Mathematical Symbols and Special Characters

COMSOL Multiphysics supports a subset of the LaTeX language for creating equations as part of the documentation or in user-developed physics interfaces and other applications. Commands include Greek and other characters, mathematical symbols and operators, arrows, text and font formats, and environments for text and mathematical typesetting. The following tables and lists contain the commands that are available for creating equations and other mathematical text.

If the LaTeX syntax is not correct or not included in the COMSOL software, the equation preview is empty, but no error appears.

Greek and Other Characters

The following table contains the supported lowercase and uppercase Greek letters and the Swedish character Å:

Table 20-15: Greek and other Characters

Command (uppercase)	Character	Command (LOWERCASE)	Character
		\alpha	α
		\beta	β
\Gamma	Γ	\gamma	γ
\Delta	Δ	\delta	δ
		\varepsilon	ε
		\epsilon
		\zeta	ζ
		\eta	η
\Theta	Θ	\theta	θ
		\vartheta
	φ	\iota	ι
		\kappa	κ
\Lambda	Λ	\lambda	λ
		\mu	μ
		\nu	ν
\Xi	Ξ	\xi	ξ
\Pi	Π	\pi	π
		\varpi
		\rho	ρ
\Sigma	Σ	\sigma	σ
		\varsigma
		\tau	τ
\Upsilon	Υ	\upsilon	υ
\Phi	Φ	\phi	φ
		\varphi	ϕ
		\chi	χ
\Psi	Ψ	\psi	ψ
\Omega	Ω	\omega	ω
\AA	Å

Accents

The following accents are available:

Table 20-16: Greek and other Characters

Command	Accent	Command	Accent
\acute	é	\bar
\breve		\check
\ddot	ë	\dot
\grave	è	\hat	ê
\tilde		\vec

Mathematical Symbols and Operators

The following mathematical symbols and operators are available:

Table 20-17: General Symbols and Mathematical Operators

Command	Symbol	Command	Symbol
\dots		\nabla	∇
\ldots		\bot	⊥
\hbar		\diamondsuit	♦
\Re	ℜ	\neg	¬
\forall	∀	\lnot	¬
\cdots	…	\imath
\Im	ℑ	\exists	∃
\prime		\triangle	Δ
\top		\heartsuit	♥
\flat		\vdots
\Diamond		\aleph	ℵ
\mho		\emptyset	Ø
\infty	∞	\angle	∠
\clubsuit	♣	\pounds	£
\ddots		\Box
\wp		\partial	∂
\surd	√	\spadesuit	♠
\dag		\ddag
\S		\P	¶
\copyright	©	\textregistered	®

The following table lists the available “big” mathematical operator as well as binary mathematical operators and relations:

Table 20-18: Big and Binary Mathematical Operators and Relations

Command	Symbol	Command	Symbol
\sum		\prod
\coprod		\int
\bigoplus		\bigcup
\bigcap		\bigsqcup
\oint		\bigotimes
\bigvee		\bigwedge
\biguplus		\bigodot
\pm	±	\cdot	·
\times	×	\cup
\sqcup		\vee	V
\lor	V	\oplus	⊕
\otimes	⊗	\lhd
\unlhd		\mp
\div	÷	\setminus	\
\cap		\sqcap
\wedge		\land
\ominus		\oslash
\rhd		\unrhd
\star		\ast	*
\circ	°	\bullet	•
\uplus		\amalg
\dagger	†	\ddagger	‡
\wr		\leq	≤
\le	≤	\ll	>>
\prec		\preceq
\subset	⊂	\subseteq	⊆
\sqsubset		\sqsubseteq
\in	∈	\vdash
\mid	|	\geq	≥
\ge	≥	\gg	>>
\succ		\succeq
\supset	⊃	\supseteq	⊇
\sqsupset		\sqsupseteq
\ni		\owns
\dashv	–	\parallel	||
\notin	∉	\equiv	≡
\doteq		\sim	~
\simeq		\approx	≈
\cong		\bowtie or \Join
\propto	∝	\models
\perp		\asymp
\neq	≠	\ne	≠
\bigtriangleup		\bigcirc
\bigtriangledown		\triangleleft
\triangleright		\diamond
\smile		\frown

Arrows

The following table includes the available types of arrows:

Table 20-19: Arrows

Command	Symbol	Command	Arrow
\leftarrow or \gets	←	\rightarrow or \to	→
\Leftarrow	⇐	\Rightarrow	⇒
\Leftrightarrow	⇔	\leftrightarrow	↔
\hookleftarrow		\leftharpoonup
\leftharpoondown		\leftrightharpoons
\nearrow		\swarrow
\leadsto		\hookrightarrow
\rightharpoonup		\rightharpoondown
\searrow		\nwarrow
\downarrowtobar		\uparrowtobar
\downtrianglefilled		\uptrianglefilled
\mapsto		\longmapsto
\longleftarrow		\longrightarrow
\longleftrightarrow		\Longleftarrow
\Longrightarrow		\Longleftrightarrow
\iff

Delimiters and Environments

The following tables includes the available delimiter, spaces, and environments:

