Green’s Theorem, flux form Consider the following regions R and vector fields F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in Green’s Theorem and check for consistency. F = <<x, y>>, R = {(x, y): x^2 + y^2 <= 4}

Step 1
a.
To compute the divergence of the vector field F.
Given F(x,y)=xi+yj.
The divergence of F is ${\displaystyle ÷F=\frac{df}{dx}+\frac{dg}{dy}}$.
Here f(x,y)=x and g(x,y)=y.
${\displaystyle ÷F=\frac{df}{dx}+\frac{dg}{dy}}$
=1+1
=0
Thus the two dimensional divergence of the vector field is 2.
Step 2
b.
To evaluate the integrals of Greens Theorem:
By Greens theorem ${\displaystyle {\int }_{c}fdx+gdy=\int {\int }_R(\frac{dg}{dx}-\frac{df}{dy})dA}$.
${\displaystyle \int {\int }_R(\frac{dg}{dx}-\frac{df}{dy})dA=\int {\int }_R(0-0)dA}$
=0