# Question in defining a tensor

I am confused with definition of tensor. Some texts give definition of tensor as

$n$th order tensor is a mapping $$T: \mathbb {R^{n_d} \times \dots \times R^{n_d} \rightarrow R}$$

$\mathbb {R^{n_d}}$ is real $n$dimensional space. We can make this definition for a special case, such as second order tensor.

I understand from this definition that tensor produces a real number. Is that a scalar (zero order tensor)?

On the other hand, some texts define a (second order) tensor as $$c_i = A_{mi}e_m$$

$A$ is second order and $c,e$ are first order tensors.

In this case, a second order tensor produces first order tensor.

So, what exactly a tensor gives out as a result - a first or second order tensor? Or am I making some mistake in understanding the definition? Can someone comment?

• See these questions: 1, 2, 3 (or more specifically, their answers). – user137731 Oct 4 '16 at 1:05