Tensorial Analysis and Continuum Mechanics

Jean-François Gouyet
LPMC, Ecole Polytechnique
Palaiseau, France, 2003

This first chapter is intended for recalling some general features of Tensors and Vectors, in the spirit of the present notebook on Continuum Mechanics, and of the notebook for tensor calculus "Tensorial".

Initial rules and definitions (Automatically Initialized)

The flavors allow to distinguish between various basis.
The change of  flavor is given by ToFlavor[new, old]. ToFlavor[bar]  is equivalent to  ToFlavor[bar, Identity].

1.   Vector and Tensors

Dot Product, Vector Components

Canonical basis

The commonly used basis is the orthonormal basis. In the following, we will generally use black indices with base indices {1,2,3} defined in the initial rules. The dimension of the considered space is NDim

The geometry of the vector space is completely defined for a set of  basis vectors (e) and the corresponding metric tensor (g), we define the values of the metric tensor. The basis vectors of the orthonormal basis are

The metric tensor, whose components are the scalar product of the basis vectors :

and for an orthonormal basis :

If we consider now a new basis, characterized by a red flavor :

again, the characteristics of this red basis is completely determined by its metric tensor

There are many ways to build . Here we start from the components of in the canonical (black) basis

which corresponds equivalently to the rules (see below "Basis Change" )

The corresponding values (RedMetric) for the metric tensor are obtained either directly from the components of the red basis,

There is an easy way to calculate the red reciprocal basis vectors. Just take the rows of  Transpose[Inverse[RedBasisVectors]]:

The red basis will be used as numerical example in the following developments.

The scalar products can of course be calculated following various ways.

We can expand the above using LinearBreakout.

EvaluateDotProducts evaluates the dot product without references to the tensor values of e.

Basis change

Change from black basis to red basis will be defined by a basis change denoted (for instance) or . Due to the symmetry between red and black, it is conventional in Tensorial to use the notation .

The operation KroneckerAbsorb[β] is appropriate here to show the basis change

The coordinate transformation from black to red basis are now completely defined : it is the matrix of the components of in the black basis,

The above basis change is precisely the one considered above in the rule called MixedBasisInnerProductsRule.

It can be shown easily that is also given by   ∂ /∂ which is of course not a delta function.
The present formulation leads to

Finally notice the possibilities of successive basis changes

Created by Mathematica  (November 27, 2007)