Thread: What is a tensor?

Can someone please explain to me in layman terms what a tensor is?

From what I understand it is like a vector but instead of having 1 value and a direction, it can have values ( a matrix ) for multiple directions ( dimensions ? ).

I am still confused as to exactly what it is can someone help?

A tensor is really just a mathematical device used to describe a quantity at a particular point in space. The description of the quantity could be simply a number, in the case of a scalar value (tensor rank 0), a vector (tensor rank 1), or something more complex in the case of higher-rank tensors. The complexity of the tensor used just depends on the level of complexity of the quantity being described.

Tensors are most easily understood by discussing the progression of tensor 'ranks'. Generally when one talks about tensors, though, one is referring to tensors of rank two or higher.

A scalar quantity is simply a number -- it has only magnitude. A scalar can be designated a tensor of rank zero.

A vector quantity has magnitude and direction. In two dimensional space, for example, it has x- and y-components, and in three dimensional space, it has 3 components. Vectors can have any number of dimensions. These components are commonly shown in a one dimensional column matrix.

v =
a
b
c
.
.
n

A vector can be designated a tensor of rank one.

A tensor of rank two is represented by a matrix:

T2 =

aa ab ac ... an
ba bb bc ... bn
ca cb cc ... cn
. . . .
. . . .
ma mb mc ... mn

A rank-three tensor is represented with a cubic matrix, with components coming out of your computer screen.

(Tensors with rank higher than three are harder to represent; the most common notation is known as Einsteinian Notation, which makes use of indices. Note that a rank-four tensor is represented by a hyper-rectangular matrix. )

Visualizing tensors is very difficult, akin to visualizing hyperdimensional objects. One way to think of tensors is in terms of fields.

A scalar field is created by simply assigning scalar quantities (numbers) to each point in space. Think of temperature -- each point in the room has a different temperature.

A vector field is created by assigning vectors to each point. An electric field is an example -- a test charge placed at a point in space will move at a certain speed and direction as represented by the vector at that point.

A tensor field has a tensor corresponding to each point space. An example is the stress on a material, such as a construction beam in a bridge.

Other examples of tensors include the strain tensor, the conductivity tensor, and the inertia tensor.

I'll attempt it. A tensor is the mathematical entity for an interaval in a manifold, represented in an invariant fashion. If one puts a manifold thru a coordinate conversion, the coordinates and vectors between them all change. The tensor does not, and hence makes a better represention of physical time-space intervals. If one thinks in tensors, relativity becomes simpler. Physicists use them al the time. And aside from some old memories of their look in symbols, that about covers what I can tell you on tensors. Hope it helps.

Just started reading "General Relativity" by Hans Stephani--almost stuck already on page 3, but here is an example of the use of a tensor in that book. The matrix representation is mine. You could copy and paste this equation into OpenOffice.org equation editor to see it more clearly.


ds^2 = g_{%alpha %beta}(x^%nu )dx^%alpha dx^%beta newline
{}=
left( matrix{g_rr(r,%theta,%phi) # g_{r%theta}(r,%theta,%phi) # g_{r%phi}(r,%theta,%phi)
## g_{%theta r}(r,%theta,%phi) # g_{%theta%theta}(r,%theta,%phi) # g_{%theta%phi}(r,%theta,%phi)
## g_{%phi r}(r,%theta,%phi) # g_{%phi%theta}(r,%theta,%phi) # g_{%phi%phi}(r,%theta,%phi)} right)
left( matrix{dr ## d%theta ## d%phi} right)
cdot
left( matrix{dr ## d%theta ## d%phi} right)newline
{} =
left( matrix {1 # 0 # 0 ## 0 # r^2 # 0 ## 0 # 0 # r^2 sin^2 %theta}right)
left( matrix{dr ## d%theta ## d%phi} right)
cdot left( matrix{dr ## d%theta ## d%phi} right) newline
{} = dr^2 + r^2 d%theta + r^2 sin^2%theta d%phi^2

I think here is an interesting application of tensors in Posts 2, 7, 8, 9.

Using a Matrix Exponent converts a simple looking tensor into a rotation or Lorentz Boost.

By far the best introduction to tensors I've yet seen

Leonard Susskind's General Relativity Lecture 4

Originally Posted by SpaceCadet

A tensor is really just a mathematical device used to describe a quantity at a particular point in space. The description of the quantity could be simply a number, in the case of a scalar value (tensor rank 0), a vector (tensor rank 1), or something more complex in the case of higher-rank tensors. The complexity of the tensor used just depends on the level of complexity of the quantity being described.

Tensors are most easily understood by discussing the progression of tensor 'ranks'. Generally when one talks about tensors, though, one is referring to tensors of rank two or higher.

[...]

Other examples of tensors include the strain tensor, the conductivity tensor, and the inertia tensor.

So, would it be proper to say that tensors are properties associated with an point in space? And that each point may have one or more of these properties at the same time?