Tensor Analysis

Phase 4: Advanced Topics

Syllabus Goal

Elementary ideas about tensors. Introduction, Cartesian Tensors, Tensor Notation.

Core Concepts

• Scalars: Rank-0 tensors (magnitude only).
• Vectors: Rank-1 tensors (magnitude and direction).
• Tensors: Rank-n geometric objects invariant under coordinate transformations.

Notation

• Einstein Summation: Repeated indices imply summation ($a_i b^i = \sum_i a_i b^i$).
• Covariant ($A_\mu$) vs Contravariant ($A^\mu$): How components transform.
• Metric Tensor ($g_{\mu\nu}$): Defines distance and lowers indices.

Applications

• General Relativity.
• Fluid Dynamics (Stress Tensor).
• Electromagnetism (Field Tensor $F_{\mu\nu}$).