Green's Theorem. $$\oint_C Mdx+Ndy = \iint_R N_y-M_x dA.$$ $$\oint_C Ndx-Mdy = \iint_R M_x+N_y dA.$$ The del operator $\vec \nabla$. $$\ds \vec\nabla = \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)$$ $$\ds \vec\nabla f = \left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right)$$ $$\ds \vec\nabla\cdot\vec F = \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\cdot (M,N,P) = M_x+N_y+P_z$$ $$\ds \vec\nabla\cdot\times F = \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\times (M,N,P) = (?,?,?)$$
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