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Path: blob/main/09aManifold_Tensors_onModules.ipynb
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9a. Tensors on modules
This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).
Reminder. Vectors and linear forms on modules
Recall that for free modules of finite rank (in notebook 6) we have introduced vectors and linear forms.
Example 9.1a
Define a free module of finite rank and some vector and linear form.
General linear forms on a 3-dimensional module can be defined as follows:
The linear form applied to a vector gives a scalar.
The value of a vector on a linear form is by definition equal to .
Now we are ready to define more general objects.
Tensors on modules
If is a vector space or a module, by a multilinear or more precisely -linear form we mean a function which is linear in each of its arguments, i.e., for
Assume that is a module.
Tensor module
Tensors of type (k,l) on are multilinear maps:
where denotes the dual module, i.e. the module of linear forms on (cf. notebook 6).
Since defines a linear form , the elements can be considered as elements of . On the other hand as the space of linear forms on is equal to . Thus, tensors generalize vectors and linear forms.
Example 9.2a
Let us show how to use tensor_module
method of SageMath Manifolds.
Module - of covariant tensors of rank
is the module of multilinear maps
For k=1 we obtain the module of linear forms or covectors on .
In we introduce the algebraic operations by
where , and .
For we define the tensor product by
ParseError: KaTeX parse error: Undefined control sequence: \label at position 107: …. , v_{k+l} ), \̲l̲a̲b̲e̲l̲{eq:tensor_prod…for .
In SageMath Manifolds the symbol of tensor product is simply .
Example 9.3a
Let us check the last formula in the case of two general tensors from and 2-dimensional module.
Tensor product has the following properties (proofs for modules where denotes a manifold are given in the next notebook)
ParseError: KaTeX parse error: Undefined control sequence: \label at position 149: …, \end{matrix} \̲l̲a̲b̲e̲l̲{eq:tensor_prod…for and for arbitrary covariant tensors (the addition is defined only for tensors of the same rank ).
Covariant tensors in components
One can check that if is a basis of the module and its dual basis (defined in notebook 6), then the elements
form a basis for and if we put
then
In fact, since then
Example 9.4a
Let us show the representation of a tensor in components.
General tensor of type (0,3) on a 2-dimensional module:
Tensor of type (0,3) on a 2-dimensional module with concrete components:
Check that the coefficient is equal to :
Module of contravariant tensors of rank
is the module of -linear forms:
In we introduce the module structure by
where , and .
For we define the tensor product by
for .
Contravariant tensors in components
One can check that if is a basis of the module and its dual basis then the elements
form a basis for and if we put
then
This follows from and for linear form and since we have
Example 9.5a
Let us show the representation of a tensor of type (3,0) in components.
General tensor of type (3,0) on a 2-dimensional module:
Tensor of type (3,0) on a 2-dimensional module with concrete components:
Check that the coefficient is equal to :
General tensors from in components
For we define the tensor product by
for and .
Generalizing the formulas (9.3a) and (9.5a) we obtain the following expression for the general tensor in components
ParseError: KaTeX parse error: Undefined control sequence: \label at position 147: …imes e^{j_l}. \̲l̲a̲b̲e̲l̲{}\tag{9.7a} \e…Very often the notation is used and then
To check that the elements ParseError: KaTeX parse error: Undefined control sequence: \label at position 96: …otimes e^{j_l} \̲l̲a̲b̲e̲l̲{} \tag{9.8a} \…
are linearly independent, assume that the linear combination
vanishes. If we apply this combination to we get .
Since as in previous cases we can check that elements of the type (9.8a) span , we have proved that these elements form a basis for .
Example 9.6a
Now we show the representation of a tensor of (2,2) type in components.
General tensor from on a 2-dimensional module ( components):
Here is a version with concrete components:
Check that the coefficient is equal to :
What's next?
Take a look at the notebook Tensors on .