Abstract Nonsense

Crushing one theorem at a time

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.

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May 31, 2011 - Posted by | Analysis | , , , , , ,

2 Comments »

  1. […] Point of Post: This is a continuation of this post. […]

    Pingback by Higher Order Partial Derivatives and the Equality of Mixed Partials (Pt. II) « Abstract Nonsense | June 1, 2011 | Reply

  2. […] where . We say that ( with ) if is open in and has partial derivatives of all types of order and that is continuous on for every possible . We say that if is continuous on . We often say […]

    Pingback by Functions of Class C^k « Abstract Nonsense | June 4, 2011 | Reply


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