Question

As per Green's theorem $\oint_C Mdx + Ndy = \iint_R (_____)dxdy$

• A. \( \frac{\partial M}{\partial x} - \frac{\partial N}{\partial y} \)
• B. \( \frac{\partial N}{\partial y} - \frac{\partial M}{\partial x} \)
• C. \( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \)
• D. \( \frac{\partial M}{\partial y} - \frac{\partial N}{\partial x} \)

Hint:

Remember the order of the partial derivatives in Green's Theorem: $\frac{\partial N}{\partial x}$ comes first, then subtract $\frac{\partial M}{\partial y}$.

Answer 1

The Correct Option is C

Green's Theorem states that for a positively oriented, simple closed curve $C$ and a region $R$ bounded by $C$, the line integral of a vector field $\mathbf{F} = M\mathbf{i} + N\mathbf{j}$ around $C$ is equal to the double integral over $R$ of the scalar quantity $\left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right)$. Thus, the correct form of Green's Theorem is: $$\oint_C Mdx + Ndy = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dxdy$$