LaTeX Reference Card
Typesetting ¶
Font Types ¶
| LaTex | Typesetting | Typical Use |
|---|---|---|
| ℜ | \Re |
Real part of a complex number |
| ℑ | \Im |
Imaginary part of a complex number |
| \mathbb{R} | \mathbb{R} |
Sets |
| \mathbb{F} | \mathbb{F} |
Vectors |
| \mathfrak{J} | \mathfrak{J} |
|
| \mathscr{I} | \mathscr{I} |
|
| \mathcal{S} | \mathcal{S} |
|
| \mathcal{G} | \mathcal{G} |
Note: Used for describing lie algebra |
| \bar{Q} | \bar{Q} |
Bar over symbol |
| \hat{v} | \hat{v} |
Unit vector |
| \vec{u} | \vec{u} |
handwritten vector notation |
| \mathbf{r} | \mathbf{r} |
vector notation used in most university-level books |
| \hat{\mathbf{n}} | \hat\mathbf{n} |
Unit vector using bold face |
| \tilde{n} | \tilde{n} |
Tilde over symbol |
Font Sizes ¶
+ $\Huge Hello!$
+ $\huge Hello!$
+ $\LARGE Hello!$
+ $\Large Hello!$
+ $\large Hello!$
+ $\normalsize Hello!$
+ $\small Hello!$
+ $\scriptsize Hello!$
+ $\tiny Hello!$
Rendering
- \Huge Hello!
- \huge Hello!
- \LARGE Hello!
- \Large Hello!
- \large Hello!
- \normalsize Hello!
- \small Hello!
- \scriptsize Hello!
- \tiny Hello!
Table ¶
\notag
\begin{array} {|l|l|}
\hline symbol & description & value & unit
\\ \hline m & \text{rod mass} & 0.15 & kg
\\ \hline m & \text{cart mass} & 0.4 & kg
\\ \hline l & \text{rod length} & 0.05 & m
\\ \hline j & \text{rod intertia} & 0.005 & kg.m^2
\\ \hline b & \text{friction constant} & 0.8 & n.m.s
\\ \hline
\end{array}
Rendering:^{Note that MWiki markdown and MyST markdown already support tables.}
Symbols ¶
Greek Letters and Math Symbol ¶
| Rendering | Code |
|---|---|
| \alpha | \alpha |
| \beta | \beta |
| \gamma | \gamma |
| \Gamma | \Gamma |
| \delta | \delta |
| \Delta | \Delta |
| \theta | \theta |
| \Theta | \Theta |
| \vartheta | \vartheta |
| \varTheta | \varTheta |
| \phi | \phi |
| \Phi | \Phi |
| \psi | \psi |
| \Psi | \Psi |
| \zeta | \zeta |
| \eta | \eta |
| \iota | \iota |
| \kappa | \kappa |
| \nu | \nu |
| \mu | \mu |
| \xi | \xi |
| \Xi | \Xi |
| \tau | \tau |
| \rho | \rho |
| \pi | \pi |
| \Pi | \Pi |
| \sigma | \sigma |
| \Sigma | \Sigma |
| \epsilon | \epsilon |
| \varepsilon | \varepsilon |
| \nabla | \nabla^{Not a greek letter, but it is a widely used symbol in calculus and fluid mechanics.} |
| \partial | \partial^{Not a greek letter. This symbol is used for partial derivatives.} |
Equality and comparison ¶
| Name | LaTeX | Symbol | |
|---|---|---|---|
| Less or equal than | \leq |
\leq | |
| Greater or equal than | \geq |
\geq | |
| Much greater than |
\gg |
\gg | |
| Much less than | \ll |
\ll | |
| Not equal to | \neq |
\neq | |
| Is equivalent to | \equiv |
\equiv | |
| Is approximately equal to | \approx |
\approx | |
| Is proportional to | \propto |
\propto | |
Sets Symbols and Logical Operators ¶
| Name | LaTeX | Symbol | |
|---|---|---|---|
| Empty set | \emptyset |
||
| Natural Numbers | \mathbb{N} |
\mathbb{N} | |
| Real Numbers | \mathbb{R} |
\mathbb{R} | |
| Complex Numbers | \mathbb{Q} |
\mathbb{Q} | |
| Quaternions | \mathbb{H} |
\mathbb{H} | |
| Element in set | \in |
\in | |
| Element not in set | \notin |
\notin | |
| Exists | \exists |
\exists | |
| For all | \forall |
\forall | |
| Union of sets | \cup |
\cup | |
| Intersection of sets | \cap |
\cap | |
| Right arrow | \Rightarrow |
\Rightarrow | |
Common Cases ¶
Fractions ¶
| Expression | Example | Description |
|---|---|---|
\frac{num}{den} |
\frac{a}{b} | |
\displaystyle \frac{num}{den} |
