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Question

# Verify Green’s theorem for the given vector field $\mathbf{F}=M(x, y) \mathbf{i}+N(x, y) \mathbf{j}$ and region D by calculating both $\oint_{\partial D} M d x+N d y$ and $\iint_{D}\left(N_{x}-M_{y}\right) d A.$ $\mathbf{F}=\left(x^{2}-y\right) \mathbf{i}+\left(x+y^{2}\right) \mathbf{j}$ is the rectangle bounded by x = 0, x = 2, y = 0, and y = 1.

Solution

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Try graphing the rectangle, where $A(0,0), \; B(2,0), \; C(1,2), and \; D(0,1)$:

\centering $\color{#4257b2}{A\rightarrow B:}$ $\; y=0, \; dy=0, \; x=0 \; to \; 2$

\centering $\color{#4257b2}{B\rightarrow C}:$ $\; x=2, \; dx=0, \; y=0 \; to \; 1$

\centering $\color{#4257b2}{C\rightarrow D}:$ $\; y=1, \; dy=0, \; x=2 \; to \; 0$

\centering $\color{#4257b2}{D\rightarrow A}:$ $\; x=0, \; dx=0, \; y=1 \; to \; 0$

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