# 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.)
${\oint }_{c}F\cdot nds=\int {\int }_{D}÷F\left(x,y\right)dA$

You can still ask an expert for help

## Want to know more about Green's, Stokes', and the divergence theorem?

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

tabuordy
Step 1
$\int {\int }_{D}f{\mathrm{\nabla }}^{2}gdA={\oint }_{C}f\left(\mathrm{\nabla }g\right).nds-\int {\int }_{D}\mathrm{\nabla }f.\mathrm{\nabla }gdA$
Add $\int {\int }_{D}\mathrm{\nabla }f.\mathrm{\nabla }gdA$ to both sides,
$\int {\int }_{D}f{\mathrm{\nabla }}^{2}gdA+\int {\int }_{D}\mathrm{\nabla }f.\mathrm{\nabla }gdA={\oint }_{C}f\left(\mathrm{\nabla }g\right).nds$
$\int \int \left(f{\mathrm{\nabla }}^{2}g+\mathrm{\nabla }f.\mathrm{\nabla }g\right)dA={\oint }_{C}f\left(\mathrm{\nabla }g\right).nds$
Step 2
Using the product rule for derivatives,
we can write $f{\mathrm{\nabla }}^{2}g+\mathrm{\nabla }f.\mathrm{\nabla }g=f\mathrm{\nabla }.\left(\mathrm{\nabla }g\right)+\mathrm{\nabla }f.\mathrm{\nabla }g=\mathrm{\nabla }.\left(f\mathrm{\nabla }g\right)$
$\int {\int }_{D}\mathrm{\nabla }.f\left(\mathrm{\nabla }g\right)dA={\oint }_{C}f\left(\mathrm{\nabla }g\right).nds$
$\int {\int }_{D}÷\left(f\mathrm{\nabla }g\right)dA={\oint }_{C}f\left(\mathrm{\nabla }g\right).nds$