Let \(C\) be a closed curve and \(D\) the enclosed region. Prove Green's identities in two dimensions.

a. First prove

\[\int_{D}(v abla \cdot \mathbf{F}+\mathbf{F} \cdot abla v) d A=\int_{C}(v \mathbf{F}) \cdot d \mathbf{s}\]

b. Let \(\mathbf{F}=abla u\) and obtain Green's first identity,

\[\int_{D}\left(v abla^{2} u+abla u \cdot abla v\right) d A=\int_{C}(v abla u) \cdot d \mathbf{s}\]

c. Use Green's first identity to prove Green's second identity,

\[\int_{D}\left(u abla^{2} v-v abla^{2} u\right) d A=\int_{C}(u abla v-v abla u) \cdot d \mathbf{s}\]