Editing Total derivative (section)

==The total derivative as a linear map==
Let <math>U \subseteq \R^n</math> be an [[open subset]].  Then a function <math>f:U \to \R^m</math> is said to be ('''totally''') '''differentiable''' at a point <math>a\in U</math> if there exists a [[linear transformation]] <math>df_a:\R^n \to \R^m</math>  such that

:<math>\lim_{x \to a} \frac{\|f(x)-f(a)-df_a(x-a)\|}{\|x-a\|}=0.</math>

The [[linear map]] <math>df_a</math> is called the ('''total''') '''derivative''' or ('''total''') '''differential''' of <math>f</math> at <math>a</math>.  Other notations for the total derivative include <math>D_a f</math> and <math>Df(a)</math>.  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 <math>df_a</math> is the best linear approximation to <math>f</math> at the point <math>a</math>.  This can be made precise by quantifying the error in the linear approximation determined by <math>df_a</math>.  To do so, write
:<math>f(a + h) = f(a) + df_a(h) + \varepsilon(h),</math>
where <math>\varepsilon(h)</math> equals the error in the approximation.  To say that the derivative of <math>f</math> at <math>a</math> is <math>df_a</math> is equivalent to the statement
:<math>\varepsilon(h) = o(\lVert h\rVert),</math>
where <math>o</math> is [[Big O notation#Little-o notation|little-o notation]] and indicates that <math>\varepsilon(h)</math> is much smaller than <math>\lVert h\rVert</math> as <math>h \to 0</math>.  The total derivative <math>df_a</math> 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 <math>f</math>.

The function <math>f</math> is differentiable if and only if each of its components <math>f_i \colon U \to \R</math> 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 <math>f</math> is differentiable at <math>a</math>, then each partial derivative <math>\partial f/\partial x_i</math> exists at <math>a</math>.  The converse does not hold: it can happen that all of the partial derivatives of <math>f</math> at <math>a</math> exist, but <math>f</math> is not differentiable at <math>a</math>.  This means that the function is very "rough" at <math>a</math>, to such an extreme that its behavior cannot be adequately described by its behavior in the coordinate directions.  When <math>f</math> is not so rough, this cannot happen.  More precisely, if all the partial derivatives of <math>f</math> at <math>a</math> exist and are continuous in a neighborhood of <math>a</math>, then <math>f</math> is differentiable at <math>a</math>.  When this happens, then in addition, the total derivative of <math>f</math> is the linear transformation corresponding to the [[Jacobian matrix]] of partial derivatives at that point.<ref>{{cite book |first1=Ralph |last1=Abraham |author-link=Ralph Abraham (mathematician) |first2=J. E. |last2=Marsden |author-link2=Jerrold E. Marsden |first3=Tudor |last3=Ratiu |author-link3=Tudor Ratiu |title=Manifolds, Tensor Analysis, and Applications |publisher=Springer Science & Business Media |year=2012 |page=78 |isbn=9781461210290 |url=https://books.google.com/books?id=b-IlBQAAQBAJ&pg=PA78 }}</ref>