Abstract Nonsense

Crushing one theorem at a time

The Total Derivative


Point of Post: In this post we discuss the notion of the total derivative for mappings f:\mathbb{R}^n\to\mathbb{R}^m which generalizes the notions of derivatives f:\mathbb{R}^n\to\mathbb{R}^m where n or m=1 as is discussed in usual calculus.

\text{ }

Motivation

\text{ }

The basic notion of multivariable analysis begins by abstractly defining what the derivative for a function f:\mathbb{R}^n\to\mathbb{R}^m means. Intuitively, the derivative should be a ‘best approximation’ near a point. The question then is how to define this abstractly. For functions f:\mathbb{R}\to\mathbb{R} what we got when we took the derivative and evaluated it at a point was a number. But, let’s look a little closer at the definition of the derivative for functions \mathbb{R}\to\mathbb{R}. Namely, a function f is differentiable at the point x_0 if the limit

\text{ }

\displaystyle \lim_{h\to 0}\frac{f(x_0+h)-f(x_0)}{h}

\text{ }

exists. In other words, if there exists a number denoted f'(x_0) such that the above limit evaluates to it. But, let’s rephrase this a little bit. Namely, we could write the fact that this above limit equals f'(x_0) could be restated as

\text{ }

\displaystyle \lim_{h\to 0}\frac{f(x_0+h)-f(x_0)-f'(x_0)h}{h}=0

\text{ }

In other words, the derivative is really the  linear transformation D(h)=f'(x_0)h. Thus, it can easily be shown that a function f:\mathbb{R}\to\mathbb{R} is differentiable at x_0 if and only if there exists a linear transformation \lambda(h) such that

\text{ }

\displaystyle \lim_{h\to 0}\frac{f(x_0+h)-f(x_0)-\lambda(h)}{h}=0

\text{ }

This is precisely how we seek to extend the notion of  derivatives to higher dimensions.

\text{ }

The Total Derivative

\text{ }

Let f:E\to\mathbb{R}^m be a function where E is an open subset of \mathbb{R}^n and let x_0\in\mathbb{R}^n. We say that f is differentiable at x_0 if there exists a linear transformation T\in\text{Hom}\left(\mathbb{R}^n,\mathbb{R}^m\right) such that the limit

\text{ }

\displaystyle \lim_{h\to\bold{0}}\frac{\left\|f(x_0+h)-f(x_0)-T(h)\right\|}{\|h\|}=0

\text{ }

(note that the norm on top is the norm in \mathbb{R}^m and the norm on bottom is the one on \mathbb{R}^n). We say that T is the derivative (or total derivative) of f at x_0. If f is differentiable at every point of some region R\subseteq E we say that f is differentiable on R. Note that it’s important that our set E is open, so that there is some open ball B_{\delta}(x_0)\subseteq E and thus for \|h\|<\delta we have that \|x_0-(x_0+h)\|=\|h\|<\delta and so x_0+h\in B_\delta(x_0)\subseteq\text{Dom}(f).

\text{ }

The justification for the ‘the’ follows from the following theorem:

\text{ }

Theorem: Let f:E\to\mathbb{R}^m be differentiable at x_0 and T,S ‘two’ derivatives of f at x_0. Then, S=T.

Proof: We know that S(\bold{0})=\bold{0}=T(\bold{0}) and so it suffices to show that S,T agree on \mathbb{R}^n-\{\bold{0}\}. To do this let x\ne 0 and note that

\text{ }

\displaystyle \begin{aligned}\lim_{h\to\bold{0}}\frac{\left\|S(h)-T(h)\right\|}{\|h\|} &=\lim_{h\to\bold{0}}\frac{\left\|\left(f(x_0h)-f(x_0)-T(h)\right)-\left(f(x_0+h)-f(x_0)-S(h)\right)\right\|}{\|h\|}\\ &\leqslant \lim_{h\to\bold{0}}\frac{\left\|f(x_0+h)-f(x_0)-T(h)\right\|}{\|h\|}+\lim_{h\to\bold{0}}\frac{\left\|f(x_0+h)-f(x_0)-S(h)\right\|}{\|h\|}\\ &=0\end{aligned}

