Tensor

Given a vector space (or a module) $V$ over field (or ring) $R$, an $(n,m)$-tensor is a multilinear form (linear in each of its arguments): $$T: \underbrace{ V^* \times\dots\times V^*}_{n \text{ copies}} \times \underbrace{ V \times\dots\times V}_{m \text{ copies}} \rightarrow R.$$ Here $V$ is the space of vectors and the dual space $V^*$ of $V$ is space of covectors.

Calculus studies tensor fields, i.e., tensors parametrized by location.

Differential forms are $(n,0)$-tensor fields.

From the definition, it follows that $$T\in T^m_n(V) = \underbrace{V \otimes\dots\otimes V}_{n \text{ copies}} \otimes \underbrace{V^* \otimes\dots\otimes V^*}_{m \text{ copies}},$$ via tensor product, which may also serve as a definition of tensor.