The vector space is introduced in Simple tensor algrbra I, several rules are provided in it. To reduce the abstract, we represent the vectors with the scalar coefficients and in , respectively. The composition of two vectors is expanded by,

**Dot product ():**

The *dot product* is also called *scalar product* or *inner product* under Cartesian coordinates and it provides the Euclidean magnitudes of these two vectors and the cosine of the angle between them.

**Cross product ():**

The *cross product* is also called *vector product*, and it represents the area of a parallelogram with the sides of these two vectors. Now, a new composition for two vectors is introduced here.

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

**Outer product ():**

Compared with the *dot product*, it seems we just change the row vector to column vector for and the column vector to row vector for . Actually, it is much more complicated than that. The *outer product* represents the linear mapping from one vector space to another spaces , this linear mapping is also called tensor product.

**Linear mapping:**

Let and be two vector spaces over the field , then if operation satisfies the following two conditions for any and any scalar ,

- Additivity: ,
- Homogeneity: .

is called a linear mapping ().

An invertible linear mapping is called an isomorphism.

To be specific, let be a vector, consider the vector product,

the component can be treated as a linear mapping . Thus, we use a new symbol here.

**Tensor product:**

Go back to the *tensor product* or *outer product*, we start from an example.

Example: Consider the projection of a vector to another vector in the direction of , this operation can be written by,note that the RHS is attributed to that the is a scalar and the

dot productsatisfies the commutativity.Such operation can be verified with the linear mapping condition and be denoted by,

The operation here is actually the same with preceding one in

outer productsection.

Now, let and be two vectors in , an arbitary vector can be mapped into the vector space , the mapping operation is denoted by . Then,

**Tensor product properties:**

Linear mapping:

Associativity:

Distributivity:

Note that the commutativity is not satisfied any more:

we will prove that the transpose of is later.

Example: Proof the distributivityConsider the definition of tensor product, we select a vector . Multiplying both sides of the equation by , the LHS can be written by,

The RHS can be written by,

Thus, we get the result.

**Contraction:**

In tensor product, the *dot product* and the *double dot product* are usually called the contraction. Let be a fourth order tensor (we will find that the *tensor product* yields the high order tensor later), then the *dot product* contraction are,

The *double dot product* contraction is the operation that combines four nearby vector under tensor product,

**Rotation:**

The rotation operation of a vector is a typical example to show the usage of vector product. In continuum mechanics, the rotation matrix and the transformation matrix are very useful to compute the stress and strain tensor in different orientations. The rotation matrix can be constructed by a rotation axis vector (unit vector) and a rotation angle , we assume the rotation follows the *right-hand rule*. The result can be written by,

To get such result, we first construct a local coordinate system. Denote are the bases of the Cartesian coordinate system as shown in the figure. is a unit vector that aligns with vector , is a unit vector that aligns with vector . Then, the last unit vector can be expressed by . The position of vector rotates to the position with a rotation angle about the axis . First, the vector can be written by,

Then, after the rotation,

Since the length of vector is the same as , expand the vector in local coordinate system,

Thus,

Note that we want to construct a mapping operation (i.e. ) that satisfies , so we need to eliminate from the preceding equation to get (i.e. ). Consider that the vector is a projection of vector on rotation axis , then,

and from equation ,

Plug and into equation and note that ,

Rearrange the equation,

Since , the corss product can be written by,

Thus,

is the vector in equation , thus we get the result.