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

Yasmin 2021-03-04 Answered

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.)
cFnds=D÷F(x,y)dA

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tabuordy
Answered 2021-03-05 Author has 91 answers
Step 1
Df2gdA=Cf(g).ndsDf.gdA
Add Df.gdA to both sides,
Df2gdA+Df.gdA=Cf(g).nds
(f2g+f.g)dA=Cf(g).nds
Step 2
Using the product rule for derivatives,
we can write f2g+f.g=f.(g)+f.g=.(fg)
D.f(g)dA=Cf(g).nds
D÷(fg)dA=Cf(g).nds
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