Implicit Function Theorem problem

[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.

 

[Ray Vickson]

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.

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
$$u-v = (h-a)S\\<br /> (w-u)=(k-b) T \\<br /> a/S = b/T$$
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 $s,t$.