# Implicit Function Theorem problem

1. May 1, 2015

### KevinMWHM

Part 1. If I want to solve the system;

u-v = (h-a)e^-s

w-u = (k-b)e^-t

ae^s = be^t

for a, b, u, in terms of the remaining variables using the implicit function theorem...

If I want to know when I can solve, can I just say f(a, b, u) can not = 0? And if I set a, b, u, = 0

Than I get k and h can not = 0.

Part 2. Calculate du/ds, da/ds, db/ds, by exhibiting a linear set of equations.

So for da/ds for example, solve equation 1 for a giving;

d(-e^s(u-v)+h)/ds?

I don’t need to solve the system.

2. May 1, 2015

### Ray Vickson

For Part 1, you don't NEED to solve the system, but the easiest solution is to just go ahead and solve it anyway! Using the notation $S = e^{-s}$ and $T = e^{-t}$, write your equations as
$$u-v = (h-a)S\\ (w-u)=(k-b) T \\ a/S = b/T$$
This is a simple linear system, from which you can easily find $a,b,u$ in terms of $v,w,h,k,S,T$.

For Part 2, you can also derive a simple linear system for $\partial a/ \partial s, \: \partial b / \partial s, \: \partial u / \partial s$ and get a solution in terms of $a,b,u,v,w,h,k,S,T$, using the definitions of $S,T$ in terms of $s,t$.

Last edited: May 1, 2015