The components of a tensor with respect to a basis is an indexed array. Theorder of a tensor is the number of indices needed. Some texts may refer to the tensor order using the termdegree orrank.
The rank of a tensor is the minimum number of rank-one tensor that must be summed to obtain the tensor. A rank-one tensor may be defined as expressible as the outer product of the number of nonzero vectors needed to obtain the correct order.
This notation is based on the understanding that whenever a multidimensional array contains a repeated index letter, the default interpretation is that the product is summed over all permitted values of the index. For example, ifaij is a matrix, then under this conventionaii is itstrace. The Einstein convention is widely used in physics and engineering texts, to the extent that if summation is not to be applied, it is normal to note that explicitly.
The classical interpretation is by components. For example, in the differential formaidxi thecomponentsai are a covariant vector. That means all indices are lower; contravariant means all indices are upper.
This refers to any tensor that has both lower and upper indices.
Cartesian tensor
Cartesian tensors are widely used in various branches ofcontinuum mechanics, such asfluid mechanics andelasticity. In classicalcontinuum mechanics, the space of interest is usually 3-dimensionalEuclidean space, as is the tangent space at each point. If we restrict the local coordinates to beCartesian coordinates with the same scale centered at the point of interest, themetric tensor is theKronecker delta. This means that there is no need to distinguish covariant and contravariant components, and furthermore there is no need to distinguish tensors andtensor densities. AllCartesian-tensor indices are written as subscripts.Cartesian tensors achieve considerable computational simplification at the cost of generality and of some theoretical insight.
This avoids the initial use of components, and is distinguished by the explicit use of the tensor product symbol.
Tensor product
Ifv andw are vectors invector spacesV andW respectively, then
is a tensor in
That is, the ⊗ operation is abinary operation, but it takes values into a fresh space (it is in a strong senseexternal). The ⊗ operation is abilinear map; but no other conditions are applied to it.
Pure tensor
A pure tensor ofV ⊗W is one that is of the formv ⊗w.
It could be written dyadicallyaibj, or more accuratelyaibjei ⊗fj, where theei are a basis forV and thefj a basis forW. Therefore, unlessV andW have the same dimension, the array of components need not be square. Suchpure tensors are not generic: if bothV andW have dimension greater than 1, there will be tensors that are not pure, and there will be non-linear conditions for a tensor to satisfy, to be pure. For more seeSegre embedding.
Tensor algebra
In the tensor algebraT(V) of a vector spaceV, the operation becomes a normal (internal)binary operation. A consequence is thatT(V) has infinite dimension unlessV has dimension 0. Thefree algebra on a setX is for practical purposes the same as the tensor algebra on the vector space withX as basis.
Hodge star operator
Exterior power
Thewedge product is the anti-symmetric form of the ⊗ operation. The quotient space ofT(V) on which it becomes an internal operation is theexterior algebra ofV; it is agraded algebra, with the graded piece of weightk being called thek-thexterior power ofV.
