# Abstract Nonsense

## Higher Order Partial Derivatives and the Equality of Mixed Partials (Pt. I)

Point of Post: In this post we discuss the notion of higher order partial derivatives and prove the classical result concerning when partial derivatives ‘commute’

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Motivation

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As was mentioned in our last post if a function $f:U\to\mathbb{R}^m$ possesses a partial derivative $D_jf(a)$ for every $a\in R$ for some open region $R\subseteq U$ we then get a function $D_jf:R\to\mathbb{R}^m$ it may then be possible that there is a open subregion $R'\subseteq R$ such that $D_iD_jf(a)$ exists for every $a\in R'$. There may then exist some open subregion $R''\subseteq R'$ such that $D_kD_iD_jf(a)$ exists for every $a\in R''$, etc. Thus, we get the notion of higher order partial derivatives. And obvious question then is when do these ‘mixed partials’ (when $i\ne j$ in $D_i D_jf$) ‘commute’ in the sense that $D_iD_jf=D_jD_if$ for every element of some suitable region.

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Higher Order Partial Derivatives

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Let $U\subseteq\mathbb{R}^n$ be open and let $f:U\to\mathbb{R}^n$. Recall that if $D_if(a)$ exists for every $a\in U$ then we get a function $D_if:U\to\mathbb{R}^n$. Suppose further that $D_jD_if(a)$ exists for some $a\in U$ then we see that $f$ has a partial derivative of order two and type $(i,j)$ and denote $D_jD_if(a)$ as $D_{i,j}f(a)$–note that the order in the ordered pair and the index is ‘reversed’. It’s supposed to indicated “Do $D_i$ first then $D_j$” considering we read left to right. If $D_{i,j}f(a)$ exists for all $a\in U$ (really, just an open subregion would do but we could just initially restrict to this subregion so we assume it’s just $U$) then we can consider the function $D_{i,j}f$ on $U$. If for some $a\in U$ we have that $D_kD_{i,j}f(a)$ exists we say that $f$ has a third order derivative of type $(i,j,k)$ at $a$ and denote it $D_{i,j,k}f(a)$. Continuing in this way we may define partial derivatives (if they exist in the manner described above) on $U$ of all orders and types. We may alternatively denote $D_{i_1,\cdots,i_p}f(a)$ as $\displaystyle \frac{\partial^p f}{\partial x_{i_p}\cdots\partial x_{i_1}}(a)$.

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Equality of Mixed Partials

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An obvious question one might ask is “how important is the order I take partial derivatives in?”. To phrase this less vaguely, suppose that $D_if(a)$ exists for all $a\in U$ and $D_{i,j}f(b)$ exists for some $b\in U$–is it true that $D_{j,i}f(b)$ exists, and if so does $D_{i,j}f(b)=D_{j,i}f(b)$? It may be surprising that both of these properties don’t always hold. It really comes down to the unfortunate fact that, in general, limiting processes don’t commute. Let’s consider an example where this fails. Consider the function $f:\mathbb{R}^2\to\mathbb{R}$ given by

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$f(x,y)=\begin{cases}\displaystyle 2xy\frac{x^2-y^2}{x^2+y^2} & \mbox{if} \quad (x,y)\ne(0,0)\\ 0 & \mbox{if} \quad (x,y)=(0,0)\end{cases}$

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Evidently for all $(x,y)\ne (0,0)$ we have that $D_xf(x,y)$ exists and is equal to $\displaystyle 2y\frac{x^2-y^2}{x^2+y^2}+2xy\frac{4xy^2}{\left(x^2+y^2\right)^2}$ and for $(x,y)=(0,0)$ we revert to the definition and see that

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$\displaystyle D_1f(0,0)=\lim_{t\to0}\frac{f(t,0)-f(0,0)}{t}=\lim_{t\to0}\frac{0}{t}=0$

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Thus, $D_1f(a)$ exists for all $a\in\mathbb{R}^2$ and

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$\displaystyle D_1f(x,y)=\begin{cases}\displaystyle 2y\frac{x^2-y^2}{x^2+y^2}+2xy\frac{4xy^2}{\left(x^2+y^2\right)^2} & \mbox{if}\quad (x,y)\ne(0,0)\\ 0 & \mbox{if} \quad (x,y)=(0,0)\end{cases}$

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Now, we’d like to show that $D_2D_1f(0,0)=D_{1,2}f(0,0)$ exists. Indeed:

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$\displaystyle D_{1,2}f(0,0)=\lim_{t\to0}\frac{D_1f(0,t)-f(0,0)}{t}=\lim_{t\to0}\frac{-2t}{t}=-2$

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But, consider using the same methodology as before that $D_2f$ exists everywhere on $\mathbb{R}^2$ and is equal to

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$\displaystyle D_2f(x,y)=\begin{cases}\displaystyle 2x\frac{x^2-y^2}{x^2+y^2}-2xy\frac{4x^2y}{(x^2+y^2)} & \mbox{if} \quad (x,y)\ne(0,0)\\ 0 & \mbox{if} \quad (x,y)=(0,0)\end{cases}$

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Moreover, we note that $D_{2,1}f(0,0)$ exists and

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$\displaystyle \lim_{t\to0}\frac{D_2f(t,0)-D_2f(0,0)}{t}=\lim_{t\to0}\frac{2t}{t}=2$

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Thus, $D_{1,2}f(0,0)$ and $D_{2,1}f(0,0)$ both exist but are unequal.

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What we’d now like to do is given sufficient conditions for the equality of order two mixed partials (those of the type $(i_1,\cdots,i_p)$ where $p>1$ and $i_q\ne i_1$ for every $q=1,\cdots,p$). In fact, we prove a strong theorem than this–one which will not only indicate the equality of mixed partials but of the existence of one mixed partial given the other. Stated more concretely:

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Theorem (Schwarz or Clairaut): Let $f:U\to\mathbb{R}^m$ ,where $U\subseteq\mathbb{R}^n$ is open, be such that $D_if(a)$ and $D_jf(a)$ exist everywhere and are continuous on $U$. If then $D_{i,j}f(b)$ exists for all $b\in U$ and is continuous for some $a\in U$ and $i,j\in[n]$ then $D_{j,i}f(a)$ exists and $D_{i,j}f(a)=D_{j,i}f(a)$.

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The proof is a little involved, so before we start it let’s get a break down of what we’re going to do. Our first trick is to only prove this for mappings $f:\mathbb{R}^2\to\mathbb{R}$ and then show afterwards that this implies our result. To prove the restricted case our best friend is going to be the mean value theorem (MVT) for functions of one variable. We’ll be able to apply this recalling our alternative way of looking at partial (and directional) derivatives as really just the derivative of a particular real valued function. So, the idea is then to use the mean value theorem to write the double iterated limit that comes out of applying the limit definition of $D_{j,i}f(a)$ and by doing some clever manipulation make the big-and-scary looking double fraction look like the compact $D_{i,j}(-,-)$ where each entry is some ‘guaranteed’ $\xi$ like thing we got after applying the mean value theorem. The idea then, is that whatever the entries are, they will be bounded above and below by the variables in the two limits and thus will have to go to ‘what we want them to’ as the variables in the limits go to zero–this is where continuity plays in.

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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, 1965. Print.

2. Apostol, Tom M. Mathematical Analysis. Reading, MA: Addison-Wesley Pub., 1974. Print.