Principles Of Tensor Calculus: Tensor Calculus Apr 2026

Tensor calculus allows us to write "coordinate-free" laws. Instead of writing separate equations for

Tensors are defined by how their components transform during a change of coordinates. There are two primary types of transformation: Contravariant ( Aicap A to the i-th power Principles of Tensor Calculus: Tensor Calculus

Contraction is the process of summing over a repeated upper and lower index (Einstein summation convention). This reduces the "rank" of a tensor. For example, contracting a vector with a covector results in a , which is a single value that everyone—regardless of their coordinate system—will agree upon. Summary of Utility Tensor calculus allows us to write "coordinate-free" laws

): Components that transform "against" the coordinate change (e.g., position or velocity). They are denoted with upper indices. Covariant ( Aicap A sub i This reduces the "rank" of a tensor

): Components that transform "with" the coordinate change (e.g., gradients of a scalar field). They are denoted with lower indices.