D.1 Generic vector identities
The rules of engagement are as follows:
- The Cartesian axes are numbered using an index , with
1, 2, and 3 for , , and
- Also, indicates the coordinate in the direction,
, , or .
- Derivatives with respect to a coordinate are indicated by
a simple subscript .
- If the quantity being differentiated is a vector, a comma is
used to separate the vector index from differentiation ones.
- Index is the number immediately following in the
cyclic sequence ...123123...and is the number
immediately preceding .
The first identity to be derived involves the “vectorial triple
To do so, first note that the -th component of
is given by
Repeating the rule, the -th component of
That writes out to
since the first and fourth terms cancel each other. The first three
terms can be recognized as the -th component of
and the last three as the -th
component of .
A second identity to be derived involves the “scalar triple
This is easiest derived from simply writing it out. The left hand
while the right hand side is
Inspection shows it to be the same terms in a different order. Note
that since no order changes occur, the three vectors may be