GATE PH 2007 • Q23
MCQ • 1 Mark
Section 1: Mathematical Physics
Linear Algebra & Vector Calculus
Subtopic: Vector Differential Operators (Grad, Div, Curl)
Question Statement:
If $\vec{F} = x\hat{i} + y\hat{j}$, then
Options:
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(A) $\nabla \cdot \vec{F} = 0$ and $|\nabla \times \vec{F}| = 0$
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(B) $\nabla \cdot \vec{F} = 2$ and $|\nabla \times \vec{F}| = 0$
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(C) $\nabla \cdot \vec{F} = 2$ and $|\nabla \times \vec{F}| = 1$
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(D) $\nabla \cdot \vec{F} = 3$ and $|\nabla \times \vec{F}| = 0$
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2007 Q.23)**
**1. Theoretical Formulation & Vector Calculus**:
- **Field & Concept**: *Section 1: Mathematical Physics* (`Vector Differential Operators (Grad, Div, Curl)`)
- Given 2D vector field:
$\vec{F} = x\hat{i} + y\hat{j}$
**2. Detailed Step-by-Step Derivation**:
- **Divergence of $\vec{F}$**:
$\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} = \frac{\partial}{\partial x}(x) + \frac{\partial}{\partial y}(y) = 1 + 1 = 2$
- **Curl of $\vec{F}$**:
$\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ x & y & 0 \end{vmatrix} = \hat{k} \left( \frac{\partial y}{\partial x} - \frac{\partial x}{\partial y} \right) = \vec{0}$
- Magnitude of the curl:
$|\nabla \times \vec{F}| = 0$
**3. Official Verification & Conclusion**:
- **Official Master Answer**: **Option (B)**: $\nabla \cdot \vec{F} = 2$ and $|\nabla \times \vec{F}| = 0$.
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