Test the vector field F to determine if it is conservative.
-$$\mathbf { F } = \left( \frac { 9 x ^ { 8 } \left( y - x ^ { 2 } \right) ^ { 9 } } { z } \right) i + \left( \frac { 5 x ^ { 9 } \left( y - x ^ { 2 } \right) ^ { 8 } } { z } \right) j - \left( \frac { x ^ { 9 } \left( y - x ^ { 2 } \right) ^ { 5 } } { z ^ { 2 } } \right) \mathbf { k }$$
A) Conservative
B) Not conservative

Question

Quizplus · Accepted Answer

We can test if the vector field is conservative by computing its curl. If the curl is zero everywhere in space, the vector field is conservative. 
\begin{align*}

abla 	imes F &= \left( \frac{d}{dx}, \frac{d}{dy}, \frac{d}{dz} ight) 	imes \left( \frac{9x^8(y-x^2)^9}{z} , \frac{5x^9(y-x^2)^8}{z}, -\frac{x^9(y-x^2)^5}{z^2} ight)\
&= \begin{pmatrix}
\frac{d}{dy} \left( -\frac{x^9(y-x^2)^5}{z^2} ight) - \frac{d}{dz} \left( \frac{5x^9(y-x^2)^8}{z} ight) \
-\frac{d}{dx} \left( -\frac{x^9(y-x^2)^5}{z^2} ight) + \frac{d}{dz} \left( \frac{9x^8(y-x^2)^9}{z} ight)\
\frac{d}{dx} \left( \frac{5x^9(y-x^2)^8}{z} ight) - \frac{d}{dy} \left( \frac{9x^8(y-x^2)^9}{z} ight)
\end{pmatrix}\
&= \begin{pmatrix}
\frac{9x^8(y-x^2)^{10}}{z^2} - \frac{45x^8(y-x^2)^8}{z^2} & \frac{-x^9(45y^2 - 70x^2y + 21x^4)}{z^2} & \frac{-x^8(y-x^2)^9}{z^3} \
\frac{x^9(45y^2 - 70x^2y + 21x^4)}{z^2} & \frac{-9x^8(y-x^2)^{10}}{z^2} + \frac{40x^9(y-x^2)^8}{z^2} & \frac{-x^9(y-x^2)^8}{z^3} \
\frac{-40x^8(y-x^2)^8}{z^2} & \frac{9x^8(y-x^2)^9}{z^2} - \frac{10x^9(y-x^2)^8}{z^2} & 0
\end{pmatrix}
\end{align*}
Therefore, $
abla 	imes F 
eq \mathbf{0}$ everywhere in space, which means that the vector field is not conservative.

Test the Vector Field F to Determine If It Is