Step 1: Concept Identification: The objective is to present Green's theorem. This theorem establishes a connection between a line integral along a closed curve C and a double integral over the region R enclosed by C.
Step 2: Core Formula: For a vector field \( \vec{F} = M(x,y)\hat{i} + N(x,y)\hat{j} \), Green's theorem states that the line integral of \( \vec{F} \) taken counterclockwise around a simple closed curve C equals the double integral of the component of the curl of \( \vec{F} \) in the z-direction over the region R bounded by C. The line integral is expressed as \( \oint_C \vec{F} \cdot d\vec{r} = \oint_C Mdx + Ndy \). The z-component of the curl is \( (\text{curl} \vec{F})_z = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \). Consequently, Green's theorem is formulated as: \[ \oint_C Mdx + Ndy = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dA \] where \( dA = dx dy \).
Step 3: Detailed Evaluation: Comparing the derived formula with the provided options:
Option (A) is incorrect due to incorrect terms and sign.
Option (B) accurately represents \( \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dx dy \), aligning with Green's theorem.
Option (C) has terms in an inverted order, leading to a sign error.
Option (D) represents the double integral of the divergence, which pertains to the flux form of Green's theorem or the 2D Divergence Theorem, not the circulation form requested.
Step 4: Conclusion: Option (B) provides the correct statement of Green's theorem.