Differentiable function of several variables

Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$. The function $f$ is differentiable at $a$ when there is a linear map $L \colon \mathbb{R}^n \to \mathbb{R}^m$ such that $$\lim_{h \to 0} \frac{\|f(a + h) - f(a) - L h\|}{\|h\|} = 0.$$ Here the limit is over $h \ne 0$ with $a + h \in U$, and $\|\cdot\|$ is the Euclidean norm. The map $L$ is then unique. It is the derivative $Df(a)$, also called the total derivative of $f$ at $a$.

Remarks

This is the Fréchet derivative for the spaces $\mathbb{R}^n$ and $\mathbb{R}^m$, where each linear map is continuous. A function that is differentiable at $a$ is continuous at $a$, and all its directional derivatives exist at $a$. The converse fails: a function can have all partial derivatives at a point and not be differentiable there, see this counterexample. For $n = m = 1$, the definition is the usual one, with $L h = f'(a) h$.

Explanations

L1 human

A differentiable function looks flat when you zoom in on its graph near the point. For $f(x, y) = x^2 + y^2$ at $(1, 1)$, the change is $f(1 + h_1, 1 + h_2) - f(1, 1) = 2 h_1 + 2 h_2 + h_1^2 + h_2^2$. The linear part is $L h = 2 h_1 + 2 h_2$, and the rest $h_1^2 + h_2^2 = \|h\|^2$ is small compared with $\|h\|$.