# It is correct to say that a tensor is simply a multidimensional array of related quantities? But what about a tensor as a transformation?

It is correct to say that a tensor is simply a multidimensional array of related quantities?

More specifically a tensor is a collection or tuples of vectors where every vector in the tuple represent a different type of information but the components of the different vectors depend of each other.

I said this because of the following sentence I read in Wikipedia:

"at the end of the 19th century, the electromagnetic field was understood as a collection of two vector fields in space. nowadays, one recognizes this as a single antisymmetric 2nd-rank tensor field in spacetime"

I understand that electric and magnetic fields are grouped into a tensor because the components the magnetic field depend of some maner of the components of the electric field and viceversa.

But what about tensors as transformation objects? if you have two vectors (not tuples of vectors) then the transformation is simply a matrix (or rank 2 tensor), but what is the necessity for tensors of rank bigger than 2? Is to transform tuples of vectors?

I only have a background in physics so please try to answer in the most simple terms.

• Be aware that the term tensor is used on data analysis and machine learning for a multidimensional array without the transformation properties that's mathematical physics tensor has. – Brian Borchers Jul 4 '18 at 16:44
• rank one tensors are vectors and linear functionals (covectors). Rank two tensors are tensor products of rank one tensors... and so on – janmarqz Jul 5 '18 at 13:30
• Hi janmarqz, and what is the practical need (in physics, engineering, etc) of having tensors of rank bigger than 2? – Sirius Fuenmayor Jul 5 '18 at 17:13
• Can you say that as a matrix resulting from the tensor product of two vectors is the relation between this 2 vectors, a tensor of rank 3 is the object that defines the relation between two matrices or rank 2 tensors? – Sirius Fuenmayor Jul 5 '18 at 17:20
• with a tensor of rank four you can describe curvature of space-times, do special and general relativity, classify solutions of several Einstein Field Eqns, build physical theories on multiple dimensions..., among a plethora of phenomena. For the 1st question. – janmarqz Jul 5 '18 at 23:05

A tensor is a "multidimensional array of quantities" in the same sense that a vector is an "array of quantities" -- that is, you're giving a description of the coordinate representation of said objects relative to a chosen basis.

The example of electromagnetism is just an example of a phenomenon where, when we move to a unified notion of space-time, that some classical quantities turn out to be the time and space components of some other quantity. Another example is how energy and classical 3-dimensional momentum turn out to be the time and space components of the special relativity momentum 4-vector

There is a rank 2 tensor field which breaks apart into components:

• The (time, time) component is zero
• The (time, space) component is the electric field vector
• The (space, time) component is the negative of the electric field vector
• The (space, space) component is the magnetic field pseudovector

(in coordinates, the magnetic field really ought to be an antisymmetric two-dimensional array of scalars — but as history goes we first discovered how to represent such things via pseudovector, and that description has stuck in classical physics)

• I'd add that it's like saying "a linear transformation is just a matrix". – Jackozee Hakkiuz Jul 4 '18 at 17:47

It's more common to refer to transformation properties:

A tensor of rank $(p, q),$ i.e. with $p$ upper indices and $q$ lower indices, on an $n$-dimensional space, is an object with $n^{pq}$ components, which transform using $p$ factors of $J$ and $q$ factors of $J^{-1},$ each acting on a different index, where $J = \partial x_{\text{new}} / \partial x_{\text{old}}$ is the Jacobian of the coordinate change.