Abstract Nonsense

Complex Differentiable and Holmorphic Functions (Pt. III)

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

May 1, 2012

Complex Differentiable and Holmorphic Functions (Pt. II)

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

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May 1, 2012

Complex Differentiability and Holomorphic Functions (Pt. I)

Point of Post: In this post we define the notion of a function $f:\Omega\to\mathbb{C}$ to be holmorphic on some domain $\Omega\subseteq\mathbb{C}$.

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Motivation

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We are going to start discussing complex analysis in preparation for later discussion on Riemann surfaces. We start this discussion, naturally, with the notion of differentiability for functions mapping $\mathbb{C}\supseteq\Omega\to\mathbb{C}$. There is a standard amount of amazement associated to functions which are differentiable in the complex sense since, as we shall see (and, as I’m sure you well-know), they are MUCH nicer then any kind of real differentiable function $U\to\mathbb{R}^2$. In particular, we shall see that any once differentiable function shall be infinitely differentiable and, moreover, locally be expressable as a power series. Think about how different this is from standard real differentiable functions, say, even just $\mathbb{R}\to\mathbb{R}$ where we can find functions that have any number $N$ of deriviatives we desire, yet it’s $N^{\text{th}}$ derivative not even be continuous, let alone differentiable. We can even find functions that are infinitely differentiable yet whose Taylor series at a point doesn’t converge to the function!

May 1, 2012

Local Homeo(Diffeo)morphisms to Global Homeo(Diffeo)morphisms

Point of Post: In this post we discuss an important consequence of the inverse function theorem which relates local diffeomorphisms to global diffeomorphisms.

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September 22, 2011

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

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

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

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

Directional Derivatives and Partial Derivatives

Point of Post: In this post we discuss the notions of directional derivatives and partial derivatives

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Motivation

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Roughly what the total derivative does is describe conditions when a function can be locally approximated very well (sublinearly) well by an affine transformation. Indeed, suppose that $f:\mathbb{R}^n\to\mathbb{R}^m$ is differentiable at $a\in\mathbb{R}^n$. By definition the limit for any $\varepsilon>0$ we may choose $\delta>0$ such that $\|h\|<\delta$ implies $\left\|f(a+h)-\left(T(h)+f(a)\right)\right\|<\varepsilon\|h\|$. Note that that we see from this that ‘locally’ here means in all possible directions (as soon as $h$ is within the open ball $B_{\delta}(a)$ the above inequality applies). Sometimes though we are only interested in the approximation, or notion of change in a particular direction. Thus arises the directional derivative which, roughly put, measures the rate of change of a function $f$ at a point $a$ towards a vector $u$. The idea is simple, namely one takes, along the line from $a$ to $u$, successive differences of the value $f(a)$ and $f(a+tu)$ and let $t$ tend to zero. When the vector is one of the elements of the canonical basis we get the partial derivatives which, as we shall see, are of huge importance in multivariable differential analysis.

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May 29, 2011

Further Properties of the Total Derivative (Pt. II)

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

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May 26, 2011