Table 20-20: Delimiters, Spaces, Environments, BOxes

Command	Symbol/Explanation	Command	Symbol/Explanation
\uparrow	↑	\updownarrow
\downarrow	↓	\Uparrow	⇑
\Updownarrow		\Downarrow	⇓
\lbrack	[	\rbrack	]
\lbrace	{	\rbrace	}
\vert	|	\backslash	\
\Vert		\lfloor	⎣
\lceil	⎡	\rfloor	⎦
\rceil	⎤	\langle	〈
\rangle	〉	\left	Delimiter sizing (see Note below)
\right	Delimiter sizing (see Note below)	\quad	Explicit horizontal spacing
\qquad	Double explicit horizontal spacing	\raisebox	Creates a box containing text; it is used to raise or lower text.
\mbox	Enclose text in a box	\phantom	Adds an invisible component to, for example, balance subscripts
\begin	Invoke the array environment (see Note below)	\end	End the array environment (see Note below)
\unicode	Display Unicode characters as supported by the font: \unicode{ÅÄÖ}, for example	\hspace	Horizontal space

The \left and \right commands must be used in pairs to provide flexible delimiters that fit the formula inside. Put the desired delimiter — ( and ), for example — immediately after the \left and \right commands. For example, \left( \frac{x}{y} \right) provides x/y as a fraction within parentheses that fit the expression’s size.

The \begin and \end commands must be used in pairs to mark the beginning and end of an environment. The only supported environment is the array. For example, \begin{array}{clcr} 1 & 2 & 3 \\ 4 & 5 & 6\end{array} creates a matrix with 2 rows and three columns.

Mathematical Function Names

The following function commands provide the function name using a Roman font:

Table 20-21: Functions

Command	Symbol	Command	Symbol
\arccos	arccos	\cos	cos
\csc	csc	\exp	exp
\ker	ker	\limsup	limsup
\arcsin	arcsin	\cosh	cosh
\deg	deg	\gcd	gcd
\lg	lg	\ln	ln
\arctan	arctan	\cot	cot
\det	det	\hom	hom
\lim	lim	\log	log
\arg	arg	\coth	coth
\dim	dim	\inf	inf
\liminf	liminf	\max	max
\sinh	sinh	\sup	sup
\tan	tan	\tanh	tanh
\min	min	\Pr	Pr
\sec	sec	\sin	sin

Special Mathematical Typesetting

There are two mathematical formula components with a special syntax: \frac for fractions and \sqrt for roots:

•	Use the syntax \frac{numerator}{denominator} to create a fraction. For example, the expression \frac{n!}{k!(n-k)!} produces the following output:

•	Use the syntax \sqrt[order]{expression} to create a root surrounding an expression. The [order] argument is optional; without it, the syntax produces a square root. For example, \sqrt[n]{1+x^2} produces the following output:

Text and Font Elements

The following syntax elements are available for creating different text elements and fonts:

Table 20-22: Various text and Font operations

Command	Explanation	Command	Explanation
\textsuperscript	Superscripts	^	Superscripts
\textsubscript	Subscripts	_	Subscripts
\overline	Overlining	\underline	Underlining
\overleftarrow	Overlining using a left-pointing arrow	\underleftarrow	Underlining using a left-pointing arrow
\overrightarrow	Overlining using a right-pointing arrow	\underrightarrow	Underlining using a right-pointing arrow
\overbrace	Overlining using a brace	\underbrace	Underlining using a brace
\textnormal	Normal text	\textbf	Boldface text
\textit	Text in italics	\textrm	Text in Roman font
\mathnormal	Normal mathematical mode (the default)	\mathbf	Mathematical boldface text
\mathit	Mathematical text in italics	\mathrm	Mathematical text in Roman font
\displaystyle	Size for equations in display mode	\textstyle	Size for equations in text mode
\scriptstyle	Size for first subscript or superscript	\scriptscriptstyle	Size for subsequent subscripts and superscripts
\emph	Emphasize text in normal (Roman) text mode	\tiny	Smallest font size in text mode
\scriptsize	Second smallest fontsize in text mode	\footnotesize	Third smallest font size in text mode
\small	Fourth smallest fontsize in text mode	\normalsize	Normal font size in text mode
\large	Fifth largest font size in text mode	\Large	Fourth largest font size in text mode
\LARGE	Third largest font size in text mode	\huge	Second largest font size in text mode
\Huge	Largest font size in text mode

The \textsuperscript and ^ syntax alternatives are identical for creating superscripts. Likewise, \textsubscript and _ are identical for creating subscripts.

Special Control Sequences and Characters

The following special control sequences and special characters are available:

Table 20-23: Special Control Sequences and Special Characters

Sequence/character	Description	Sequence/character	Description
\#	# character	\:	Medium space
\$	$ character	\;	Thick space
\%	% character	\!	Negative thin space
\&	& character	$, or /[ to start and /] to end	Start and end mathematical mode in text mode
\_	_ character	%	Insert comments
\{	{ character	&	Separate items in arrays
\}	} character	~	Nonbreaking space
\|	|| character	_	Subscript
\<space>	Space	^	Superscript
\,	Thin space	\, {, }, [, ]	Command syntax elements