^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] << [[Schutz - 2.21 - Index Notation|2.21 Index Notation]] | [[Schutz - 2.23 - Examples of Tensors|2.23 Examples of Tensors]] >> # Tensors and Tensor Fields > **Tensor of Type $(N, N')$** > Consider a point $P \in M$ where $M$ is some manifold. A *tensor of type $(N, N')$* at $P$ is defined to be a multilinear function which takes as an argument $N$ one-forms and $N'$ vectors, whose value is a real number. This is a generalization of [[Schutz - 2.16 - One-Forms#^7e7ffc|one-forms]]. ^472d5f If we want to speak of e.g. a (2,2) tensor $\overline{F}$ without specifying its arguments, we may write $\overline{F}(., .; . , .)$. > **Tensor Field of Type $(N, N')$** > A *tensor field* of type $(N, N')$ is a rule which assigns a [[Schutz - 2.22 - Tensors and Tensor Fields#^472d5f|tensor of type (N,N')]] to every point in $U \subset M$. A differentiable tensor-field is defined as for [[Schutz - 2.19 - The Gradient and the Pictorial Representation of a One-Form#^89c474|differentiable one-forms.]] See also [[Schutz - 2.20 - Basis One-Forms and Components of One-Forms#^d5f715|§2.20]] Note that vectors, as a special case of tensors, are $(1, 0)$ tensors, while one-forms are $(0, 1)$ tensors. By convention, a scalar function on the manifold is taken to be a type $(0,0)$ tensor (see [[Schutz - 2.8 - Basis Vectors and Basis Vector Fields|§2.8]]).