Consider the contour C shown in the figure below. For the vector
F⃗ = (x + 2y)êx + (2x + 4y)êy, the integral ∮ F⃗ ⋅ dl⃗
C =________.
Here dl⃗ represents an infinitesimal length along the contour C.
Options:
(A) 0
(B) 2
(C) 4
(D) 6
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.11)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to evaluate the closed counter-clockwise line integral $\oint_C \vec{F} \cdot d\vec{r}$ around contour $C$ for the 2D vector field:
$\vec{F}(x, y) = (x + 2y)\hat{e}_x + (2x + 4y)\hat{e}_y$
where $F_x = x + 2y$ and $F_y = 2x + 4y$.
**2. Step-by-Step Mathematical Derivation**:
1. **Check for Conservativeness via Curl**:
By Green's Theorem in the plane, for any simple closed curve $C$ enclosing region $R$:
$\oint_C (F_x dx + F_y dy) = \iint_R \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) dA$
2. **Evaluate the 2D Curl / Integrand**:
$\frac{\partial F_y}{\partial x} = \frac{\partial}{\partial x}(2x + 4y) = 2$
$\frac{\partial F_x}{\partial y} = \frac{\partial}{\partial y}(x + 2y) = 2$
$\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 2 - 2 = 0$
3. **Integration Result**:
Since the curl of $\vec{F}$ is identically zero everywhere in the $xy$-plane, the vector field is **conservative** (an exact gradient field: $\vec{F} = \nabla \phi$ with $\phi = \frac{1}{2}x^2 + 2xy + 2y^2$).
The circulation of any conservative field around any closed loop is strictly zero:
$\oint_C \vec{F} \cdot d\vec{r} = 0$
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 0**: ✅ **CORRECT**. The curl evaluates to $2 - 2 = 0$, making the field irrotational and the closed line integral vanish identically.
- **(B) 2**: ❌ **INCORRECT**. Result of computing $\partial F_y/\partial x$ alone without subtracting $\partial F_x/\partial y$.
- **(C) 4**: ❌ **INCORRECT**. Result of adding the partial derivatives ($2 + 2 = 4$) instead of subtracting them.
- **(D) 6**: ❌ **INCORRECT**. Spurious distractor from summing the coefficients.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Always test $\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}$ first for 2D line integrals around closed loops. If zero, Green's theorem immediately yields 0 without needing any parameterization of the loop boundary.
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