Total derivative
The total derivative as a differential form

When the function under consideration is real-valued, the total derivative can be recast using differential forms. For example, suppose that f: R n → R {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} } is a differentiable function of variables x 1, …, x n {\displaystyle x_{1},\ldots,x_{n}}. The total derivative of f {\displaystyle f} at a {\displaystyle a} may be written in terms of its Jacobian matrix, which in this instance is a row matrix:

The linear approximation property of the total derivative implies that if

is a small vector (where the T {\displaystyle {\mathsf {T}}} denotes transpose, so that this vector is a column vector), then

Heuristically, this suggests that if d x 1, …, d x n {\displaystyle dx_{1},\ldots,dx_{n}} are infinitesimal increments in the coordinate directions, then

In fact, the notion of the infinitesimal, which is merely symbolic here, can be equipped with extensive mathematical structure. Techniques, such as the theory of differential forms, effectively give analytical and algebraic descriptions of objects like infinitesimal increments, d x i {\displaystyle dx_{i}}. For instance, d x i {\displaystyle dx_{i}} may be inscribed as a linear functional on the vector space R n {\displaystyle \mathbb {R} ^{n}}. Evaluating d x i {\displaystyle dx_{i}} at a vector h {\displaystyle h} in R n {\displaystyle \mathbb {R} ^{n}} measures how much h {\displaystyle h} points in the i {\displaystyle i} th coordinate direction. The total derivative d f a {\displaystyle df_{a}} is a linear combination of linear functionals and hence is itself a linear functional. The evaluation d f a (h) {\displaystyle df_{a}(h)} measures how much f {\displaystyle f} points in the direction determined by h {\displaystyle h} at a {\displaystyle a}, and this direction is the gradient. This point of view makes the total derivative an instance of the exterior derivative.

Suppose now that f {\displaystyle f} is a vector-valued function, that is, f: R n → R m {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}}. In this case, the components f i {\displaystyle f_{i}} of f {\displaystyle f} are real-valued functions, so they have associated differential forms d f i {\displaystyle df_{i}}. The total derivative d f {\displaystyle df} amalgamates these forms into a single object and is therefore an instance of a vector-valued differential form.
combination:two_responses