Definition:Tensor

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Definition

A tensor can be informally defined as an abstract object with a set of components which are functions of position in $n$-dimensional space.


Let points have $n$ coordinates in a certain Cartesian coordinate system:

$x^i = \tuple {x^1, x^2, \ldots, x^n}$

Let $x^i$ have corresponding coordinates in a different Cartesian coordinate system:

$\overline x^i = \tuple {\overline x^1, \overline x^2, \ldots, \overline x^n}$

A set of $n$ components, denoted $A^i$, that are functions of the $n$ coordinates $x^i$ will become a set of $n$ components, denoted $\overline A^i$, that are functions of the $n$ coordinates $\overline x^i$ on a change of coordinates from the first to the second system.


Similarly, $A^{ij}$, $A^{ijk}$, and so on, denote sets of $n^2$, $n^3$ components, and so on.


Covariant Tensor

Let $F$ be a tensor of type $\tuple {k, 0}$:

$\ds F : \underbrace{V \times \ldots \times V}_{\text{$k$ times}} \to \R$


Then $F$ is called a covariant $k$-tensor (on $V$).


Contravariant Tensor

Let $F$ be a tensor of type $\tuple {0, l}$:

$\ds F : \underbrace { {V^*} \times \ldots \times {V^*} }_{\text {$l$ times} } \to \R$

Then $F$ is called a contravariant $l$-tensor (on $V$).


Mixed Tensor

Let $V$ and $V^*$ be a vector space and its dual.


Then a (mixed) tensor $F$ of type $\tuple {k, l}$ is a multilinear map such that:

$F : \underbrace {V \times \ldots \times V}_{\text {$k$ times} } \times \underbrace { {V^*} \times \ldots \times {V^*} }_{\text {$l$ times} } \to \R$


Notation

The components of a tensor are by convention identified by means of superscripts, rather than the subscripts that are general in other branches of mathematics.

Thus we have such constructs as:

$A^i := \set {A^1, A^2, \ldots, A^n}$

and:

$x^i := \set {x^1, x^2, \ldots, x^n}$

instead of what would be more usual in the language of vector spaces:

$\mathbf A := \set {A_1, A_2, \ldots, A_n}$

and:

$\mathbf x := \set {x_1, x_2, \ldots, x_n}$


Hence such suffixes are not treated as exponents in tensor notation.

This may cause pain and suffering in the form of confusion for new students, but it is what it is and it is probably unproductive to fight against it.


Just to add to the confusion:

contravariant tensors use superscripts:
$A^i := \set {A^1, A^2, \ldots, A^n}$
covariant tensors use subscripts:
$A_i := \set {A_1, A_2, \ldots, A_n}$
mixed tensors use both subscripts and superscripts:
$A^i_j$


Examples

Rank $0$

A tensor of rank $0$ is a scalar.


Rank $1$

A tensor of rank $1$ is a vector.


Also see

  • Results about tensors can be found here.


Sources