Previously, the vector space \(\mathbb{V}\) and three vector product operations were considered in Simple tensor algebra I and Simple tensor algebra II. But we do not talk much about the tensor. Here, the tensor will be introduced.

So, what is a tensor? The following figure shows a tensor in a 2D Cartesian coordinate system.

The vector is a tensor, and it is a first-order (1st-order) tensor. Actually, the scalar is a tensor as well, we can consider the scalar as a zeroth-order (0th-order) tensor.

Let \( a, b\) be two scalars in \( \mathbb{R}\), we put these two scalars (0th-order tensor) together \( \{ a, b \}\) to form a vector (1st-order tensor). So, if we can put two vectors \( \boldsymbol g, \boldsymbol f \) together, then we will get a 2nd-order tensor. The tensor product \( \otimes \) plays such role here (\( \boldsymbol g \otimes \boldsymbol f \)).

This is not a tutorial, please refer to the books for tensor algebra.