Tensor/Old

Tensors are quantities that describe a transformation between coordinate systems. They were resurrected from obscurity by Einstein in order to formulate General relativity in such a way that the physical laws that were described were independent of the coordinate system chosen.

See Tensor/Alternate and Tensor for alternative treatments of tensors.

Throughout we will use the Einstein summation convention in which repeated indices imply a sum over components. Capitalized indices will likewise be summed. The reader should already be familiar with vector spaces and their properties.

An example in 3 space follows:

Let A be a vector and {E₁,E₂,E₃} be a set of linearly independent vectors that span a 3 dimensional space (all 3 dimensional spaces are spanned if any is as a result of the properties of a vector space).

A=A¹E₁ + A²E₂ + A³E₃

This is rather painful to do as we have to find the projection of A into all of these vectors and solve n equations in n unknowns. We can solve this in a computationally (and eventually much clearer fashion) by finding reciprocal vectors for our set of vectors {E_i} with i from 1 to n.

The reciprocal vectors can be found as such. We take

E¹ = E₂ cross E₃ * 1/V where V is the volume of the parallepiped defined by the three E vectors. This volume can be found by finding the triple product (which can be written as E₁ dot E₂ cross E₃, or by finding the determinant of the matrix of the three vectors |E_IJ| with I,J = 1,2,3 (The determinant actually finds a volume!).

Excercise: What can you say about three vectors where the triple product equals 0 (or alternatively has a matrix determinant equal to 0)?

E² is likewise equal to E₃ cross E₁ * 1/V and
E³ = E₁ cross E₂ * 1/V.
This can be generalized to an nth dimensional space, keeping in mind that the operations must be done in clockwise cyclic[?] order.

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In linear algebra and abstract algebra, the tensor product of two vector spaces V and W is a vector space T, together with a bilinear operator B: V x W -> T, such that for every bilinear operator C: V x W -> X there exists a unique linear operator L: T -> X with C = L o B

The tensor product is up to isomorphism uniquely specified by this requirement. Using a rather involved construction, one can show that the tensor product for any two vector spaces exists. The space T is generated by the image of B.

A tensor product can be defined similarly between a right module M and a left module N over the same ring R -- the only thing that changes is the definition of the bilinear operator. If M is a right module over a ring S, then the tensor has a natural right module structure over S. Similarly, if N is a left module over T, then the tensor has a left module structure over T. R must be specified, and matters for the definition of the tensor.

For free modules[?], the dimension of the Tensor is the product of the dimensions of the modules. It is possible to generalize the definition into a tensor of any number of spaces -- but needless. From universality, the tensor is associative. So the tensor of V, W and X is the tensor of V and the tensor of W and X.

See also : Tensor