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