Definition
Tensor Analysis is the study of tensors, which are geometric objects that describe linear relations between vectors, scalars, and other tensors. They generalize the concepts of scalars and vectors to higher dimensions and provide a coordinate-independent way to express physical laws.
Why It Matters
Tensors are the ‘true’ language of the universe, independent of the observer’s coordinate system. Without tensor analysis, we could not describe gravity (General Relativity) or the stress within a complex material, making modern physics and advanced engineering impossible.
Core Concepts
- Rank/Order: A scalar is a rank-0 tensor, a vector is a rank-1 tensor, and a matrix is a rank-2 tensor.
- Invariance: Tensors maintain their properties regardless of the coordinate system used, making them essential for General Relativity and Fluid Dynamics.
- Covariance and Contravariance: Describes how the components of a tensor change when the coordinate system is transformed.
- Einstein Summation Convention: A notation system used to simplify complex tensor equations.