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Let $a_1,a_2,a_3,a_4 \in \mathbb{R}$ be given so that $(a_1,a_2) $ and $(a_3,a_4)$ are linearly independent vectors in the plane. Let $\alpha$ be the boundary of the parallelogram with vertices $(0,0), (a,b),(c,d),(a+c,b+d) $ oriented counterclockwise. How can I obtain

$$ \int_{\alpha} x dy - y dx $$

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Green's theorem says $\int_C Ldx+Mdy=\int\int_D\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}dA$, where C is a closed loop bounding a region without singularities.

Use Green's theorem to calculate the line integral as it's a closed loop, then it's easy exercise of multiple integral calculus.

In this case $\int_C -ydx+xdy=\int\int_D\frac{\partial x}{\partial x}-\frac{\partial (-y)}{\partial y}dA=\int\int_D2dxdy$

$D$ is the region bounded by the parallelogram, you can calculate the integral by Fubini's Theorem. Since it's a parallelogram with opposite sides parallel, change the coordinate to make the integration to be over a rectangular region is even better.

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