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This page is going to serve as a general introduction to the concept of tensors.

Tensors[edit | edit source]

A tensor is essentially a generalization of vectors and matrices that readers should be familiar with from linear algebra. A vector (with one dimension) is a rank-1 tensor, and a matrix (with two dimensions) is a rank-2 tensor. The rank of a tensor is the number of indices that are required to find an element from within that tensor.

Along with a rank, a tensor also has a size. For instance, a vector with three elements is a "Rank 1 tensor (3)", while a matrix that has 2 rows and 4 columns is a "Rank 2 tensor (2, 4)".

A tensor field is a generalization of the tensor concept, where each element in the tensor may be a variable or a scalar-valued function. In general, we will use the terms "tensor" and "tensor field" interchangeably. A vector field, like those that we have been studying so far, can be considered a specific case of tensor field, where each point in the field has an associated rank 1 tensor (a vector).

Tensors are, in the most basic geometrical terms, a relationship between other tensors. For instance, a rank-2 tensor is a linear relationship between two vectors, while a rank-3 tensor is a linear relationship between two matrices, and so on.

Symmetric and Antisymmetric Tensors[edit | edit source]

Covariant and Non-Covariant Tensors[edit | edit source]

Tensor Product[edit | edit source]