From 1385e8530e2a9dacf09f5eca00106bd5b0667cd1 Mon Sep 17 00:00:00 2001 From: Michael Griebling Date: Wed, 18 Jan 2023 15:49:33 -0500 Subject: [PATCH] ... --- EXAMPLES.md | 105 ---------------------------------------------------- 1 file changed, 105 deletions(-) diff --git a/EXAMPLES.md b/EXAMPLES.md index 0c4e54a..d9523e0 100644 --- a/EXAMPLES.md +++ b/EXAMPLES.md @@ -16,10 +16,6 @@ SwiftMath: x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} ``` -MathJax: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ - -SwiftMath: - ![Quadratic Formula](img/quadratic.png) ## Standard Deviation @@ -27,10 +23,6 @@ SwiftMath: \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2} ``` -MathJax: $$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2}$$ - -SwiftMath: - ![Standard Deviation](img/standard.png) ## De Morgan's laws @@ -38,10 +30,6 @@ SwiftMath: \neg(P\land Q) \iff (\neg P)\lor(\neg Q) ``` -MathJax: $\neg(P\land Q) \iff (\neg P)\lor(\neg Q)$ - -SwiftMath: - ![De Morgan](img/demorgan.png) ## Log Change of Base @@ -49,10 +37,6 @@ SwiftMath: \log_b(x) = \frac{\log_a(x)}{\log_a(b)} ``` -MathJax: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$ - -SwiftMath: - ![Log Base Change](img/log.png) ## Cosine addition @@ -60,10 +44,6 @@ SwiftMath: \cos(\theta + \varphi) = \cos(\theta)\cos(\varphi) - \sin(\theta)\sin(\varphi) ``` -MathJax: $\cos(\theta + \varphi) = \cos(\theta)\cos(\varphi) - \sin(\theta)\sin(\varphi)$ - -SwiftMath: - ![Cos Sum](img/trig.png) ## Limit e^k @@ -71,10 +51,6 @@ SwiftMath: \lim_{x\to\infty}\left(1 + \frac{k}{x}\right)^x = e^k ``` -MathJax: $\lim_{x\to\infty}\left(1 + \frac{k}{x}\right)^x = e^k$ - -SwiftMath: - ![Limit](img/limit.png) ## Calculus @@ -82,10 +58,6 @@ SwiftMath: f(x) = \int\limits_{-\infty}^\infty\hat f(\xi)\,e^{2 \pi i \xi x}\,\mathrm{d}\xi ``` -MathJax: $$f(x) = \int\limits_{-\infty}^\infty\!\hat f(\xi)\,e^{2 \pi i \xi x}\,\mathrm{d}\xi$$ - -SwiftMath: - ![Calculus](img/calculus.png) ## Stirling Numbers of the Second Kind @@ -93,10 +65,6 @@ SwiftMath: {n \brace k} = \frac{1}{k!}\sum_{j=0}^k (-1)^{k-j}\binom{k}{j}(k-j)^n ``` -MathJax: ${n \brace k} = \frac{1}{k!}\sum_{j=0}^k (-1)^{k-j}\binom{k}{j}(k-j)^n$ - -SwiftMath: - ![Stirling Numbers](img/stirling.png) ## Gaussian Integral @@ -104,10 +72,6 @@ SwiftMath: \int_{-\infty}^{\infty} \! e^{-x^2} dx = \sqrt{\pi} ``` -MathJax: $\int_{-\infty}^{\infty} \! e^{-x^2} dx = \sqrt{\pi}$ - -SwiftMath: - ![Gauss Integral](img/gaussintegral.png) ## Arithmetic mean, geometric mean inequality @@ -115,10 +79,6 @@ SwiftMath: \frac{1}{n}\sum_{i=1}^{n}x_i \geq \sqrt[n]{\prod_{i=1}^{n}x_i} ``` -MathJax: $\frac{1}{n}\sum_{i=1}^{n}x_i \geq \sqrt[n]{\prod_{i=1}^{n}x_i}$ - -SwiftMath: - ![AM-GM](img/amgm.png) ## Cauchy-Schwarz inequality @@ -126,10 +86,6 @@ SwiftMath: \left(\sum_{k=1}^n a_k b_k \right)^2 \le \left(\sum_{k=1}^n a_k^2\right)\left(\sum_{k=1}^n b_k^2\right) ``` -MathJax: $\left(\sum_{k=1}^n a_k