multi-index derivative of a power
Proof. The proof follows from the corresponding rule for
the ordinary derivative; if i,k are in 0,1,2,…, then
didxixk={k!(k-i)!xk-iifi≤k,0otherwise. | (1) |
Suppose i=(i1,…,in), k=(k1,…,kn), and x=(x1,…,xn). Then we have that
∂ixk | = | ∂|i|∂xi11⋯∂xinnxk11⋯xknn | ||
= | ∂i1∂xi11xk11⋅⋯⋅∂in∂xinnxknn. |
For each r=1,…,n, the function xkrr only depends on xr.
In the above, each
partial differentiation ∂/∂xr therefore
reduces to the corresponding
ordinary differentiation d/dxr.
Hence, from equation 1, it follows that ∂ixk vanishes
if ir>kr for any r=1,…,n. If this is not the case, i.e.,
if i≤k as multi-indices, then for each r,
dirdxirrxkrr=kr!(kr-ir)!xkr-irr, |
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 |