I know how to formulate the weak form of an equation with a second derivative, but am not sure for one with only a first order derivative. With the following equation $$0=c\nabla N$$ I can multiply a test function $A$ to it: $$0=c\nabla N \cdot A$$ and then integrate over it: $$0=c\left(\nabla N, A\right) $$ Now I can either say that I am done, or rewrite it: $$\begin{split} 0&=c\left(\nabla N, A\right)\\ &=c\left(\left(N, A\right) - \left(N, \nabla A\right)\right) \end{split}$$
In terms of calculation: What is the more useful form? Both forms involve exactly one derivative, thus I assume that they are equal in terms of uncertainty. Is that correct?