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.

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


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


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