\displaystyle \frac{b}{a} | |
\dfrac{num}{den} |
\dfrac{a}{b} | typesets the numerator and denominator in display style, it is equivalent to add \displaystyle to a latex commadn. |
\left(\dfrac{a}{b}\right) |
\left(\dfrac{a}{b}\right) | |
\tfrac{num}{den} |
uses text style (inline math) | |
\cfrac{num}{den} |
continous fraction | |
a \over b |
a \over b | |
a \atop b |
a \atop b | |
n \choose k |
n \choose k |
Complex Numbers ¶
| Name | Symbol | LaTeX |
|---|---|---|
| Real Numbers | \mathbb{R} | \mathbb{R} |
| Complex Numbers | \mathbb{Q} | \mathbb{Q} |
| Imaginary Unit 'i' (Europe) | \mathrm{i} | \mathrm{i} |
| Imaginary Unit 'j' (Europe) | \mathrm{j} | \mathrm{j} |
| Imaginary Unit 'i' (Italic) | \textit{i} | \textit{i} |
| Imaginary Unit 'j' (Italic) | \textit{j} | \textit{j} |
| Imaginary Unit (i - imath) | \imath | \imath{i} |
| Imaginary Unit (j - jmath) | \jmath | \jmath{j} |
| Real part of z | \Re\{z\} | \Re\{z\} |
| Imaginary part of z | \Im\{z\} | \Im\{z\} |
| Argument or phase angle of z | \arg z | \arg z |
| Argument or phase angle of z | \angle z | \angle z |
| Argument or phase angle of z | \angle\{z\} | \angle\{z\} |
| Magnitude of complex number of z | \rho = | z| | \rho = | z | |
| Complex number (Europe) | z = x + \mathrm{i} y | z = x + \mathrm{i} y |
| Complex number (Europe) | z = x + \mathrm{j} y | z = x + \mathrm{j} y |
| Complex number (Italic) | z = x + \textit{i} y | z = x + \textit{i} y |
| Complex number (Italic) | z = x + \textit{j} y | z = x + \textit{j} y |
| Complex number in polar form | z = \rho e^{j \phi} | z = \rho e^{j \phi} |
| Complex number in polar form | z = \rho \angle \phi | z = \rho \angle \phi |
| Complex number in polar form | z = \rho\ \angle\{\phi\} | z = \rho \angle \{ \phi \} |
Note:
- The imaginary unit 'j' is preferred in electrical engineering-related fields, instead of 'i' because the symbol 'i' is used in those fields for denoting electrical current.
- In Europe, it considered as good typography the imaginary unit as an upright 'i' or 'j' written in LaTeX as \mathrm{i} (\mathrm{i}) or j \mathrm{j} (\mathrm{j}). The letter 'i' in written in italic \textit{i} (\textit{i}) style when used as an index or summation index.
- Shortcut for writing imaginary unit \renewcommand{\i}{{\mathrm{i}}, then the imaginary unit can be written in LaTeX as just 'z + \i y'.
- (Usenet comp.text.tex, 2001)
Matrices ¶
Basic Matrix (bmatrix)
\begin{bmatrix}
a_{11} & a_{12}
\\ a_{21} & a_{22}
\end{bmatrix}
Basic matrix (PMatrix)
\begin{pmatrix}
A & B & C \cr
d & e & f \cr
1 & 2 & 3 \cr
\end{pmatrix}
Basic matrix (BMatrix)
\begin{Bmatrix}
x & y \\
z & v
\end{Bmatrix}
Matrix with N elements and dots
\begin{bmatrix}
d_1 & c_1 & 0 & \cdots & 0 \\
a_1 & d_2 & c_2 & \cdots & 0 \\
0 & a_2 & d_3 & \cdots & 0 \\
\vdots & \vdots & \ddots & \ddots & c_{n-1} \\
0 & 0 & \cdots & a_{n-1} & d_{n}
\end{bmatrix}
Matrix written in indices notation
R = \begin{bmatrix}
R_{11} & R_{12} & R_{13}
\\ R_{21} & R_{22} & R_{23}
\\ R_{31} & R_{32} & R_{33}
\end{bmatrix}
Matrix that consists of column vectors (Rotation Matrix in this case)
A rotation 3 by matrix R consists of 3 column vectors \mathbf{u}, \mathbf{v} and \mathbf{w}, which are basis vectors of the rotated reference frame expressed in the fixed frame. The unit vector\mathbf{u} is the x axis of the rotated frame, \mathbf{v} is the y axis of the rotated frame and \mathbf{w} is its z axis.
