Consider a two dimensional Cartesian coordinate system in which a rank 2
contravariant tensor is represented by the matrix (0 1
1 0). The coordinate system
is rotated anticlockwise by an acute angle 𝜃 with the origin fixed. Which one of
the following matrices represents the tensor in the new coordinate system?
Options:
(A) ( 0 cos 2𝜃
- sin 2𝜃 0 )
(B) ( sin 2𝜃 cos 2𝜃
cos 2𝜃 - sin 2𝜃)
(C) (sin 2𝜃 -cos 2𝜃
cos 2𝜃 sin 2𝜃 )
(D) (sin 2𝜃 0
0 - cos 2𝜃)
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.17)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Mathematical Physics** within *Section 1: Mathematical Physics* (1 Mark, 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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