Editing Total derivative (section)

==The total derivative as a differential form==
When the function under consideration is real-valued, the total derivative can be recast using [[differential form]]s.  For example, suppose that <math>f \colon \R^n \to \R</math> is a differentiable function of variables <math>x_1, \ldots, x_n</math>.  The total derivative of <math>f</math> at <math>a</math> may be written in terms of its Jacobian matrix, which in this instance is a row matrix:
:<math>D f_a = \begin{bmatrix} \frac{\partial f}{\partial x_1}(a) & \cdots & \frac{\partial f}{\partial x_n}(a) \end{bmatrix}.</math>
The linear [[approximation property]] of the total derivative implies that if
:<math>\Delta x = \begin{bmatrix} \Delta x_1 & \cdots & \Delta x_n \end{bmatrix}^\mathsf{T}</math>
is a small vector (where the <math>\mathsf{T}</math> denotes transpose, so that this vector is a column vector), then
:<math>f(a + \Delta x) - f(a) \approx D f_a \cdot \Delta x = \sum_{i=1}^n \frac{\partial f}{\partial x_i}(a) \cdot \Delta x_i.</math>
Heuristically, this suggests that if <math>dx_1, \ldots, dx_n</math> are [[infinitesimal]] increments in the coordinate directions, then
:<math>df_a = \sum_{i=1}^n \frac{\partial f}{\partial x_i}(a) \cdot dx_i.</math>

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 form]]s, effectively give analytical and algebraic descriptions of objects like infinitesimal increments, <math>dx_i</math>.  For instance, <math>dx_i</math> may be inscribed as a [[linear functional]] on the vector space <math>\R^n</math>.  Evaluating <math>dx_i</math> at a vector <math>h</math> in <math>\R^n</math> measures how much <math>h</math> "points" in the <math>i</math>th coordinate direction.  The total derivative <math>df_a</math> is a linear combination of linear functionals and hence is itself a linear functional.  The evaluation <math>df_a(h)</math> measures how much <math>f</math> points in the direction determined by <math>h</math> at <math>a</math>, and this direction is the [[gradient]].  This point of view makes the total derivative an instance of the [[exterior derivative]].

Suppose now that <math>f</math> is a [[vector-valued function]], that is, <math>f \colon \R^n \to \R^m</math>.  In this case, the components <math>f_i</math> of <math>f</math> are real-valued functions, so they have associated differential forms <math>df_i</math>.  The total derivative <math>df</math> amalgamates these forms into a single object and is therefore an instance of a [[vector-valued differential form]].