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Michael Griebling
2023-01-18 15:49:33 -05:00
parent 69ac453bd0
commit 1385e8530e
1 changed files with 0 additions and 105 deletions
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EXAMPLES.md
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@@ -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)