multi-index derivative of a power
Proof. The proof follows from the corresponding rule for
the ordinary derivative; if are in , then
(1) |
Suppose , , and . Then we have that
For each , the function only depends on .
In the above, each
partial differentiation therefore
reduces to the corresponding
ordinary differentiation .
Hence, from equation 1, it follows that vanishes
if for any . If this is not the case, i.e.,
if as multi-indices, then for each ,
and the theorem follows.
Title | multi-index derivative of a power |
---|---|
Canonical name | MultiindexDerivativeOfAPower |
Date of creation | 2013-03-22 13:42:01 |
Last modified on | 2013-03-22 13:42:01 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 9 |
Author | matte (1858) |
Entry type | Theorem |
Classification | msc 05-00 |