the following vector field is a gradient field. $\vec { F } = y ^ { 3 } z ^ { 2 } \vec { i } + \left( 4 y ^ { 2 } x z ^ { 2 } + 4 y z ^ { 3 } \right) \vec { j } + \left( 2 x y ^ { 3 } z + 3 y ^ { 2 } z ^ { 2 } \right) \vec { k }$
A)Only at the origin
B)No
C)Yes

Question

Quizplus · Accepted Answer

The curl of a gradient field is always zero. Calculating the curl of $\vec{F}$ and finding it to be non-zero except possibly at isolated points (like the origin) indicates that $\vec{F}$ is not a gradient field everywhere, hence the correct answer is that the vector field is not a gradient field.

Use the Curl Test to Determine Whether the Following Vector