Editing Derivative (section)

=== Total derivative, total differential and Jacobian matrix ===
{{Main|Total derivative}}

When <math> f </math> is a function from an open subset of <math> \R^n </math> to {{tmath|1= \R^m }}, then the directional derivative of <math> f </math> in a chosen direction is the best linear approximation to <math> f </math> at that point and in that direction. However, when {{tmath|1= n > 1 }}, no single directional derivative can give a complete picture of the behavior of <math> f </math>. The total derivative gives a complete picture by considering all directions at once. That is, for any vector <math> \mathbf{v} </math> starting at {{tmath|1= \mathbf{a} }}, the linear approximation formula holds:{{sfn|Davvaz|2023|p=[https://books.google.com/books?id=ofzKEAAAQBAJ&pg=PA266 266]}}
<math display="block">f(\mathbf{a} + \mathbf{v}) \approx f(\mathbf{a}) + f'(\mathbf{a})\mathbf{v}.</math>
Similarly with the single-variable derivative, <math> f'(\mathbf{a}) </math> is chosen so that the error in this approximation is as small as possible. The total derivative of <math> f </math> at <math> \mathbf{a} </math> is the unique linear transformation <math> f'(\mathbf{a}) \colon \R^n \to \R^m </math> such that{{sfn|Davvaz|2023|p=[https://books.google.com/books?id=ofzKEAAAQBAJ&pg=PA266 266]}}
<math display="block">\lim_{\mathbf{h}\to 0} \frac{\lVert f(\mathbf{a} + \mathbf{h}) - (f(\mathbf{a}) + f'(\mathbf{a})\mathbf{h})\rVert}{\lVert\mathbf{h}\rVert} = 0.</math>
Here <math> \mathbf{h} </math> is a vector in {{tmath|1= \R^n }}, so the norm in the denominator is the standard length on <math> \R^n </math>. However, <math> f'(\mathbf{a}) \mathbf{h} </math> is a vector in {{tmath|1= \R^m }}, and the norm in the numerator is the standard length on <math> \R^m </math>.{{sfn|Davvaz|2023|p=[https://books.google.com/books?id=ofzKEAAAQBAJ&pg=PA266 266]}} If <math> v </math> is a vector starting at {{tmath|1= a }}, then <math> f'(\mathbf{a}) \mathbf{v} </math> is called the [[pushforward (differential)|pushforward]] of <math> \mathbf{v} </math> by <math> f </math>.{{sfn|Lee|2013|p=72}}

If the total derivative exists at {{tmath|1= \mathbf{a} }}, then all the partial derivatives and directional derivatives of <math> f </math> exist at {{tmath|1= \mathbf{a} }}, and for all {{tmath|1= \mathbf{v} }}, <math> f'(\mathbf{a})\mathbf{v} </math> is the directional derivative of <math> f </math> in the direction {{tmath|1= \mathbf{v} }}.  If <math> f </math> is written using coordinate functions, so that {{tmath|1= f = (f_1, f_2, \dots, f_m) }}, then the total derivative can be expressed using the partial derivatives as a [[matrix (mathematics)|matrix]]. This matrix is called the [[Jacobian matrix]] of <math> f </math> at <math> \mathbf{a} </math>:{{sfn|Davvaz|2023|p=[https://books.google.com/books?id=ofzKEAAAQBAJ&pg=PA267 267]}}
<math display="block">f'(\mathbf{a}) = \operatorname{Jac}_{\mathbf{a}} = \left(\frac{\partial f_i}{\partial x_j}\right)_{ij}.</math>
<!-- The existence of the total derivative <math> f(\mathbf{a}) </math> is strictly stronger than the existence of all the partial derivatives, but if the partial derivatives exist and are continuous, then the total derivative exists, is given by the Jacobian, and depends continuously on {{tmath|1= \mathbf{a} }}.

The definition of the total derivative subsumes the definition of the derivative in one variable. That is, if <math> f </math> is a real-valued function of a real variable, then the total derivative exists if and only if the usual derivative exists. The Jacobian matrix reduces to a {{nowrap|1=1×1}} matrix whose only entry is the derivative {{tmath|1= f'(x) }}. This {{nowrap|1=1×1}} matrix satisfies the property that
<math display="block"> f(a+h) \approx f(a) + f'(a)h.</math>
Up to changing variables, this is the statement that the function <math>x \mapsto f(a) + f'(a)(x-a)</math> is the best linear approximation to <math> f </math> at {{tmath|1= a }}.{{cn|date=January 2024}}

The total derivative of a function does not give another function in the same way as the one-variable case. This is because the total derivative of a multivariable function has to record much more information than the derivative of a single-variable function. Instead, the total derivative gives a function from the [[tangent bundle]] of the source to the tangent bundle of the target.{{cn|date=January 2024}}

The natural analog of second, third, and higher-order total derivatives is not a linear transformation, is not a function on the tangent bundle, and is not built by repeatedly taking the total derivative. The analog of a higher-order derivative, called a [[jet (mathematics)|jet]], cannot be a linear transformation because higher-order derivatives reflect subtle geometric information, such as concavity, which cannot be described in terms of linear data such as vectors.  It cannot be a function on the tangent bundle because the tangent bundle only has room for the base space and the directional derivatives. Because jets capture higher-order information, they take as arguments additional coordinates representing higher-order changes in direction. The space determined by these additional coordinates is called the [[jet bundle]].  The relation between the total derivative and the partial derivatives of a function is paralleled in the relation between the {{nowrap|1=<math> k </math>-}}th order jet of a function and its partial derivatives of order less than or equal to {{tmath|1= k }}.{{cn|date=January 2024}}

By repeatedly taking the total derivative, one obtains higher versions of the [[Fréchet derivative]], specialized to {{tmath|1= \R^p }}. The {{nowrap|1=<math> k </math>-}}th order total derivative may be interpreted as a map
<math display="block"> D^k f: \mathbb{R}^n \to L^k(\mathbb{R}^n \times \cdots \times \mathbb{R}^n, \mathbb{R}^m), </math>
which takes a point <math> \mathbf{x} </math> in <math> \R^n </math> and assigns to it an element of the space of {{nowrap|1=<math> k </math>-}}linear maps from <math> \R^n </math> to <math> \R^m </math> &mdash; the "best" (in a certain precise sense) {{nowrap|1=<math> k </math>-}}linear approximation to <math> f </math> at that point. By precomposing it with the [[Diagonal functor|diagonal map]] <math> \Delta </math>, <math> \mathbf{x} \to (\mathbf{x}, \mathbf{x}) </math>, a generalized Taylor series may be begun as
<math display="block">\begin{align}
 f(\mathbf{x}) & \approx f(\mathbf{a}) + (D f)(\mathbf{x-a}) + \left(D^2 f\right)(\Delta(\mathbf{x-a})) + \cdots\\
 & = f(\mathbf{a}) + (D f)(\mathbf{x - a}) + \left(D^2 f\right)(\mathbf{x - a}, \mathbf{x - a})+ \cdots\\
 & = f(\mathbf{a}) + \sum_i (D f)_i (x_i-a_i) + \sum_{j, k} \left(D^2 f\right)_{j k} (x_j-a_j) (x_k-a_k) + \cdots
\end{align}</math>
where <math> f(\mathbf{a}) </math> is identified with a constant function, <math> x_i - a_i </math> are the components of the vector <math> \mathbf{x}- \mathbf{a} </math>, and <math> (Df)_i </math> and <math> (D^2 f)_{jk} </math> are the components of <math> Df </math> and <math> D^2 f </math> as linear transformations.{{cn|date=January 2024}}-->