Equations#
This page is a notebook. So you can use binder to play with it, or download it to reuse it.
The Markdown parser included in the Jupyter Notebook is MathJax-aware. This means that you can freely mix in mathematical expressions using the MathJax subset of Tex and LaTeX. Some examples from the MathJax demos site are reproduced below, as well as the Markdown+TeX source.
LaTeX Math#
Jupyter Notebooks’ Markdown cells support LateX for formatting mathematical equations. To tell Markdown to interpret your text as LaTex, surround your input with dollar signs like this:
$z=\dfrac{2x}{3y}$
\(z=\dfrac{2x}{3y}\)
n equation can be very complex:
$F(k) = \int_{-\infty}^{\infty} f(x) e^{2\pi i k} dx$
\(F(k) = \int_{-\infty}^{\infty} f(x) e^{2\pi i k} dx\)
If you want your LaTex equations to be indented towards the center of the cell, surround your input with two dollar signs on each side like this:
$$2x+3y=z$$
Example: The Lorenz Equations#
Source#
\begin{align}
\dot{x} & = \sigma(y-x) \\
\dot{y} & = \rho x - y - xz \\
\dot{z} & = -\beta z + xy
\end{align}
Display#
\(\begin{align} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{align}\)
The Cauchy-Schwarz Inequality#
Source#
\begin{equation*}
\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)
\end{equation*}
Display#
\(\begin{equation*} \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) \end{equation*}\)
A Cross Product Formula#
Source#
\begin{equation*}
\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{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}
\end{equation*}
Display#
\(\begin{equation*} \mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix} \end{equation*}\)
The probability of getting (k) heads when flipping (n) coins is#
Source#
\begin{equation*}
P(E) = {n \choose k} p^k (1-p)^{ n-k}
\end{equation*}
Display#
\(\begin{equation*} P(E) = {n \choose k} p^k (1-p)^{ n-k} \end{equation*}\)
An Identity of Ramanujan#
Source#
\begin{equation*}
\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} =
1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
{1+\frac{e^{-8\pi}} {1+\ldots} } } }
\end{equation*}
Display#
\(\begin{equation*} \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\ldots} } } } \end{equation*}\)
A Rogers-Ramanujan Identity#
Source#
\begin{equation*}
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 $|q|<1$}.
\end{equation*}
Display#
Maxwell’s Equations#
Source#
\begin{align}
\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{align}
Display#
\(\begin{align} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{align}\)
Equation Numbering and References#
Equation numbering and referencing will be available in a future version of the Jupyter notebook.
Inline Typesetting (Mixing Markdown and TeX)#
While display equations look good for a page of samples, the ability to mix math and formatted text in a paragraph is also important.
Source#
This expression $\sqrt{3x-1}+(1+x)^2$ is an example of a TeX inline equation in a [Markdown-formatted](https://daringfireball.net/projects/markdown/) sentence.
Display#
This expression \(\sqrt{3x-1}+(1+x)^2\) is an example of a TeX inline equation in a Markdown-formatted sentence.
Other Syntax#
You will notice in other places on the web that $$ are needed explicitly to begin and end MathJax typesetting. This is not required if you will be using TeX environments, but the Jupyter notebook will accept this syntax on legacy notebooks.
Source#
$$
\begin{array}{c}
y_1 \\\
y_2 \mathtt{t}_i \\\
z_{3,4}
\end{array}
$$
$$
\begin{array}{c}
y_1 \cr
y_2 \mathtt{t}_i \cr
y_{3}
\end{array}
$$
$$\begin{eqnarray}
x' &=& &x \sin\phi &+& z \cos\phi \\
z' &=& - &x \cos\phi &+& z \sin\phi \\
\end{eqnarray}$$
$$
x=4
$$