A. Vector and Tensor Notation
FOR THE SAKE of convenience and to better describe the structure of the moment
equations, a few symbols will be introduced. Let
,
,
,
be vectors in a (real) space of dimension .
|
(A.1) |
Then, the symbol
|
(A.2) |
represents the set of all pairs of the components , of two vectors, where the
indices and range independently over all the admissible values. Hence,
eqn. (A.2) defines a tensor, whose elements are the quantities
. The
definition can be extended to the triplets
|
(A.3) |
and so on. The following notation is adopted [4]:
where the numerical subscript refers to the rank of the tensor. The rank equals the
number of vectors defining it. For instance in eqn. (A.2) and in
eqn. (A.3). Hence the tensor has in general different elements. From
eqn. (A.2) it follows, that
.
In the special case
, however, the tensor is
completely symmetric and any permutation of indices in an element leaves the tensor unchanged.
In this case, the number of elements which are different from each other is smaller than
, and the tensor is invariant when the order of the vectors is changed.
It is useful to expand the tensor product to scalars as well, and to identify it with the
normal product,
|
(A.6) |
In the following, the scalar product will be used as well. If necessary for the sake of
clarity, it will be expressed explicitly like, for example in
In general, the scalar product depends on the order of the tensor, and becomes invariant only
when all tensors involved are symmetric.
Let now , , , be spatial coordinates, which will also be
referred to as
, and let the tensors be functions of
as well.
Introducing the nabla operator
|
(A.10) |
the divergence of a rank tensor is defined as
|
(A.11) |
where it is intended, that the multiplication by
is symbolic. It is seen, that
is a tensor of rank . Consistently, the gradient of a rank
tensor is defined as the rank tensor
|
(A.12) |
From eqn. (A.11) and eqn. (A.12) useful identities can be derived. For any
, the following relationships hold:
Another definition which will be used is the statistical average of a tensor
|
(A.15) |
where
is the definition domain of
in the -space and is a
distribution function.
M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF