Recall that for a function [itex]f:A\subset \mathbb{R}^n\rightarrow \mathbb{R}^m[/itex], the derivative of f at x is defined as the linear map L:R^n-->R^m such that ||f(x+h)-f(x)-L(h)||=o(||h||)(adsbygoogle = window.adsbygoogle || []).push({});

if such a linear map exists.

We can show that for certain geometries of the set A, when the derivative exists, it is unique. This is the case for instance if A is a closed disk or more simply, if A is any open set. However, for certain sets, the derivative may exists and not be unique. For instance, if A is a singleton, then any linear map does the trick.

I arrive at a strange conclusion if I assume that a function [itex]f:A\subset \mathbb{R}^n\rightarrow \mathbb{R}^m[/itex] is differentiable at x with L_1, L_2 two distinct derivatives of f at x and follow the proof that the matrix representation of the derivative is the Jacobian matrix.

By definition, we have that [itex]||f(x+h)-f(x)-L_k(h)||=o(||h||) [/itex] (k=1,2), which implies by the sandwich theorem that [itex]|f_j(x+h)-f_j(x)-(L_k)_j(h)_|=o(||h||)[/itex], for each component j=1,...,m. In particular,

[tex]\lim_{t\rightarrow 0}\left|\frac{f_j(x_1,...,x_i+t,...,x_n)-f_j(x_1,...,x_n)-(L_k)_j(0,...,t,...,0)}{t} \right|=0 [/tex]

from which it follows by definition of the partial derivatives that

[tex]\frac{\partial f_j}{\partial x_i}(x)=(L_k)_j(e_i) [/tex]

But this is absurd since L_1 and L_2 are assumed distincts so there is at least a couple (i,j) for which [itex](L_1)_j(e_i)\neq (L_2)_j(e_i) [/itex], leading to the contradiction

[tex]\frac{\partial f_j}{\partial x_i}(x)\neq \frac{\partial f_j}{\partial x_i}(x) [/tex]

Does anyone sees where I'm mistaken in my reasoning??

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Something strange about uniqueness of the derivative in higher dimensions

**Physics Forums | Science Articles, Homework Help, Discussion**