Suppose that \(\mathcal{S}_{1}\) and \(\mathcal{S}_{2}\) are surfaces with the same oriented boundary curve \(C\). In each case, does the condition guarantee that the flux of \(\mathbf{F}\) through \(\mathcal{S}_{1}\) is equal to the flux of \(\mathbf{F}\) through \(\mathcal{S}_{2}\) ?

(a) \(\mathbf{F}=abla f\) for some function \(f\)

(b) \(\mathbf{F}=\operatorname{curl}(\mathbf{G})\) for some vector field \(\mathbf{G}\)