MY5400 MIDTERM REVIEW, FALL 2011, PATRICK K BOWEN
In tensor notation, indices with only one
occurrence are
free indices
, and those with
two occurrences are
dummy indices
.
More
than two are not allowed.
Free indices can
assume any value from 1
3, and dummy
indices must be summed over.
The
Kronecker Delta
has a simple form:
{
The Kronecker Delta will
switch the
indices
, in the term that comes after it (
will switch the existing index “i" with new
index “j” that
is:
The Kronecker Delta can be factored “out”
of any vector or matrix, say to turn
.
The
Permutation Symbol
,
, will assume
a value of 1, -1, or 0 depending on the index
order and value.
This index will assume
values:
{
The even/odd permutation in determined by
drawing a circle in which 1,2,3 are ordered
clockwise
.
If the index is, say, 2,3,1, then
the permutation will be
even
, because the
indices follow the clockwise arrangement.
If the index is 1,3,2, then the permutation
will be
odd
because the indices run counter
to the circle direction.
If the index is 1,3,1,
then the index will assume a value of zero.
Remember
that
indices
in
different
equations are
arbitrary
, that is:
Recall that the
directional cosine
of a
vector is the vector’s components divided
by the vector’s length:
⃑
||⃑ ||
⃑
√
The
dot product
is the amount of vector
⃑
in the direction of vector
⃑
, or:
⃑ ⃑ ||⃑ ||||⃑ ||
(
⃑⃑
⃑⃑ )
The
cross product
is the vector (
⃑⃑⃑
)
orthogonal to the plane formed by
⃑
and
⃑
in a right-handed manner:
⃑ ⃑
(
⃑⃑
⃑⃑ )
⃑⃑⃑
The
product
of
two permutation indices
can be written instead in terms of Kronecker
Deltas:
In this identity, the dummy index must be
written first.
The two permutation indices
can be rewritten as long as the index sense