linguistic:translation, quality:correction, structural:summarization
The total derivative as a linear map

Let U ⊆ R n {\displaystyle U\subseteq \mathbb {R} ^{n}} be an open subset. Then a function f: U → R m {\displaystyle f:U\to \mathbb {R} ^{m}} is said to be (totally) differentiable at a point a ∈ U {\displaystyle a\in U} if there exists a linear transformation d f a: R n → R m {\displaystyle df_{a}:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} such that

The linear map d f a {\displaystyle df_{a}} is called the (total) derivative or (total) differential of f {\displaystyle f} at a {\displaystyle a}. Other notations for the total derivative include D a f {\displaystyle D_{a}f} and D f (a) {\displaystyle Df(a)}. A function is (totally) differentiable if its total derivative exists at every point in its domain.

Conceptually, the definition of the total derivative expresses the idea that d f a {\displaystyle df_{a}} is the best linear approximation to f {\displaystyle f} at the point a {\displaystyle a}. This can be made precise by quantifying the error in the linear approximation determined by d f a {\displaystyle df_{a}}. To do so, write

where ε (h) {\displaystyle \varepsilon (h)} equals the error in the approximation. To say that the derivative of f {\displaystyle f} at a {\displaystyle a} is d f a {\displaystyle df_{a}} is equivalent to the statement

where o {\displaystyle o} is little-o notation and indicates that ε (h) {\displaystyle \varepsilon (h)} is much smaller than ‖ h ‖ {\displaystyle \lVert h\rVert } as h → 0 {\displaystyle h\to 0}. The total derivative d f a {\displaystyle df_{a}} is the unique linear transformation for which the error term is this small, and this is the sense in which it is the best linear approximation to f {\displaystyle f}.

The function f {\displaystyle f} is differentiable if and only if each of its components f i: U → R {\displaystyle f_{i}\colon U\to \mathbb {R} } is differentiable, so when studying total derivatives, it is often possible to work one coordinate at a time in the codomain. However, the same is not true of the coordinates in the domain. It is true that if f {\displaystyle f} is differentiable at a {\displaystyle a}, then each partial derivative ∂ f / ∂ x i {\displaystyle \partial f/\partial x_{i}} exists at a {\displaystyle a}. The converse does not hold: it can happen that all of the partial derivatives of f {\displaystyle f} at a {\displaystyle a} exist, but f {\displaystyle f} is not differentiable at a {\displaystyle a}. This means that the function is very "rough" at a {\displaystyle a}, to such an extreme that its behavior cannot be adequately described by its behavior in the coordinate directions. When f {\displaystyle f} is not so rough, this cannot happen. More precisely, if all the partial derivatives of f {\displaystyle f} at a {\displaystyle a} exist and are continuous in a neighborhood of a {\displaystyle a}, then f {\displaystyle f} is differentiable at a {\displaystyle a}. When this happens, then in addition, the total derivative of f {\displaystyle f} is the linear transformation corresponding to the Jacobian matrix of partial derivatives at that point.