Abstract Nonsense

Crushing one theorem at a time

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

Further Properties of the Total Derivative (Pt. I)


Point of Post: In this post we finally prove the majority of the basic theorems regarding the total derivative (differentiable functions form a vector space, etc.)

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Motivation

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So we now finish proving the ‘obvious’ facts one would hope that the total derivative would share with the normal derivative, such as the sum of differentiable functions is differentiable, the derivative of a sum is the sum of the derivatives, the product of two real valued differentiable functions is differentiable, the derivative of such a product is the product ‘rule’, the derivative of a vector valued function is differentiable if and only if each of its coordinate functions is, etc. A lot of the work is already done because of the corollaries of the total derivative of a multilinear function

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May 25, 2011 Posted by | Analysis | , , , , , , | 4 Comments

The Chain Rule


Point of Post: In this post we discuss the chain rule of total derivatives which generalizes the normal chain rule.

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Motivation

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If the total derivative is the generalization of the normal derivative for functions \mathbb{R}\to\mathbb{R} we’ve made it out to be one would hope that it shares most of the nice attributes of the regular derivative. In particular, one of the nicest properties of the normal derivatives for real valued real functions is the chain rule. Here we prove that an analogous theorem holds for differentiable mappings \mathbb{R}^n\to\mathbb{R}^m.

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May 24, 2011 Posted by | Analysis, Uncategorized | , , , , | 8 Comments

The Total Derivative of a Multilinear Function (Pt. II)


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

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May 23, 2011 Posted by | Analysis | , , , , , , | 3 Comments

The Total Derivative of a Multilinear Function (Pt. I)


Point of Post: In this post we prove that a multilinear form is differentiable everywhere and compute its derivative.

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Motivation

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Now that we have the definition of the derivative for mappings f:\mathbb{R}^n\to\mathbb{R}^m it’s time to get our hands a little dirty and compute something. In particular we aim at proving that the wide sweeping class of multilinear function on spaces of the form \mathbb{R}^{n_1}\times\cdots\mathbb{R}^{n_p} are everywhere differentiable and compute their derivative. From this we will be able to recover as a corollary a lot of particulary (and important) functions are differentiable, in particular linear trnasofrmations, the functions of the form \mathbb{R}\times\cdots\times\mathbb{R}\to\mathbb{R}given by (x,\cdots,z)\mapsto x\cdots z and (x,\cdots,z)\mapsto x+\cdots+z, the usual inner product on \mathbb{R}^n, and the determinant.

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May 23, 2011 Posted by | Analysis | , , , , , | 7 Comments

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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Motivation

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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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May 22, 2011 Posted by | Analysis | , , , , | 13 Comments