Posts Tagged ‘calculus’

YAKC: Differential (One) Forms

August 11, 2011

In my previous post in this series, I introduced the idea of a derivative, and we realized a number of rules regarding them. In this post, we’re going to give some thought as to what derivatives are, look at them from from a rather different perspective, and realize several more rules regarding them. (more…)

You Already Know Calculus: Derivatives

July 31, 2011

Calculus is made to be a whole big hoopla in high school and first year university. It’s supposedly the hardest math class in high school, notoriously complicated and unintuitive.

I blame this on bad education, not just because I’ve observed so much bad math education at the high school level, but because I can’t see any other way anyone could conclude that calculus is difficult. Because I believe that everyone already knows calculus. They just never connect what they already know to the symbols they’re manipulating in math class.

(I’m writing this series of posts — yes, this is only the first of a number of posts! — in a didactic form, but I think they may be of interest to mature mathematicians. While they are (will be?)  informal and unrigorous, they provide intuitive reasons for why everything in basic calculus is true. The results in it are certainly valuable to me: they’re the result of me spending some time trying to answer ‘why’ everything is true at a compelling level. If nothing else, they may be useful in teaching calculus.)

So what is calculus? It’s the mathematical study of rates of change, nothing more and nothing less. We call the rate of change of a function its derivative. (more…)

Derivative of the Derivative?

January 17, 2010

Recently I had a very odd thought. Can we do calculus on functions that take and return functions (eg, (\mathbb{R} \to \mathbb{R}) \to (\mathbb{R} \to \mathbb{R}))?

Well, we’d need to have a conceptual idea of space between functions. Well, we could borrow one from the real numbers by constructing a trivial set of functions that are isomorphic to the real numbers: \{x \to kx | k \in \mathbb{R}\}.

Great. So we have an organized set of functions. It seems appropriate to call this a function space. Now we just need a functions that acts on it so we can do calculus.

Because it would be ironic, let’s choose \frac{d}{dx} as our function.

Hm… what happens when we apply \frac{d}{dx} to functions of the form x \to kx? \frac{d}{dx}(x \to kx)=x\to k.

Blast it! That’s outside our function space! What will we do? We’ll add another function space \{x \to k | k \in \mathbb{R}\}.

The cartesian product of those two earily similar to the second degree real polynomial space, P^2(\mathbb{R})… Let’s consider them to be the same thing!

So, it seems reasonable to say that the Jacobian of the n-dimensional version of this is:

J\left(\frac{d}{dx}\right) = \frac{d}{dx_{(1,2,3....)}}\left(\frac{d}{dx}\right)_{1,2,3...}= ~ \small \left[\begin{array}{cccc} 0&1&0&\ldots\\ 0&0&2&\ldots\\ 0&0&0&\ldots\\ \vdots&\vdots&\vdots&\ddots\\ \end{array}\right] ~~~ (in ~P^n(\mathbb{R})

Just some random thoughts. I mentioned this to a math professor and he said some stuff had been done on it in the field of functional analysis, but a quick search didn’t show much…

Limits and the Infinitesimal Number

January 4, 2010

I’ve been thinking about the infinitesimal number, \delta = \frac{1}{\infty}, recently. In particular, that one could use it to evaluate limits.

What is a limit, really? I’ve been reading some topology recently and I think that it really is a function that returns an accumulation point (hint: these are alternatively known as limit points). More specifically, I believe that \lim_{x\to a} f(x) is an attempt to find a value y such that (a,y) is a limit point of the graph of the the function f.

But I’ve digressed since the simpler, “It’s the value as we approach the point” is perhaps more useful to us…

Consider \lim_{x\to a} f(x). How is this defferent from f(x\pm\delta)? The difference is that we’re looking for the hypothetical value that the function is becoming (also the value of the point which any open set containing it intersected with the graph is not null), not its value when it is infinitly close. Consider \lim_{x\to 2} x: the difference is 2 versus 2\pm \delta. So, we need to get rid of the infintesimal difference. Let \mathbb{R}(x) represent the rounding of anumber x to the nearest real number. Then,

\lim_{x\to a} f(x) = \mathbb{R}\cdot f(x\pm\delta)

Does this have any applications? I beleive it may provide a more elegant way to present Calculus.

Why can’t I take the first principles defenition of a derivative and:

\frac{dy}{\rlap{---}dx} = \lim _{h\to 0} \frac{f(x+h) -f(x)}{\rlap{--}h}

? Because the defenition of a limit would give us zero, that being the closest real number. But instead, we could just say that a value is close to delta and use that. Not useing h because I see no reason to use a random symbol when there is a logical one.

dy = f(x+dx) -f(x)|_{dx\simeq\delta}

And we have a differential form, let’s add vectors (for more dimensions):

dy = f(\vec{x}+\vec{dx}) -f(\vec{x})|_{|\vec{dx}|\simeq\delta}

and for yet more clarity:

dy = y(\vec{x}+\vec{dx}) -y(\vec{x})|_{|\vec{dx}|\simeq\delta}

which is far less cumborsome than

l\lim_{\vec{dx \to 0}} \frac{f(\vec{x}+\vec{dx}) -f(\vec{x})}{|\vec{dx}|}=0

And has almost identical properties (I suppose that since it is + \delta it will be forward facing, eg, d|x|(0) = \delta instead of undefined).

Just some random thoughts.

dy = f(\vec{x}+\vec{dx}) -f(\vec{x})|_{|\vec{dx}|\simeq\delta}

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