I am working on a homework problem that requires me to find the curl of another vector field–which I can easily find if the $\nabla \cdot F=0$. However, I am stuck on the portion that requires me to solve for that vector field, given that the $\nabla \cdot F$ is indeed zero.
In other words, I am being asked to backtrack, and instead of finding the curl between two vectors, I am asked to find that vector field whose curl is given to me.
To be more specific, the vector given (which is indeed the curl of another v.f.) is:
$<-\sin(x), \cos(y), z\cos(x) + z\sin(y)>$.
Given that its divergence is equal to zero, how do I find the vector field that outputs the given v.f. above; how do I reverse the curl in order to find $F$ in $\nabla \times F$? I am not sure whether to integrate each component (as it is the antiderivative).
Is there another formula I can refer to? I am aware that I can simply set the equation up as such:
$<-\sin(x), \cos(y), z\cos(x) + z\sin(y)> \text{ }= \left(\frac{dF3}{dy} - \frac{dF2}{dz}\right)i - \left(\frac{dF3}{dx} - \frac{dF1}{dz}\right)j + \left(\frac{dF2}{dx} - \frac{dF1}{dy}\right)k$
But how do I find the actual, F1, F2, and F3 components for the desired vector field? Is integration involved?
Please advise. Thank you in advance for your help.
if needed, the questions reads, "Which of the following vector fields is the curl of another vector field? Find the vector field if it exists.
*a) $F = <x\cos(y), z\sin(y), xyz>$
*b) $F = <-\sin(x), \cos(y), z\cos(x) + z\sin(y)>$
and I only need help with letter b).