In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in material object, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a magnitude and a direction, like velocity), a tensor field is a generalization of a scalar field and a vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor A is defined on a vector fields set X(M) over a module M, we call A a tensor field on M. A tensor field, in common usage, is often referred to in the shorter form "tensor". For example, the Riemann curvature tensor refers a tensor field, as it associates a tensor to each point of a Riemannian manifold, a topological space.

Let ${\displaystyle M}$ be a manifold, for instance the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$.

Definition. A tensor field of type ${\displaystyle (p,q)}$ is a section

$${\displaystyle T\ \in \ \Gamma (M,V^{\otimes p}\otimes (V^{*})^{\otimes q})}$$

where ${\displaystyle V=TM}$ to be the tangent bundle of ${\displaystyle M}$ (whose sections are called vector fields or contra variant vector fields in Physics) and ${\displaystyle V^{*}=T^{*}M}$ is its dual bundle, the cotangent space (whose sections are called 1 forms, or covariant vector fields in Physics), and ${\displaystyle \otimes }$ is the tensor product of vector bundles.

Equivalently, a tensor field is a collection of elements ${\displaystyle T_{x}\in V_{x}^{\otimes p}\otimes (V_{x}^{*})^{\otimes q}}$ for every point ${\displaystyle x\in M}$, where ${\displaystyle \otimes }$ now denotes the tensor product of vectors spaces, such that it constitutes a smooth map ${\displaystyle T:M\rightarrow V^{\otimes p}\otimes (V^{*})^{\otimes q}}$. The elements ${\displaystyle T_{x}}$ are called tensors.

Locally in a coordinate neighbourhood ${\displaystyle U}$ with coordinates ${\displaystyle x^{1},\ldots x^{n}}$ we have a local basis (Vielbein) of vector fields ${\displaystyle \partial _{1}={\frac {\partial }{\partial x^{n}}}\ldots \partial _{n}={\frac {\partial }{\partial x_{n}}}}$, and a dual basis of 1 forms ${\displaystyle dx^{1},\ldots dx^{n}}$ so that ${\displaystyle dx^{i}(\partial _{j})=\partial _{j}x^{i}=\delta _{j}^{i}}$. In the coordinate neighbourhood ${\displaystyle U}$ we then have $${\displaystyle T_{x}=T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n})\partial _{i_{1}}\otimes \cdots \otimes \partial _{i_{p}}\otimes dx^{j_{1}}\otimes \cdots \otimes dx^{j_{q}}}$$ where here and below we use Einstein summation conventions. Note that if we choose different coordinate system ${\displaystyle y^{1}\ldots y^{n}}$ then ${\displaystyle {\frac {\partial }{\partial x^{i}}}={\frac {\partial y^{k}}{\partial x^{i}}}{\frac {\partial }{\partial y^{k}}}}$ and ${\displaystyle dx^{j}={\frac {\partial x^{j}}{\partial y^{\ell }}}dy^{\ell }}$ where the coordinates ${\displaystyle (x^{1},\ldots ,x^{n})}$ can be expressed in the coordinates ${\displaystyle (y^{1},\ldots y^{n}}$ and vice versa, so that

$${\displaystyle {\begin{aligned}T_{x}&=T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n}){\frac {\partial }{\partial x^{i_{1}}}}\otimes \cdots \otimes {\frac {\partial }{\partial x^{i_{p}}}}\otimes dx^{j_{1}}\otimes \cdots \otimes dx^{j_{q}}\\&=T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n}){\frac {\partial y^{k_{1}}}{\partial x^{i_{1}}}}\cdots {\frac {\partial y^{k_{p}}}{\partial x^{i_{p}}}}{\frac {\partial x^{j_{1}}}{\partial y^{\ell _{1}}}}\cdots {\frac {\partial x^{j_{q}}}{\partial y^{\ell _{q}}}}{\frac {\partial }{\partial y^{k_{1}}}}\otimes \cdots \otimes {\frac {\partial }{\partial y^{k_{p}}}}\otimes dy^{\ell _{1}}\otimes \cdots \otimes dy^{\ell _{q}}\\&=T_{\ell _{1},\cdots \ell _{q}}^{k_{1},\ldots ,k_{p}}(y^{1},\ldots y^{n}){\frac {\partial }{\partial y^{k_{1}}}}\otimes \cdots \otimes {\frac {\partial }{\partial y^{k_{p}}}}\otimes dy^{\ell _{1}}\otimes \cdots \otimes dy^{\ell _{q}}\\\end{aligned}}}$$ i.e. $${\displaystyle T_{\ell _{1},\cdots \ell _{q}}^{k_{1},\ldots ,k_{p}}(y^{1},\ldots y^{n})=T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n}){\frac {\partial y^{k_{1}}}{\partial x^{i_{1}}}}\cdots {\frac {\partial y^{k_{p}}}{\partial x^{i_{p}}}}{\frac {\partial x^{j_{1}}}{\partial y^{\ell _{1}}}}\cdots {\frac {\partial x^{j_{q}}}{\partial y^{\ell _{q}}}}}$$ The system of indexed functions ${\displaystyle T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n})}$ (one system for each choice of coordinate system) connected by transformations as above are the tensors in the definitions below.