3. The Cross Product

Initialization

The following values are taken as an example of a (red) basis frame :

Completely antisymmetric symbol

Remark : AntiSymmetric includes the factor 1/ n!

Definition of the operator CrossProductExpansion which calculate the ScalarTripleProduct of the basis set, and the Cross Product of two basis vectors.:

CrossProductExpansion[e_,e_][expr] expand the triple scalar products and the cross products expressed in a given basis e into LeviCivita symbols e. The considered space is three-dimensional.

In an orthonormal (blue-)basis, is simply a PermutationSymbol [e] and the cross product of two basis vectors is given by

Determinants

The two kinds of PermutationSymbolRule[ε] εddd[i,j,k] or εuuu[i,j,k] also allow to calculate determinants (in the base system) :

and of course we can verify the equivalence of the above detA with the build in Det function :

Determinants of the metric tensors g

Transformation tensors β

Example : Basis change from the black to the red basis (see Chapter 1)

Determinants of the transformation tensors β

Introduction of a determinant Δ of a transformation matrix β between the initial flavor, flavor1, to the final flavor, flavor2.

Note : I shall have to check the homogeneity of the notations between initial and final situations (see ToFlavor for instance, for which it is the reverse).

Introduction of the elementary volume formed by the basis set (for the red flavor) associated to g

From the relation between the two basis =

Basis Change of any tensor

Examples :

We can now extend the antisymmetric tensor ε ddd [ i , j , k ] to general coordinate systems and define a tensor eddd [ i , j , k ] and equivalently a tensor euuu [ i , j , k ] . These tensors are called permutation tensors or Levi-Civita tensors.

Levi-Civita tensors

We also introduce here, to be complete, the standard notation used in the Standard Packages.

We can define for convenience :

The "black" basis is orthonormal. But not the red's one :

Consider the three following vectors :

ScalarTripleProduct is defined in the package Calculus`VectorAnalysis` . We define here the operation MetricTripleProduct which groups all the operation performed above

Definition of an element of surface and of volume

The volume element defined by the three vectors {dr, ds, dt} is :

Its explicit value is :

In another reference frame this volume will remain unchanged. So we must be very careful that only the LeviCivita (e here) symbol is a tensor.

So, the correct transformation is :

Definition of an area element by the vector dA normal to it :

Note the possibility to use both FullForm (Tensor[...]) and Shortcut form to write an identity,

the two forms below being not litterally identical :

Pressure p of a fluid or a gas

Its product with an area dA,on which it acts,is a force dF normal to the area,i.e.it has the direction of dA,with components:

Weight of a volume element dV

The weight dW of a volume element dV is a vector (direction of the gravity field), the product of the volume and the specific weight ϒ :

hence

dW also remains invariant by any change of frame. Explicitly we have,

Created by Mathematica (November 27, 2007) |