Tensor

TensorIt is a multilinear function that can be used to represent linear relationships between vectors, scalars, and other tensors. Basic examples of these linear relationships include inner products, outer products, linear maps, and Cartesian products.

Coordinates in N-dimensional space have N ^ r components, each of which is a function of the coordinates. When the coordinates are transformed, these components will also undergo linear transformations according to certain rules, where r is called the rank or order of the tensor.

In the isomorphic sense, let the zero-order tensor (r = 0) be a scalar, the first-order tensor (r = 1) be a vector, and the second-order tensor (r = 2) be a matrix.

Depending on the transformation method, tensors can be divided into three categories: "covariant tensors" with index down, "contravariant tensors" with index up, and "mixed tensors" with both index up and index down.