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edited Apr 13, 2017 at 12:21

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Tensor : Multidimensional array :: Linear transformation : Matrix.

The short of it is, tensors and multidimensional arrays are different types of object; the first is a type of function, the second is a data structure suitable for representing a tensor in a coordinate system.

In the sense you're asking, mathematicians usually define a "tensor" to be a multilinear function: a function of several vector variables that is "linear in each variable separately". A "tensor field" is a "tensor-valued function".

In more detail, if $M$ is a smooth manifold, there are a tangent bundle $TM$, a cotangent bundle $T^{*}M$, and "tensor bundles" obtained by taking tensor products of these bundles. A tensor field on $M$ is a section of a tensor bundle. (The answers to the question question linked by John Mangual flesh out the details.)

Physicists call tensor fields "tensors". I'll (continue to) use mathematician-speak.

In a coordinate chart (i.e., a piece of $M$ diffeomorphic to an open set in a Euclidean space, together with a coordinate system), the tangent and cotangent bundles are trivialized by coordinate vectors and coordinate $1$-forms, and a tensor field is represented as a multidimensional array of functions.

The "transformation rules" for multidimensional arrays amount to the chain rule for vector fields and $1$-forms, i.e., to transition functions for tensors corresponding to changes of coordinates in $M$.

Tensor : Multidimensional array :: Linear transformation : Matrix.

The short of it is, tensors and multidimensional arrays are different types of object; the first is a type of function, the second is a data structure suitable for representing a tensor in a coordinate system.

In the sense you're asking, mathematicians usually define a "tensor" to be a multilinear function: a function of several vector variables that is "linear in each variable separately". A "tensor field" is a "tensor-valued function".

In more detail, if $M$ is a smooth manifold, there are a tangent bundle $TM$, a cotangent bundle $T^{*}M$, and "tensor bundles" obtained by taking tensor products of these bundles. A tensor field on $M$ is a section of a tensor bundle. (The answers to the question linked by John Mangual flesh out the details.)

Physicists call tensor fields "tensors". I'll (continue to) use mathematician-speak.

In a coordinate chart (i.e., a piece of $M$ diffeomorphic to an open set in a Euclidean space, together with a coordinate system), the tangent and cotangent bundles are trivialized by coordinate vectors and coordinate $1$-forms, and a tensor field is represented as a multidimensional array of functions.

The "transformation rules" for multidimensional arrays amount to the chain rule for vector fields and $1$-forms, i.e., to transition functions for tensors corresponding to changes of coordinates in $M$.

Tensor : Multidimensional array :: Linear transformation : Matrix.

The short of it is, tensors and multidimensional arrays are different types of object; the first is a type of function, the second is a data structure suitable for representing a tensor in a coordinate system.

In the sense you're asking, mathematicians usually define a "tensor" to be a multilinear function: a function of several vector variables that is "linear in each variable separately". A "tensor field" is a "tensor-valued function".

In more detail, if $M$ is a smooth manifold, there are a tangent bundle $TM$, a cotangent bundle $T^{*}M$, and "tensor bundles" obtained by taking tensor products of these bundles. A tensor field on $M$ is a section of a tensor bundle. (The answers to the question linked by John Mangual flesh out the details.)

Physicists call tensor fields "tensors". I'll (continue to) use mathematician-speak.

In a coordinate chart (i.e., a piece of $M$ diffeomorphic to an open set in a Euclidean space, together with a coordinate system), the tangent and cotangent bundles are trivialized by coordinate vectors and coordinate $1$-forms, and a tensor field is represented as a multidimensional array of functions.

The "transformation rules" for multidimensional arrays amount to the chain rule for vector fields and $1$-forms, i.e., to transition functions for tensors corresponding to changes of coordinates in $M$.

answered Feb 5, 2015 at 13:24

Andrew D. Hwang

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Tensor : Multidimensional array :: Linear transformation : Matrix.

The short of it is, tensors and multidimensional arrays are different types of object; the first is a type of function, the second is a data structure suitable for representing a tensor in a coordinate system.

In the sense you're asking, mathematicians usually define a "tensor" to be a multilinear function: a function of several vector variables that is "linear in each variable separately". A "tensor field" is a "tensor-valued function".

In more detail, if $M$ is a smooth manifold, there are a tangent bundle $TM$, a cotangent bundle $T^{*}M$, and "tensor bundles" obtained by taking tensor products of these bundles. A tensor field on $M$ is a section of a tensor bundle. (The answers to the question linked by John Mangual flesh out the details.)

Physicists call tensor fields "tensors". I'll (continue to) use mathematician-speak.

In a coordinate chart (i.e., a piece of $M$ diffeomorphic to an open set in a Euclidean space, together with a coordinate system), the tangent and cotangent bundles are trivialized by coordinate vectors and coordinate $1$-forms, and a tensor field is represented as a multidimensional array of functions.

The "transformation rules" for multidimensional arrays amount to the chain rule for vector fields and $1$-forms, i.e., to transition functions for tensors corresponding to changes of coordinates in $M$.