Why must the following condition be satisfied before we can use Green's Theorem:
L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there.
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$\begingroup$Because the right hand side of greens theorem is the integral of $\frac{\partial L}{\partial y}-\frac{\partial M}{\partial x}$, or vice versa (swapping the position of $L$ and $M$), so you need the partial derivatives to be continuous so that this is well defined. Also we need the region to be open, because differentiation is not define on the end point of a closed set, as we need to know what is happening "in every direction".
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