R = \begin{bmatrix}
| & | & |
\\ \mathbf{u} & \mathbf{v} & \mathbf{w}
\\ | & | & |
\end{bmatrix}
\in \mathbb{R}^{3 \times 3}
Jacobian Matrix
\begin{align}
J(\mathbf{x}) &=
\frac{\partial \mathbf{f}}{\partial \mathbf{x}} =
\begin{bmatrix}
\dfrac{\partial f_1}{\partial x_1}
& \dfrac{\partial f_1}{\partial x_2}
& \cdots
& \dfrac{\partial f_n}{\partial x_n} \\
\dfrac{\partial f_2}{\partial x_1}
& \dfrac{\partial f_2}{\partial x_2}
& \cdots
& \dfrac{\partial f_n}{\partial x_n} \\
\vdots & \vdots & \ddots & \ddots \\
\dfrac{\partial f_n}{\partial x_1}
& \dfrac{\partial f_n}{\partial x_2}
& \cdots
& \dfrac{\partial f_n}{\partial x_n}
\end{bmatrix}
\\ \text{where}
\\
\mathbf{x} &= \begin{bmatrix}
x_1 & \cdots & x_n
\end{bmatrix}^T
\\ \mathbf{f}(\mathbf{x}) &=
\begin{bmatrix}
f_1(\mathbf{x}) & \cdots & f_m(\mathbf{x})
\end{bmatrix}^T
\end{align}
Matrix with Row Labels (Labeled Matrix)
\begin{array}{cc}
&
\begin{array}{ccccc} 1 & 2 & 3 & 4 & 5\\
\end{array}
\\
A =
&
\left[
\begin{array}{ccccc}
h & e & l & l & o \\
m & s & e & * & * \\
m & e & t & a & *
\end{array}
\right]
\end{array}
Matrix with Rows and Columns Labels
\begin{array}{c c}
& \begin{array}{c c c} a & b &c \\ \end{array} \\
\begin{array}{c c c}p\\q\\r \end{array} &
\left[
\begin{array}{c c c}
.1 & .1 & 0 \\
.4 & 1 & 0 \\
.8 & 0 & .4
\end{array}
\right]
\end{array}
Default Latex Macros ¶
MWiki comes has many default useful LaTeX macros that makes it easier and faster to write complex math expressions. The LaTeX macros can be edited by clicking at the main menu's item 'Edit Macros' or by opening the link:
Note that this hyperlink does not work in the Wikis or notes exported to html, such as this one. It is only possible to edit macros by starting MWiki in server mode.
Macros for first order derivative ¶
The default macro \dt{} allows writing derivatives of fucntions or vector-valued functions with respect to time in Leibniz's notation.
f(x) = \dt{} y = \dt{y}
Output:
Instead of
f(x)= \frac{d }{dt} y = \frac{d y}{dt}
Macro for second order derivative ¶
The acceleration vector of a point-mass is the second order derivative of its position vector \mathbf{r} = (x(t), y(t), z(t)) with respect to time t.
\mathbf{a}(t) = \ddt{\mathbf{r}}
Output:
Instead of
\mathbf{a}(t) = \frac{d^2 \mathbf{r}}{dt^2}
Macros for partial derivative ¶
First Order Partial Derivative
Partial derivative of a function u(x, y) with respect to x.
\pd{u}{x}
Output:
Instead of
\frac{\partial u}{\partial x}
Second order partial derivative
\pdd{u}{x}
Output:
Instead of
\frac{\partial^2 u}{\partial x^2}
Macros for writing vectors ¶
The macro \b{VECTOR} allows writing vectors in bold face font in a more concise form than \mathbf{VECTOR}. For instance,
\b{u} = \b{v} + \b{w}
Output:
Instead of
\mathbf{u} = \mathbf{v} + \mathbf{w}
Macro for vectors with greek letters
Example:
\bs{\theta} = \begin{bmatrix}
\theta_1 \\ \vdots \\ \theta_n
\end{bmatrix}
Instead of
Where \bs{\theta} (\bs{\theta}) is a vector that consists of n generalized coordinates, joint angles, of a robotic system or mechanism.
First-Order Vector Derivative
Deriatives of vectors with dot notation can be written using the macro \bdot{VECTOR}. Example:
\b{v} = \frac{d}{dt} \b{r} = \bdot{r}
Output:
Instead of
\mathbf{v} = \frac{d}{dt} \mathbf{r} = \dot{\mathbf{r}}
The formula states that the velocity vector \b{v}(t) is the derivative of the positon vector \bdot{r}(t) with respect to time.
Vector (greek letter) derivative
Example:
\bsdot{\theta} =
\begin{bmatrix}
\dot{\theta}_1 \\ \vdots \\ \dot{\theta}_n
\end{bmatrix}
Instead of
\dot{\boldsymbol{\theta}} =
\begin{bmatrix}
\dot{\theta}_1 \\ \vdots \\ \dot{\theta}_n
\end{bmatrix}
where \bsdot{\theta}(t) (\bsdot{\theta}) is the vector of the generalized velocities of each joint of a robotic manipulator or mechanism.