\text{ }

and so

\text{ }

\displaystyle \lim_{h\to\bold{0}}\frac{\left\|S(h)-T(h)\right\|}{\|h\|}=0

\text{ }

but since evidently \lim_{t\to 0}tx=\bold{0} we can easily deduce that

\text{ }

\displaystyle \begin{aligned}0 &=\lim_{h\to\bold{0}}\frac{\left\|S(h)-T(h)\right\|}{\|h\|}\\ &=\lim_{t\to 0}\frac{\|S(tx)-T(tx)\|}{\|tx\|}\\ &=\lim_{t\to 0}\frac{\left\|S(x)-T(x)\right\|}{\|x\|}\\ &=\frac{\left\|S(x)-T(x)\right\|}{\|x\|}\end{aligned}

\text{ }

and so S(x)=T(x). Since x\in\mathbb{R}^n-\{\bold{0}\} was arbitrary the conclusion follows. \blacksquare

\text{ }

Given a function f:E\to\mathbb{R}^n which is differentiable at some point x_0\in E we denote the derivative of f at x_0 by D_f(x_0). We define the Jacbobian matrix of f at x_0 to be the matrix representation of D_f(x_0) with respect to the usual ordered basis. We denote the Jacobian of f at x_0 as \text{Jac}_f(x_0).

\text{ }

\text{ }

References:

1. Spivak, Michael. Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus. New York: W.A. Benjamin, 196

Advertisements

May 22, 2011 - Posted by | Analysis | , , , ,

13 Comments »

  1. […] that we have the definition of the derivative for mappings it’s time to get our hands a little dirty and compute […]

    Pingback by The Total Derivative of a Multilinear Function (Pt. I) « Abstract Nonsense | May 23, 2011 | Reply

  2. […] every we can apply the triangle inequality to the definition of the derivative to conclude. To do this we note that since there exists (unless is empty in […]

    Pingback by The Total Derivative of a Multilinear Map (Pt. II) « Abstract Nonsense | May 23, 2011 | Reply

  3. […] in discrete mathematics: pure linear algebra, graph theory, etc. That said, we have recently seen that the generalization of the derivative is itself a linear operator. It makes sense then that […]

    Pingback by Linear Operators and the Operator Norm « Abstract Nonsense | May 24, 2011 | Reply

  4. […] would hope that if the total derivative is as similar to the normal derivative as we’d hope that many of the nice qualities pass […]

    Pingback by Differentiability Implies Continuity « Abstract Nonsense | May 24, 2011 | Reply

  5. […] the total derivative is the generalization of the normal derivative for functions we’ve made it out to be one […]

    Pingback by The Chain Rule « Abstract Nonsense | May 24, 2011 | Reply

  6. […] we now finish proving the ‘obvious’ facts one would hope that the total derivative would share with the normal derivative, such as the sum of differentiable functions is […]

    Pingback by Further Properties of the Total Derivative (Pt. I) « Abstract Nonsense | May 25, 2011 | Reply

  7. […] what the total derivative does is describe conditions when a function can be locally approximated very well (sublinearly) […]

    Pingback by Directional Derivatives and Partial Derivatives « Abstract Nonsense | May 29, 2011 | Reply

  8. […] the question remains how the total derivative, which we said was a measure in-all-directions of local change and approximation to a function, […]

    Pingback by Relationship Between the Notions of Directional and Total Derivatives (Pt.I) « Abstract Nonsense | June 2, 2011 | Reply

  9. […] a vector that matches the curve well near the point it’s defined. In more big boy language, recall that the total derivative (i.e. an affine transformation ) is the affine transformation which best […]

    Pingback by Curves and the Implicit Function Theorem « Abstract Nonsense | September 15, 2011 | Reply

  10. […] Let , with open, be such that the total derivative is invertible for each . Then, is a local diffeomorphism and so, in particular, […]

    Pingback by Local Homeo(Diffeo)morphisms to Global Homeo(Diffeo)morphisms « Abstract Nonsense | September 22, 2011 | Reply

  11. […] or just surface if it admits a topological atlas such that for every the map is smooth and the total derivative is injective for each . Such a topological atlas is called a smooth regular atlas or just […]

    Pingback by Surfaces « Abstract Nonsense | October 7, 2011 | Reply

  12. […] functions is commutation with rotations. In particular, similar to how we defined the total derivative we shall define complex differentiable functions to be those that are locally approximatable by […]

    Pingback by Complex Differentiability and Holomorphic Functions (Pt. I) « Abstract Nonsense | May 1, 2012 | Reply

  13. […] on (in the maximal atlas!) such that . We want to show that has the property that it’s derivative at each point has positive determinant. But, the basic idea is simple enough. We know that is […]

    Pingback by Compact Riemann Surfaces are Topologically $latex g$-holed Tori « Abstract Nonsense | October 2, 2012 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: