Re: Inverse Function

*To*: mathgroup at smc.vnet.net*Subject*: [mg121472] Re: Inverse Function*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Fri, 16 Sep 2011 05:48:42 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201109150840.EAA29206@smc.vnet.net>

On 09/15/2011 03:40 AM, Giuseppe Benedetti wrote: > Hello to everyone, > I have a function of two variables g[k,s] and I want to compute (symbolically) its inverse with respect to the first variable k (ie keeping s constant). I am therefore looking for a function G[g,s], which I then need to differentiate (twice) with respect to the second variable s. Does anybody know how to do it??? > Thank you in advance. > > Giuseppe. > Start with the identity G[g[k, s], s] - k == 0 Differentiate once and twice with respect to k and likewise with respect to x. All these derivatives are zero. Use that to solve for D[0,2][G] in terms of derivatives of g. In[21]:= Derivative[0, 2][G][g[k, s], s] /. Solve[{D[G[g[k, s], s] - k, k] == 0, D[G[g[k, s], s] - k, {k, 2}] == 0, D[G[g[k, s], s] - k, s] == 0, D[G[g[k, s], s] - k, {s, 2}] == 0}, {Derivative[2, 0][G][g[k, s], s], Derivative[0, 2][G][g[k, s], s], Derivative[1, 0][G][g[k, s], s], Derivative[0, 1][G][g[k, s], s]}] Out[21]= {-2*Derivative[0, 1][g][k, s]* Derivative[1, 1][G][g[k, s], s] - (Derivative[0, 2][g][k, s] - (Derivative[0, 1][g][k, s]^2* Derivative[2, 0][g][k, s])/Derivative[1, 0][g][k, s]^2)/ Derivative[1, 0][g][k, s]} In principle I would have thought we'd also need the second order mixed derivative, but it seems to have worked without that. Daniel Lichtblau Wolfram Research

**References**:**Inverse Function***From:*Giuseppe Benedetti <beppeben@libero.it>