b_k \right)^2 \le \left(\sum_{k=1}^n a_k^2\right)\left(\sum_{k=1}^n b_k^2\right)$ - -SwiftMath: - ![Cauchy Schwarz](img/cauchyschwarz.png) ## Cauchy integral formula @@ -137,10 +93,6 @@ SwiftMath: f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_\gamma\frac{f(z)}{(z-z_0)^{n+1}}dz ``` -MathJax: $f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_\gamma\frac{f(z)}{(z-z_0)^{n+1}}dz$ - -SwiftMath: - ![Cauchy Integral](img/cauchyintegral.png) ## Schroedinger's Equation @@ -149,11 +101,6 @@ i\hbar\frac{\partial}{\partial t}\mathbf\Psi(\mathbf{x},t) = -\frac{\hbar}{2m}\n + V(\mathbf{x})\mathbf\Psi(\mathbf{x},t) ``` -MathJax: $i\hbar\frac{\partial}{\partial t}\mathbf\Psi(\mathbf{x},t) = -\frac{\hbar}{2m}\nabla^2\mathbf\Psi(\mathbf{x},t) -+ V(\mathbf{x})\mathbf\Psi(\mathbf{x},t)$ - -SwiftMath: - ![Schroedinger](img/schroedinger.png) ## Lorentz Equations @@ -167,14 +114,6 @@ equations. \end{gather} ``` -MathJax: $\begin{gather} -\dot{x} = \sigma(y-x) \\ -\dot{y} = \rho x - y - xz \\ -\dot{z} = -\beta z + xy" -\end{gather}$ - -SwiftMath: - ![Lorentz](img/lorentz.png) ## Cross product @@ -186,14 +125,6 @@ SwiftMath: \end{vmatrix} ``` -MathJax: $\vec \bf V_1 \times \vec \bf V_2 = \begin{vmatrix} -\hat \imath &\hat \jmath &\hat k \\ -\frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ -\frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 -\end{vmatrix}$ - -SwiftMath: - ![Cross Product](img/cross.png) ## Maxwell's Equations @@ -208,15 +139,6 @@ multiple equations. \end{eqalign} ``` -MathJax: $begin{eqalign} -\nabla \cdot \vec{\bf E} & = \frac {\rho} {\varepsilon_0} \\ -\nabla \cdot \vec{\bf B} & = 0 \\ -\nabla \times \vec{\bf E} &= - \frac{\partial\vec{\bf B}}{\partial t} \\ -\nabla \times \vec{\bf B} & = \mu_0\vec{\bf J} + \mu_0\varepsilon_0 \frac{\partial\vec{\bf E}}{\partial t} -\end{eqalign}$ - -SwiftMath: - ![Maxwell's Equations](img/maxwell.png) ## Matrix multiplication @@ -235,19 +157,6 @@ c\alpha + d\gamma & c\beta + d \delta \end{pmatrix} ``` -MathJax: $\begin{pmatrix} -a & b\\ c & d -\end{pmatrix} -\begin{pmatrix} -\alpha & \beta \\ \gamma & \delta -\end{pmatrix} = -\begin{pmatrix} -a\alpha + b\gamma & a\beta + b \delta \\ -c\alpha + d\gamma & c\beta + d \delta -\end{pmatrix}$ - -SwiftMath: - ![Matrix Multiplication](img/matrixmult.png) ## Cases @@ -258,13 +167,6 @@ f(x) = \begin{cases} \end{cases} ``` -MathJax: $f(x) = \begin{cases} -\frac{e^x}{2} & x \geq 0 \\ -1 & x < 0 -\end{cases}$ - -SwiftMath: - ![Cases](img/cases.png) ## Splitting long equations @@ -275,11 +177,4 @@ SwiftMath: \left( o_t - \hat{\mu}_m^{(s)} \right) ^T \cal C_m^{(s)-1} \right) ``` -MathJax: $\frak Q(\lambda,\hat{\lambda}) = --\frac{1}{2} \mathbb P(O \mid \lambda ) \sum_s \sum_m \sum_t \gamma_m^{(s)} (t) +\\ -\quad \left( \log(2 \pi ) + \log \left| \cal C_m^{(s)} \right| + -\left( o_t - \hat{\mu}_m^{(s)} \right) ^T \cal C_m^{(s)-1} \right)$ - -SwiftMath: - ![Long equation](img/long.png)