## 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’

*Motivation*

As was mentioned in our last post if a function possesses a partial derivative for every for some open region we then get a function it may then be possible that there is a open subregion such that exists for every . There may then exist some open subregion such that exists for every , etc. Thus, we get the notion of higher order partial derivatives. And obvious question then is when do these ‘mixed partials’ (when in ) ‘commute’ in the sense that for every element of some suitable region.

*Higher Order Partial Derivatives*

Let be open and let . Recall that if exists for every then we get a function . Suppose further that exists for some then we see that has *a partial derivative of order two and type *and denote as –note that the order in the ordered pair and the index is ‘reversed’. It’s supposed to indicated “Do first then ” considering we read left to right. If exists for all (really, just an open subregion would do but we could just initially restrict to this subregion so we assume it’s just ) then we can consider the function on . If for some we have that exists we say that has a *third order derivative of type at * and denote it . Continuing in this way we may define partial derivatives (if they exist in the manner described above) on of all orders and types. We may alternatively denote as .

*Equality of Mixed Partials*

An obvious question one might ask is “how important is the order I take partial derivatives in?”. To phrase this less vaguely, suppose that exists for all and exists for some –is it true that exists, and if so does ? 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 given by

Evidently for all we have that exists and is equal to and for we revert to the definition and see that

Thus, exists for all and

Now, we’d like to show that exists. Indeed:

But, consider using the same methodology as before that exists everywhere on and is equal to

Moreover, we note that exists and

Thus, and both exist but are unequal.

What we’d now like to do is given sufficient conditions for the equality of order two *mixed partials* (those of the type where and for every ). 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:

**Theorem (Schwarz or Clairaut): ***Let ,where is open, be such that and exist everywhere and are continuous on . If then exists for all and is continuous for some and then exists and .*

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 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 and by doing some clever manipulation make the big-and-scary looking double fraction look like the compact where each entry is some ‘guaranteed’ 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.

**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.

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

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