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.) oint_c F * n ds = int int_D div F(x,y)dA

Question
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.)
\(\displaystyle\oint_{{c}}{F}\cdot{n}{d}{s}=\int\int_{{D}}\div{F}{\left({x},{y}\right)}{d}{A}\)

Answers (1)

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 \(\displaystyle{f}\nabla^{{2}}{g}+\nabla{f}.\nabla{g}={f}\nabla.{\left(\nabla{g}\right)}+\nabla{f}.\nabla{g}=\nabla.{\left({f}\nabla{g}\right)}\)
\(\displaystyle\int\int_{{D}}\nabla.{f{{\left(\nabla{g}\right)}}}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}\)
\(\displaystyle\int\int_{{D}}\div{\left({f}\nabla{g}\right)}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}\)
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