WordPress Design and Latex

Posted on January 5, 2012 by Tim — No Comments ↓

Newton and Leibniz

The $\int$ ntegrated Technology Services Calgary website design is to be honest a bit hasty. I really needed to throw this site up as fast as possible so I’m not really totally in love with it. I look at it and think what does it need?

I took Engineering at the University of Calgary and really enjoyed differential and Integral Calculus when I studied and tutored so I wanted to include that in my website theme. During the process I had some problems locating an integral sign ( $\int_{0}^\infty$ ). I even installed some Microsoft nightmare math package and now I am stuck having to update endless C++ modules and .net madness. I wish I hadn’t done that.

Cool thing is I found I can write my math expressions in $\LaTeX$ . Never really heard of it before but I know now that Wikipedia uses it. Here’s an example of it in an article I was reading about the Schrödinger equation at Wikipedia and it looks like REAL MATH but the version of $\LaTeX$ they are using converts the syntax to PNG files that are uploaded to their media server.

Each of these examples would appear with a white background but I tweaked them with the #fcfcfc triplet, that’s the background color of my page.

BTW here is his equation -

$\displaystyle \langle \psi|\hat{H}|\psi\rangle = \int \psi^*(\bold{r}) \left[ - \frac{\hbar^2}{2m} \nabla^2\psi(\bold{r}) + V(\bold{r})\psi(\bold{r})\right] d^3\bold{r} =$
$\displaystyle \int \left[ \frac{\hbar^2}{2m}|\nabla\psi|^2 + V(\bold{r}) |\psi|^2 \right] d^3\bold{r}$

Here are a few examples, when I figure out what the are I’ll let you know;

$\displaystyle P_\nu^{-\mu}(z)=\frac{\left(z^2-1\right)^{\frac{\mu}{2}}}{2^\mu \sqrt{\pi}\Gamma\left(\mu+\frac{1}{2}\right)}\int_{}\frac{\left(1-t^2\right)^{\mu -\frac{1}{2}}}{\left(z+t\sqrt{z^2-1}\right)^{\mu-\nu}}dt$ with gamma then maybe a statistical thermodynamics equation

$\displaystyle G_{ab}^{(1)} = -\frac{1}{2}\partial^c\partial_c \bar{\gamma}_{ab} + \partial^c\partial_{(b}\bar{\gamma}_{a)c} -\frac{1}{2}\eta_{ab}\partial^c\partial^d\bar{\gamma}_{cd} = 8\pi T_{ab}$ this is a second order partial differential equation perhaps to do with gravitation or Gibb’s bulk function

$\displaystyle\bar{M}=\frac{kT}{\Xi}\sum_N\left(\frac{\partial Q_N}{\partial D}\right)_{V,T}\lambda^N$ some sort of thermodynamics

$\displaystyle e^{\i \pi} + 1 = 0$ This is a negative vector on the x-axis and plus a scalar of one (positive x) and you get nothing.

My banner image has a weird spiral (in green at the origin) known as the Zeta Function. Here is a multiterm evaluation.

$\displaystyle\zeta(s) = \sum_{n=1}^\infty n^{-s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots \;\;\;\;\;\;\;\sigma = \mathfrak{R}(s) > 1. \!$

Here are a few Equations I know.

$\displaystyle \Phi_E \equiv \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A}$ this is Gauss’s Law for Electric Fields

$\displaystyle F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt$ this is the Laplace Transform. Of all the transforms I used in University this was often found in Electrical Engineering math applications. You do the transform and the inverse transform and end up with solutions to differential equations that even had initial conditions after doing the inverse, it was like magic.

In classical Physics frequency is 1/t and is known as Omega.

Formally: $\omega = \frac{1}{t}$

Inverting back and forth from time space f(t) to frequency space f( $\omega$ ) like this was usually done for our math problems in Electrical Engineering.

More…..

$\displaystyle \pi = \int_{-1}^1\frac{1}{\sqrt{1-x^2}}\,dx$ here’s Pi expressed as an Integral.

This is one graphical representation of pi. ( $\pi$ is the white area, the green curve is the above function.)
You can see why pi is transcendental as the x intercepts are vertical asymptotes that extend to infinity.

pi is a transcendental number. It never repeats and is a totally random number string with no pattern. The curve above is infinite in its own way.

The white area is 3.141596 square units.

The area below y = 1 in the white area is (pi – 2) or just over half of pi’s area (2 square units; two 1×1 squares). The thin glass edges that extend up above y = 1 never end because the function has asymptotes x = 1, x = -1 so the positive y-axis extends to infinity.
I find the relationship between pi’s numerically infinite random string
3.141592653589793238462643383279502884197169399375105820
(y) and the graphically infinitely long finite area (it is after all just over 3) very intriguing.

A circle of radius one has an area of pi. The white area you see in the graphic is also pi and is actually a one to one transformation because the area is preserved but the arrangement of points on the plane is different.

$\begin{array}{*{30}c} {x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}}} & {{\rm{when}}} & {ax^2 + bx + c = 0} \\ \end{array}$ is the solution for x in a quadratic equation.

$f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty (\mathcal{F}f)(t)\, e^{itx}\,dt$ Fourier Inverse Transform; also know as Fourier inversion theorems. I’m not sure, not a hard-core mathematician but my guess is no one understands them completely so they are just theoretical. Just so happens that there was a great deal of interest in this equation. It’s kind of like a link between digital and analog, discreet and random and so on. Most of the physics Transforms usually oscillate between time space and frequency space. BTW it looks like a bell curve.

There are other types of Transforms.
I’ll leave you with a table of Transfo rms at WikipediA

Next 2D transforms…….

About Tim

Tim Helmer is a web developer living in Calgary Alberta. When he's not busy hammering out code, he's out in the mountains getting big air.

Tagged with: Calculus, Calgary WordPress Website Development, Math, Physics, Website Design, WordPress Calgary Business
Posted in Curious Posts

WordPress Design and Latex

Here are a few Equations I know.

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