# Tensors

Let us define a one-form as a linear function that takes a vector and maps it to a real number.

What do we mean by linear? We mean that f(a+b) = f(a) + f(b) and f(ka) = kf(a).

Now let us define a tensor as a linear function that takes zero or more vectors and zero or more one-forms and maps them to a real number. In particular let us designate a tensor to be of rank (m,n) if it takes m one-forms and n vectors.

A tensor of rank (0,0) is just a constant mapping to a real number. So a real number can be viewed as a tensor of rank (0,0).

By our definition of one-form above, a tensor of rank (0,1) is just a one-form.

Now, a tensor of rank (1,0) is a linear mapping from one-forms to the real numbers. For every such mapping, there is a vector that is equivalent to this mapping. If the vector is v then the mapping is the mapping that takes the one-form f to the real number f(v). In other words, a tensor of rank (1,0) can be thought of as a constant mapping to a vector.

A tensor of rank (0,2) takes two vectors and maps them linearly to a real number. This is actually equivalent to taking mapping from a vector to a one-form. (Does the tensor have to be symmetric for this to be true?) If you give a tensor of rank (0,2) just one vector you have a tensor that needs one (more) vector to make a real number - in other words, you have a one-form. See currying tensors.

A tensor of rank (2,0) takes two one-forms and maps them linearly to a real number. (Is equivalent to saying that a tensor of rank (2,0) is a constant mapping to two vectors?)

A tensor of rank (1,1) can be viewed as a mapping from a one-form and vector to a real number or as a mapping from a vector to a vector or as a mapping from a one-form to a one-form.

In fact, a tensor or rank (m, n) can be viewed as:

• a mapping from m one-forms and n vectors to a real number
• a mapping from n vectors to m vectors
• a mapping from m one-forms to n one-forms