## Wednesday, September 21, 2011

### Testing Latex Rendering

Just some random text

$1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \quad\quad \text{for} |q|<1.$

\begin{aligned} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}

embedded latex

More MathJax tests. Embedding formulae inside sentences

The formula $$\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \quad\quad \text{for} |q|<1.$$ embedded in the middle of a sentence

This expression $$\sqrt{3x-1}+(1+x)^2$$ is an example of an inline equation.

The question is Prove Universal Instantiation i.e

Prove $$(\forall x |: P) \Rightarrow P [ x := E ]$$

LHS $$\hspace{2 cm } (\forall x |: P)$$

= $$\hspace{0.5 cm }$$ notation expansion

$$\hspace{2 cm } (\forall x | true : P)$$

= $$\hspace{0.5 cm }$$ true $$\equiv a \vee \neg a$$

$$\hspace{2 cm } (\forall x | (x = E) \vee (x \neq E) : P)$$

= $$\hspace{0.5 cm }$$ Axiom: Range Split $$(\star | R \vee S: P) \equiv (\star | R : P) \star (\star | S : P)$$ here $$\star = \wedge.$$ notated as $$\forall$$

$$\hspace{2 cm } (\forall x | (x = E) : P) \wedge (\forall x | (x \neq E) : P)$$

$$\Rightarrow \hspace{0.5 cm } Theorem: a \wedge b \Rightarrow a$$

$$\hspace{2 cm } (\forall x | (x = E) : P)$$

= $$\hspace{0.5 cm }$$ Single Point Axiom $$(\star y | (y = F) : Q) \equiv$$ Q [ y := F ]

$$\hspace{2 cm }$$ P [ x := E ] = RHS.
QED