Abstract Nonsense

Crushing one theorem at a time

Curves and the Implicit Function Theorem


Point of Post: In this post we discuss the notion of smooth curves in \mathbb{R}^n and the implicit function theorem.

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Motivation

To begin our discussion of geometry it seems prudent to discuss perhaps the simplest of all smooth geometric objects–curves. Everyone has an intuitive notion of a curve (at least in two or three space). Namely, a curve can be thought of as a length of string that is twisted this way and that, in a smooth manner. But, of course in mathematics one must always back up intuitive notions with concrete, sold definitions. That said, in our case one quickly realizes that there is not one immediate definition of curve. Indeed, there are two canonical ways of defining a curve which, from our point of view, are ordered in terms of ‘importance’ (i.e. we prefer one notion over the other). To see the difference between these two notions consider probably the simplest (closed) curve one could imagine in \mathbb{R}^2–the unit circle \mathbb{S}^1. Ask any kid off the street how one defines the unit circle and you are most likely to get the immediate answer “Oh! It’s just the set of points (x,y)\in\mathbb{R}^2 such that x^2+y^2=1” (or, perhaps the set of all z\in\mathbb{C} with |z| =1). Or, the parabola is another perfectly good curve which could be described as the set (x,y) such that y=x^2. Thus, one should start to wonder if perhaps the correct notion of a curve is the ‘locus’ of a single or multiple functions in Euclidean space. To be more concrete, for functionsf_1,\cdots,f_n:\mathbb{R}^n\to\mathbb{R} define \mathbb{V}(f_1,\cdots,f_n) to be the set f_1^{-1}(\{0\})\cap\cdots\cap f_n^{-1}(\{0\}) (so that \mathbb{S}^1=\mathbb{V}(x^2+y^2-1)). Perhaps then a good definition of a ‘curve’ is a set of the form \mathbb{V}(f_1,\cdots,f_n) for some sufficiently well-behaved functions f_1,\cdots,f_n. That said, there is another natural notion of curve which is equally naturally as the definition as the locus of a set of ‘nice’ functions. Namely, a curve can be thought of as a ‘path’, or the trace of  a moving particle, or more importantly the function defining the path. To be precise, a curve could also be defined as a sufficiently nice mapping \gamma:I\to\mathbb{R}^n for some (possibly infinite) non-empty interval I\subseteq\mathbb{R}. There is a large connotational difference between curves thought of as the locus of a set of functions and as a ‘path’. In particular, a ‘path’ has notions of how quick one traverses the path, whether they turn around, etc. whereas the loci of functions is just a set. That said, there seems to be a pretty obvious ‘connection’, namely taking a ‘path’ \gamma:I\to\mathbb{R}^n and loci. Namely, it seems intuitively obvious that at least the locus \mathbb{V}(f_1,\cdots,f_n) of a set of functions and the image \gamma(I) of some ‘path’ \gamma are the same ‘objects’ (i.e. just sets). That said, a little thought shows that they are definitively not in one-to-one correspondence. For example, consider the hyperbola \mathbb{V}(x^2-y^2-1). This is a perfectly nice ‘curve’, that said there evidently does not exist a sufficiently nice (e.g. continuous) ‘path’ \gamma:I\to\mathbb{R}^2 with \gamma(I)=\mathbb{V}(x^2-y^2-1) since the right hand side is not connected and the left hand side necessarily is. That said, there is hope to find a ‘path’ that has image equal to part of the hyperbola. In particular, if one restricts \mathbb{V}(x^2-y^2-1) to points with positive y-coordinates then the path \gamma:\mathbb{R}\to\mathbb{R}^2 having \gamma:t\mapsto (\cosh(t),\sinh(t)) is a perfectly (C^\infty) nice ‘path’ with \gamma(I) equal to the aforementioned branch of the hyperbola. Thus, one wonders if perhaps there is some condition on a curve (or a point of a curve) that gurantees that the curve is locally equivalent to the image of a ‘path’.  In fact, there is a theorem to this effect, but it is perhaps more of a sophisticated answer than one would expect–in particular being a stronger version of the inverse function theorem. Roughly the theorem states that if one has a level curve, and if the ‘derivative’ of the defining functions is non-zero at some point then in some neighborhood of that point the level curve is the graph of a function! Not to point out the obvious, but math is full of simple questions with startling complicated answers–this is perhaps one of the most profound of these examples, providing an integral link between algebra of functions and their geometry.

