Total differntial

Definition

Let a function $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$. The total derivative of $f$ at a point $a$, $d{f}_a:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is defined as $d{f}_a={\sum }_{i=1}^n\frac{\partial f}{\partial x_i}(a)\cdot d{x}_i$

The best linear approximation near the point of the function.

Examples

$u=u(x,y)=u=u(x(t),y(t))=u(t)$ by Chain Rule, $\frac{du}{dt}=\frac{\partial u}{\partial x}\frac{dx}{dt}+\frac{\partial u}{\partial y}\frac{dy}{dt}\Rightarrow du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy$