multi-index derivative of a power

Theorem If i,k are multi-indices in n, and x=(x1,,xn), then


Proof. The proof follows from the corresponding rule for the ordinary derivativePlanetmathPlanetmath; if i,k are in 0,1,2,, then

didxixk={k!(k-i)!xk-iifik,0otherwise. (1)

Suppose i=(i1,,in), k=(k1,,kn), and x=(x1,,xn). Then we have that

ixk = |i|x1i1xninx1k1xnkn
= i1x1i1x1k1inxninxnkn.

For each r=1,,n, the function xrkr only depends on xr. In the above, each partial differentiation /xr therefore reduces to the corresponding ordinary differentiationMathworldPlanetmath 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 ik as multi-indices, then for each r,


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