# 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}

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.
$F=⟨x,y⟩,R=\left\{\left(x,y\right):{x}^{2}+{y}^{2}\le 4\right\}$
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Step 1
a.
To compute the divergence of the vector field F.
Given F(x,y)=xi+yj.
The divergence of F is $÷F=\frac{df}{dx}+\frac{dg}{dy}$.
Here f(x,y)=x and g(x,y)=y.
$÷F=\frac{df}{dx}+\frac{dg}{dy}$
=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 ${\int }_{c}fdx+gdy=\int {\int }_{R}\left(\frac{dg}{dx}-\frac{df}{dy}\right)dA$.
$\int {\int }_{R}\left(\frac{dg}{dx}-\frac{df}{dy}\right)dA=\int {\int }_{R}\left(0-0\right)dA$
=0