# Abstract Nonsense

## 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.

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Motivation

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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

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

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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

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

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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

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

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

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The Total Derivative

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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

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

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(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 $x_0+h\in B_\delta(x_0)\subseteq\text{Dom}(f)$.

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

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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

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\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}

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and so

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

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

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\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}

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

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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 $\text{Jac}_f(x_0)$.

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References:

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

May 22, 2011 -

## 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