Second order vector derivative
The macro \bddot{VECTOR} can be used for writing the second order derivative of a vector with respect to time. For instance,
\b{a} = \bdot{v} = \bddot{r} = \frac{d^2 \b{r}}{dt^2}
Output:
Instead of
\mathbf{a} = \dot{\mathbf{v}}
= \ddot{\mathbf{r}}
= \frac{d^2 \mathbf{r}}{dt^2}
where \bddot{r}(t) (\bddot{r}(t) is the second order derivative of the position vector \b{r} with respect to time.
Unit vectors
Unit vectors are vectors with unit magnitude, which are expressed using the hat notation with bold face. For instance,
\b{r} = x(t) \bhat{x} + y(t) \bhat{y} + z(t) \bhat{z}
Output
where \bhat{x} (\bhat{x}) is the unit vector related to X axis, \bhat{y} is the unit vector of the Y axis and \bhat{z} is the unit vector of the Z axis.
Note: Source Code of LateX Macros for vectors
% Vector notation, short for mathbf (bold face)
% Exmaple: \b{r} = (x, y, z)
\newcommand{\b}[1]{ \mathbf{#1} }
% Unit vector, example unit direction vector
% for X axis, \bhat{x}
\newcommand{\bhat}[1]{ \hat{\mathbf{#1}} }
% Vector derivative with respect to time
\newcommand{\bdot}[1]{ \dot{\mathbf{#1}} }
% Second order vector derivative with respect to time
\newcommand{\bddot}[1]{ \ddot{\mathbf{#1}} }
Macros for expressing vectors in other reference frames ¶
The LaTeX macros in this section are useful for expressing vectors in other reference frames, which is a common need in mechanical engineering, robotics and computer graphics.
Vectors expressed in other frame ¶
The velocity vector \b{v}(t) measured or expressed in the body frame B is denoted by \fb{v}{B} (\fb{v}{B}). The same vector expressed in the inertial frame N is \fb{v}{N}. The matrix {}^B_NR \in \text{SO}(3) (Special Orhtogonal Group) is a rotation matrix (orthogonal matrix), or orientation matrix, that expresses the instanteous orientation of the inertial frame N with repsect to the body frame B. Note that the macro \fb{} stands for "frame bold".
\fb{v}{B} = {}^B_NR \, \fb{v}{N}
Output:
Instead of
{}^B\mathbf{v} = {}^B_NR \, {}^N\mathbf{v}
Vectors (Greek letter) expressed in other frame ¶
Consider the angular velocity vector \bs{\omega}_1(t) expressed in the inertial frame N.
\fbs{\omega}{N}_1
Output:
Instead of
{}^N\boldsymbol{\omega}_1
Unit Vector expressed in other frame ¶
The direction vector \fbhat{z}{0}_2 (unit vector) is the axis of the joint 2 of a robotic manipulator expressed in the base frame 0.
\| \fbhat{z}{0}_2 \| = 1
Output:
Instead of
\| {}^0\hat{\mathbf{z}}_2 \| = 1
Skew-symmetric or cross-product matrix operator ¶
The skew-symmetric operator or cross-product matrix operator \skew{} is commonly used for obtaining the cross product matrix of a vector \b{r} = (x, y, z) such that \skew{\b{r}} \b{f} = \b{r} \times \b{f}, where \times is the cross-product operator. This operator is needed because, unlike matrices, the cross-product operator is not associative.
\skew{\b{r}} = \begin{bmatrix}
0 & -z & y
\\ z & 0 & -x
\\ -y & x & 0
\end{bmatrix}
Output:
Instead of
\skew{\b{r}} = \begin{bmatrix}
0 & -z & y
\\ z & 0 & -x
\\ -y & x & 0
\end{bmatrix}
Macros for Laplace Transform ¶
Example:
$$
\laplace{ \ddot{f}(t) } = -\dot{f}(0) - s f(0) + s ^ 2 \laplace{ f(t) }
$$
Output:
where \laplace{ g(t) } is the laplace transform of the the function g(t).
Macro for Inverse Laplace Transform ¶
Example:
\ilaplace{ \frac{1}{s} F(s) } = \int_{0}^{\tau} f(\tau) \, d\tau
Macro for Computation Complexity ¶
Example:
+ The computational complexity of this algorithm is $\BigO{n}$.
+ The computational complexity of the second algorithm is $\BigO{n^3}$.
Rendering:
- The comptuational complexity of this algorithm is \BigO{n}.
- The computational complexity of the second algorithm is \BigO{n^3}.