What is Cauchy Stress Tensor?

Recap: Tensors and Transformation Properties

As we have seen in here, tensors can be defined, in a linear algebraic sense, as multilinear maps that take in vectors and covectors and output scalars:

Definition: (M, N) Tensor

An (M, N) tensor at point is a multilinear map:

Where and are the tangent space and cotangent space at point respectively.

These tangent spaces and cotangent spaces have basis vectors and dual basis that transform in a specific way under coordinate transformations. For example, if we define , then the basis vectors of the tangent space transform as:

Where is the Jacobian matrix of the transformation .

By contrast, the dual basis of the cotangent space transforms as:

Tensors components are transformed according to how their basis vectors and dual basis transform. For example, a (1, 1) tensor can be expressed in components as:

Under the coordinate transformation , the components transform