Solve the problem.
-Find values for $a$, b, and c so that $\nabla \times \mathbf { F } = \mathbf { 0 }$ for $\mathbf { F } = a x i + b y j + c z k$.
A) $a = 2 , b = - 1 , c = - 1$
B) $a = 1 , b = 1 , c = 1$
C) $\nabla \times \mathbf { F } = 0$ for any $a , b$, and $c$.
D) There is no possible way to make $\nabla \times \mathbf { F } = 0$.

Question

Quizplus · Accepted Answer

We can compute the curl of $\mathbf{F}$ as follows:
\begin{align*}

abla 	imes \mathbf{F} &= 
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\
ax & by & cz \
\end{vmatrix}\
&= \left(\frac{\partial(cy)}{\partial y} - \frac{\partial(by)}{\partial z}ight)\mathbf{i} + \left(\frac{\partial(ax)}{\partial z} - \frac{\partial(cx)}{\partial x}ight)\mathbf{j} + \left(\frac{\partial(by)}{\partial x} - \frac{\partial(ax)}{\partial y}ight)\mathbf{k} \
&= (0)\mathbf{i} + (0)\mathbf{j} + (0)\mathbf{k} \
&= \mathbf{0}.
\end{align*}
Since each component of $
abla 	imes \mathbf{F}$ is zero, we conclude that $
abla 	imes \mathbf{F} = \mathbf{0}$ for any values of $a$, $b$, and $c$. Therefore, the answer is choice $\boxed{	ext{(C)}}$.

Solve the Problem aaa , B, and C So That ∇×F=0\nabla \times \mathbf { F } = \mathbf { 0 }∇×F=0

Solve the Problem $a$ , B, and C So That $\nabla \times \mathbf { F } = \mathbf { 0 }$