Question

If R is a closed region in the xy-plane bounded by a simple closed curve C and if M(x, y) and N(x, y) are continuous functions of x and y having continuous derivative in R, then

• A. \( \oint_C Mdx + Ndy = \iint_R \left( \frac{\partial M}{\partial x} - \frac{\partial N}{\partial y} \right) dx dy \)
• B. \( \oint_C Mdx + Ndy = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dx dy \)
• C. \( \oint_C Mdx + Ndy = \iint_R \left( \frac{\partial M}{\partial y} - \frac{\partial N}{\partial x} \right) dx dy \)
• D. \( \oint_C Mdx + Ndy = \iint_R \left( \frac{\partial M}{\partial x} + \frac{\partial N}{\partial y} \right) dx dy \)

Hint:

A simple way to remember the order of terms in Green's theorem is to think of the vector field \( \vec{F} = (M, N) \). The integrand is \( \frac{\partial(\text{second component})}{\partial(\text{first variable})} - \frac{\partial(\text{first component})}{\partial(\text{second variable})} \), which is \( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \). This is the 2D "curl".

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question asks for the statement of Green's theorem. Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region R bounded by C.
Step 2: Key Formula or Approach:
Green's theorem states that for a vector field \( \vec{F} = M(x,y)\hat{i} + N(x,y)\hat{j} \), the counterclockwise line integral of \( \vec{F} \) along a simple closed curve C is equal to the double integral of the curl of \( \vec{F} \) over the region R enclosed by C.
The line integral is \( \oint_C \vec{F} \cdot d\vec{r} = \oint_C Mdx + Ndy \).
The component of the curl of \( \vec{F} \) in the k-direction is \( (\text{curl} \vec{F})_z = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \).
Therefore, the theorem is: \[ \oint_C Mdx + Ndy = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dA \] where \( dA = dx dy \).
Step 3: Detailed Explanation:
We compare the formula from Step 2 with the given options.
Option (A) has the wrong terms and sign.
Option (B) correctly states \( \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dx dy \). This matches the theorem.
Option (C) has the terms in the wrong order, resulting in a sign error.
Option (D) represents the double integral of the divergence of the vector field, which is related to the flux form of Green's theorem (or the 2D Divergence Theorem), not the circulation form asked here.
Step 4: Final Answer:
The correct statement of Green's theorem is given in option (B).