Maths - Tensors

Tensors allow us to define fields and transformations in a way that is independent of coordinate systems. We can use curvilinear coordinate systems and tensors allowed Einstein to formulate special relativity.

Tensors are also a genralisation of scalars, vectors, matrices and hypermatricies allowing us to define these linear quantities with any number of indicies.

As usual I'll start with the different ways to approach the subject. It often seems to be the case with these powerful mathematical concepts that we can think of them in different ways.

In order to be a tensor, a quantity needs to be a linear function. It also needs to change, according to certain rules, in response to a change of frame.

In this subject, even more than most, there seem to be different definitions of the concepts. If you are reading different articles and books it is important to compensate for the different ways that things are defined. One of the differences is whether a tensor is defined as a mapping, such as when mapping from one coordinate system to another, this is usually the approach of mathematicians. Alternatively a tensor can be defined as a particular field such as how an electromagnetic field varies in space, this is the approach usually taken by physicists.

Another difference is whether tensors are defined in terms of incidies, a coordinate based approach or whether an approach independent of coordinates is used. The latter approach is considered more modern but it may be harder to learn and may not handle all situations. These pages use the first approach. The Geometric Algebra pages also use a coordinate free approach and, in some ways, they are complimentary to this subject.

Multi-linear mappings

Tensors can be used to represent linear (or multi-linear) mappings (such as transformations) in 'n' dimensions. Physicists also use tensor fields to map each point in space (manifold) to a tensor (such as scalar or vector) value.

A simple example of a linear transform might be:

x' = 3*x + 4*y
y' = 5*x + 3*y

Which could, of course, be represented by a matrix equation. In geometric terms this could be interpreted as mapping points (x,y) to another set of points (x',y'). Alternatively it could be interpreted as representing the same point in different coordinate systems.

This subject is often concerned with mapping between different coordinate systems and also the properties which are independent of coordinate systems.

Although tensors can represent general (multi-) linear mappings most of the complexity and power of the subject seems to come when we constrain the tensors to represent various types of symmetry.

There are two approaches to tensors:

Classical Approach

In classical terms concept of tensors is defined by an array of components in such a way that it can be used in any number of dimensions and any 'rank'. Where:

rank
representation
0 scalar
1 vector
2 matrix (n * vectors)
3 hyper-matrix

We tend to use two types of indicies to distinguish between contravariant and covariant indicies.

Grade 1 Tensors - Vectors

In order to give a vector a definate value we need to assign numerical values to it. To do this it can be represented by a linear combination of basis vectors, so vector 'a' could be represented by:

a = a1 e1 + a2 e2 + a3 e3 ...

simarly vector 'b', in the same coordinate system, could be represented by:

b = b 1 e1 + b 2 e2 + b 3 e3 ...

where:

Grade 2 Tensors - Matricies

 

Terminology

Dual vectors

- reflects the relationship between row vectors and column vectors

- when multiplied by its corresponding vector, generate a real number, by systematically multiplying each component from the dual vector and the vector together and summing the total.  If the space a vector lives in is shrunk, a contravariant vector shrinks, but a covariant vector gets larger.

In the way that tensors are 'n' dimensional and also in the way that they are developed and used by physicists they are related to spinors.

There also seems to be a relationship to Clifford / Geometric algebra, if only the terminology used. Is this just a coincidence? or is there some higher truth here?

 


Tensors

A set of components that obeys some transformation law in n-dimentional space.

A set of components - each of which is defined as a function of position in some co-ordinate system - which obeys some given transformation law.

They are arrays of numbers, or functions, that transform according to certain rules under a change of coordinates.

Tensors were originally used in physics

A tensor may be defined at a single point or collection of isolated points of space (or space-time), or it may be defined over a continuum of points. In the latter case, the elements of the tensor are functions of position and the tensor forms what is called a tensor field. This just means that the tensor is defined at every point within a region of space (or space-time), rather than just at a point, or collection of isolated points.

Order (or rank) Mathmatical Representation Physics Application
0 Scalar An example of a scalar field would be the density of a fluid as a function of position
1 Vector An example of a vector field is provided by the description of an electric field in space.
2 Matrix inertia matrix
3    
4   Riemann curvature tensor

Note, not all transforms and not all matricies are tensors. To be a tensor they must transform according to certain rules under a change of coordinates.

For instance objects called spinors. Spinors differ from tensors in how the values of their elements change under coordinate transformations. For example, the values of the components of all tensors, regardless of order, return to their original values under a 360-degree rotation of the coordinate system in which the components are described. By contrast, the components of spinors change sign under a 360-degree rotation, and do not return to their original values until the describing coordinate system has been rotated through two full rotations = 720-degrees.

The tensor notion is quite general, and applies to all of the above examples; scalars and vectors are special kinds of tensors. The feature that distinguishes a scalar from a vector, and distinguishes both of those from a more general tensor quantity is the number of indices in the representing array. This number is called the rank of a tensor. Thus, scalars are rank zero tensors (no indices at all), and vectors are rank one tensors.

A tensor may vary covariantly with respect to some variables and contravariantly with respect to others when the coordinate axes are rotated.

Contravariant Tensors

A tensor may be Contravariant or Covariant depending on how the corresponding numbers transform relative to a change in the frame of reference

Contravariant indices are written as superscripts

Covariant Tensors

covariant indices are written as subscripts

Valence of a tensor

The valence of a tensor is the pair $ (p,q)$, where $ p$is the number contravariant and $ q$the number of covariant indices, respectively.

Tensor Notation Summary

notation example meaning
subscript e1 coordinate basis
superscript x1 coordinate value
Einstein Summation Convention eixi=e1x1+e2x2+e3x3=∑eixi When the same index appears twice in an expression, once raised and once lowered, a sum is implied.
partial derivatives a=∂/∂a  

More on notation here.

 


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see also:

Tensors - Geometry

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Terminology and Notation

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