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September 15, 2011 Posted by | Analysis, Differential Geometry | , , , , , | 5 Comments

The Inverse Function Theorem (Proof)


Point of Post: This is a continuation of this post.

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September 8, 2011 Posted by | Analysis | , , , , | 4 Comments

The Inverse Function Theorem (Preliminaries)


Point of Post: In this post we give motivation for the inverse function theorem.

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Motivation

In this post we discuss one of the most fundamental analytic-geometric facts in all of multivariable analysis–the inverse function theorem. The theorem really has its humble roots back in single variable analysis with an observation about regular points (points where the derivative is non-zero) of continuously differentiable functions. Namely, it is a common theorem that if f:(a,b)\to\mathbb{R} is continuously differentiable and f'(c)\ne 0 for some c\in(a,b)  then there exists some neighborhood U\subseteq (a,b) containing c for which f is bijective and it’s inverse is continuously differentiable and moreover that \displaystyle \left(f^{-1}\right)'(f(c))=\frac{1}{f'(c)}. There, the theorem was easy to prove (see any basic analysis textbook for a proof). So, since we are doing multivariable analysis an obvious question is “does this result extend to maps f:\mathbb{R}^n\to\mathbb{R}^m?” Well, the first problem in answering this question is formulating exactly what this ‘theorem’ would say in higher dimensions. Let’s  rephrase this theorem in a language a little more amenable to total derivatives. We begin with that f'(c)\ne 0 means. In particular,  (using the notation used above) we see that since D_f(c)(x)=f'(c)x we have that f'(c)\ne 0 if and only if D_f(c)\in\text{GL}\left(\mathbb{R}\right). Thus, it seems that the natural extension would be we want to consider f:U\to\mathbb{R}^m, with U\subseteq\mathbb{R}^n open, with some distinguished point c\in U such that D_f(c):\mathbb{R}^n\to\mathbb{R}^m is an isomorphism. In particular we should make the concession that we would like to only consider maps (with the above notation) where m=n. From this, we can see that we have a visually similar condition that takes f'(c)\ne 0 to (recalling that we are only considering maps \mathbb{R}^n\to\mathbb{R}^n) the condition \det\text{Jac}_f(c)\ne 0. It’s pretty intuitive that we should replace continuously differentiable with C^1(U) (in the multivariable sense). Lastly, we see that \displaystyle \left(f^{-1}\right)'(f(c))=\frac{1}{f'(c)} seems naturally translatable to D_{f^{-1}}(x)=D_f^{-1}(f^{-1}(x)) or, in the more common form, \text{Jac}_{f^{-1}}(x)=\text{Jac}_f^{-1}(f^{-1}(x)). Thus, we can finally create a single-variable to multivariable dictionary for this theorem

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\begin{array}{c|c}\mathbb{R}\to\mathbb{R} & \mathbb{R}^n\to\mathbb{R}^n\\ \hline & \text{ }\\ \text{continuously differentiable} & C^1(U)\\ & \\ f'(c)\ne 0 & D_f(c)\in\text{GL}\left(\mathbb{R}^n\right)\\ & \\ \displaystyle \left(f^{-1}\right)'(f(c))=\frac{1}{f'(c)} & D_{f^{-1}}(f(c))=D_f^{-1}(c)\end{array}

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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.

September 8, 2011 Posted by | Analysis | , , , , | 4 Comments

The Mean Value Theorem for Multivariable Maps


Point of Post: In this post we state and prove the multidimensional analogue of the mean value theorem.

