Math Test

This page will be used to test Wikidot's mathematical abilities :-)

Let's start with some basic math..

(1)
\begin{align} ax^2 + bx + c = 0 \implies x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \end{align}
(2)
\begin{align} \sin^2 x + \cos^2 x = 1 \end{align}

Now, to some calculus

(3)
\begin{align} y(t) = \frac{dx(t)}{dt} \end{align}
(4)
\begin{align} x(t) = x(0) + \int_{0}^{t}y(\tau)d\tau \end{align}

Second Order Linear System

(5)
\begin{align} m\frac{d^2x}{dt^2} + 2b\frac{dx}{dt} + kx = f(t) \end{align}

Euler's Equidimensional equation

(6)
\begin{equation} x^{2}y'' + py' + qy = 0 \end{equation}

reduces to (via a change of variable $x = e^{z}$ (checking inline $\LaTeX$))

(7)
\begin{align} \frac{d^{2}y}{dz^2} + (p-1)\frac{dy}{dz} + qy = 0 \end{align}

Lets try partial derivatives,

Schrodinger's equation:

(8)
\begin{align} -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi = i\hbar \frac{\partial \Psi}{\partial t} \end{align}
(9)
\begin{align} \vec{\nabla} = \sum_{i}\frac{1}{h_{i}}\frac{\partial}{\partial x_{i}}\hat{e_{i}} \end{align}
(10)
\begin{align} \vec{\nabla}\times{\bf A} = \frac{1}{h_{1}h_{2}h_{3}}\left|\begin{array}{ccc}{h_{1}e_{1}&h_{2}e_{2}&h_{3}e_{3}\\\frac{\partial}{\partial x_{1}}&\frac{\partial}{\partial x_{2}}&\frac{\partial}{\partial x_{1}}\\h_{1}A_{1}&h_{2}A_{2}&h_{3}A_{3}}\end{array}\right| \end{align}
(11)
\begin{align} H|\psi\rangle = E|\psi\rangle \end{align}
(12)
\begin{align} \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} \end{align}

Some miscellaneous stuff (to be categorized later):

From Wikidot Docs: $E = mc^2$ is much more popular than $G_{\mu\nu} - \Lambda g_{\mu\nu} = \kappa T_{\mu\nu}$

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