A𝛼 and B𝛽 (𝛼, 𝛽 = 1,2,3, ⋯ , n) are contravariant and covariant vectors,
respectively. By convention, any repeated indices are summed over. Which of the
following expression is/are tensors?
Options:
(A) A𝛼B𝛽
(B) A𝛼B𝛽
A𝛼B𝛼
(C) A𝛼
B𝛽
(D) A𝛼 + B𝛽
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.51)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Mathematical Physics** within *Section 1: Mathematical Physics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Mathematical Physics:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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