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Motivation

Anyone who has taken a basic analysis course knows that the mean value theorem (MVT) is a very important, and widely used tool. In fact, we’ve had several occasions to use it in our study of multidimensional analysis. So, the obvious question is “is there a multidimensional analogue?” Well, in the strictest sense, there isn’t. By this I mean if one literally transposes the usual MVT to higher dimensions one arrives at something like “Let f be everywhere differentiable on U and let x,y\in U. Then, there exists some \xi\in \overline{xy} (where \overline{xy} is the line segment connecting x and y) such that f(x)-f(y)=D_f(\xi)(x-y)” Unfortunately, this is wildly false as the map f:\mathbb{R}\to\mathbb{R}^2:t\mapsto (\cos(t),\sin(t)) clearly shows. So, what then is the correct formulation? We play our old trick of thinking of a map \mathbb{R}^n\to\mathbb{R}^m as secretly \mathbb{R}\to\mathbb{R}^n\to\mathbb{R}^m by first mapping t\mapsto a+tb for some vectors a,b\in\mathbb{R}^n and then evaluating our map there (more explicitly something of the form t\mapsto a+tb\mapsto f(a+tb)). From there if we could find some way of going back into \mathbb{R} we’d have a map \mathbb{R}\to\mathbb{R}^n\to\mathbb{R}^m\to\mathbb{R} which, being an honest to god real valued real variable map, can have the one dimensional case of the MVT applied to it. So, the question remains as to what kind of maps \mathbb{R}^m\to\mathbb{R} we want to consider? Well, we know from the above that we’re going to have to apply the chain rule to find the derivative of the map \mathbb{R}\to\mathbb{R} and so we don’t want to pick something so crazy that we are left knowing nothing new. No, we’ll restrict our maps into \mathbb{R} to be the simplest (in terms of derivatives), namely we’ll consider \varphi\in\text{Hom}\left(\mathbb{R}^m,\mathbb{R}\right) and so our map really looks like \varphi\circ f\circ g (where g(t)=a+tb).

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June 11, 2011 Posted by | Analysis | , , , , , , , , , | 3 Comments

The Geometry of the Derivative for Real Valued Mappings (Pt. I)


Point of Post: In this post I’d like to discuss some of the geometric aspects of what the total and partial derivatives mean including the idea of approximating lines and tangent planes.

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Motivation

As usual in math it’s helpful to have a picture to backup the ideas. In this post we discuss what it geometrically looks like when a mapping f:\mathbb{R}^n\to\mathbb{R} is differentiable at a point in terms of tangent planes. This of course generalize the notion that a mapping \mathbb{R}\to\mathbb{R} is differentiable at a point if it has a tangent line there.

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June 9, 2011 Posted by | Analysis | , , , , , | Leave a comment

Functions of Class C^k


Point of Post: In this post we define the notion of classes of differentiability and discuss what the membership in some of these classes implies about the function.

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Motivation

Since differentiability and related notions are our focus for right now, it seems prudent that we should define some kind of notation which shortens saying things like “f has partial derivatives of all types of order 7“, ” f has partial derivatives of all types of order 1 and each partial derivative is continuous”, or ” f has partial derivatives of all orders and all types”. This is taken up by the notion of C^k classes which, put simply tells you how well-behaved the function is.

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June 4, 2011 Posted by | Analysis, Uncategorized | , , , , , , , , | 9 Comments

Relationship Between the Notions of Directional and Total Derivatives (Pt.II)


Point of Post: This is a continuation of this post.

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June 2, 2011 Posted by | Analysis | , , , , , , | 1 Comment

Relationship Between the Notions of Directional and Total Derivatives (Pt.I)


Point of Post: In this post we show the relationship between total and directional derivatives, and in doing so finally find an explicit formula for the total derivative in terms of the partial derivatives.

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Motivation

So the question remains how the total derivative, which we said was a measure in-all-directions of local change and approximation to a function, and directional derivatives which we said was some kind of measure of change in a specified direction that ignored all others. Some things seem intuitively obvious. For example, one feels that functions which have total derivatives at a certain point should morally be obligated to possess directional derivatives in all directions. Moreover, it seems not too bizarre that the total derivative should be able to be expressed, in some way, by some combination of directional derivatives. I mean, it makes sense that the change in an arbitrary direction should have something to do with the way it’s “component” directions act. In fact, we shall prove both of these things–namely that total differentiability implies the existence of directional derivatives in all directions, and that the Jacobian can be expressed entirely in terms of partial derivatives. The surprising thing we shall show is that a fairly strong converse holds–namely that if all the partial derivatives are ‘nice’ (in a precise sense to define soon) then we are guaranteed total differentiability

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June 2, 2011 Posted by | Analysis | , , , , , | 2 Comments

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


Point of Post: This is a continuation of this post.

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June 1, 2011 Posted by | Analysis, Uncategorized | , , , , , , , | 2 Comments

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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May 31, 2011 Posted by | Analysis | , , , , , , | 2 Comments