# Use Green's Theorem in the form of this equation to prove Green's first identity, where D and C satisfy the hypothesis of Green's Theorem and the appr

Use Green's Theorem in the form of this equation to prove Green's first identity, where D and C satisfy the hypothesis of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity grad g · n = Dng occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)
$$\displaystyle\oint_{{c}}{F}\cdot{n}{d}{s}=\int\int_{{D}}\div{F}{\left({x},{y}\right)}{d}{A}$$

2021-03-05
Step 1
$$\displaystyle\int\int_{{D}}{f}\nabla^{{2}}{g}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}-\int\int_{{D}}\nabla{f}.\nabla{g}{d}{A}$$
Add $$\displaystyle\int\int_{{D}}\nabla{f}.\nabla{g}{d}{A}$$ to both sides,
$$\displaystyle\int\int_{{D}}{f}\nabla^{{2}}{g}{d}{A}+\int\int_{{D}}\nabla{f}.\nabla{g}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}$$
$$\displaystyle\int\int{\left({f}\nabla^{{2}}{g}+\nabla{f}.\nabla{g}\right)}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}$$
Step 2
Using the product rule for derivatives,
we can write $$\displaystyle{f}\nabla^{{2}}{g}+\nabla{f}.\nabla{g}={f}\nabla.{\left(\nabla{g}\right)}+\nabla{f}.\nabla{g}=\nabla.{\left({f}\nabla{g}\right)}$$
$$\displaystyle\int\int_{{D}}\nabla.{f{{\left(\nabla{g}\right)}}}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}$$
$$\displaystyle\int\int_{{D}}\div{\left({f}\nabla{g}\right)}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}$$