Real analysis/Differentiation in Rn

Let $f$ be a real multivariate function defined on an open subset $\mathrm{\Omega }$ of ${ℝ}^n$

$f:\mathrm{\Omega }⟶ℝ$.

Then the partial derivative at some parameter $(x_1,...,x_n)$ with respect to the coordinate $x_i$ is defined as the following limit

$\underset{h\to 0}{\lim }\frac{f(x_1,\dots ,x_i+h,\dots ,x_n)-f(x_1,\dots ,x_i,\dots ,x_n)}{h}=\frac{\partial f}{\partial x_i}$.

$f$ is said to be differentiable at this parameter $(x_1,...,x_n)$ if the difference $f(x_1,...,x_i+h,...,x_n)-f(x_1,...,x_i,...,x_n)$ is equivalent up to first order in h to a linear form L (of h), that is

$f(x_1,...,x_i+h,...,x_n)-f(x_1,...,x_i,...,x_n)=L\times h+o(\Vert h\Vert ).$

The linear form L is then said to be the differential of $f$ at $(x_1,...,x_n)$, and is written as $Df{|}_{(x_1,\dots ,x_n)}$ or sometimes $\mathrm{d}f(x_1,\dots ,x_n)$.

In this case, where $f$ is differentiable at $(x_1,\dots ,x_n)$, by linearity we can write

$\mathrm{d}f=\frac{\partial f}{\partial x_1}\mathrm{d}{x}_1+\dots +\frac{\partial f}{\partial x_n}\mathrm{d}{x}_n$

$f$ is said to be continuously differentiable if its differential is defined at any parameter in its domain, and if the differential is varying continuously relative to the parameter $(x_1,...,x_n)$, that is if it coordinates (as a linear form) $\frac{\partial f}{\partial x_1}$ are varying continuously.

In case partial derivatives exists but $f$ is not differentiable, and sometimes not even continuous exempli gratia

$f:(x,y)\mapsto \frac{(xy{)}^2}{(x^2+y^2)}$

(and $f(0,0)=0$) we say that $f$ is separably differentiable.

The partial derivatives can be arranged in a matrix called the Jacobian Matrix:

$\left[\begin{array}{ccc}\frac{\partial y_1}{\partial x_1}& \cdots & \frac{\partial y_1}{\partial x_n}\\ ⋮& \ddots & ⋮\\ \frac{\partial y_m}{\partial x_1}& \cdots & \frac{\partial y_m}{\partial x_n}\end{array}\right].$