GATE PH 2026 • Q14
MCQ • 1 Mark
Section 1: Mathematical Physics
Differential equations
Subtopic: First and Second Order ODEs
Question Statement:
For a scalar field 𝜓(r ) and a vector field A (r ), ∇⃗⃗ × (A 𝜓) is equivalent to the expression
Options:
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(A) 𝜓(∇⃗⃗ × A ) - A × (∇⃗⃗ 𝜓)
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(B) 𝜓(∇⃗⃗ × A ) + A × (∇⃗⃗ 𝜓)
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(C) Null vector
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(D) A × (∇⃗⃗ 𝜓) - 𝜓(∇⃗⃗ × A )
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.14)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 1: Mathematical Physics* | **Topic**: *Linear Algebra & Vector Calculus* (`Matrices, Eigenvalues & Eigenvectors`)
- **Question Type**: **MCQ** | **Marks**: **1 Mark**
- **Theoretical Framework**: Governed by foundational physical principles in **Linear Algebra & Vector Calculus** (`Matrices, Eigenvalues & Eigenvectors`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `𝜓(∇⃗⃗ × A ) + A × (∇⃗⃗ 𝜓)`**.
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