If the curl of $a+b$ is zero. Can one find $b$, knowing $a$?

Question

If I have an equation saying that $\operatorname{curl}(a+b)=0$ and I know $a$ then is there a way to find out $b$?

Since $\operatorname{curl}(a+b)=\operatorname{curl}(a)+\operatorname{curl}(b)$, it follows that $\operatorname{curl}(b)=-\operatorname{curl}(a)$. But doesn't that mean that $b$ can take on any value as long as its curl is negative of $\operatorname{curl}(a)$, i.e. $b$ is not unique?

Answer 1

Curl is distributive.

$\mathrm{curl}( A + B ) = \mathrm{curl}( A ) + \mathrm{curl}( B )$

So if $\mathrm{curl}( A + B ) = 0$, you get:

$0 = \mathrm{curl}( A ) + \mathrm{curl}( B )\ \ \ \ \ \ \implies \ \ \ \ \ \ \mathrm{curl}( B ) = - \mathrm{curl}( A )$

The solution to $\mathrm{curl}( B ) = - \mathrm{curl}( A )$ is not unique.

In general, it's usually pretty hard to solve for B in the equation $\mathrm{curl}( B ) = f $, for some arbitrary $f$.