A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula or a mathematical expression. More formally, a mathematical symbol is any grapheme used in mathematical formulas and expressions. As formulas and expressions are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics. The most basic symbols are the decimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the Latin alphabet. The decimal digits are used for representing numbers through the Hindu–Arabic numeral system. Historically, upper-case letters were used for representing points in geometry, and lower-case letters were used for variables and constants. Letters are used for representing many other types of mathematical object. As the number of these types has increased, the Greek alphabet and some Hebrew letters have also come to be used. For more symbols, other typefaces are also used, mainly boldface a {\displaystyle \mathbf {a} } , A {\displaystyle \mathbf {A} } , b {\displaystyle \mathbf {b} } , B {\displaystyle \mathbf {B} } , ..., script typeface A {\displaystyle {\mathcal {A}}} , B {\displaystyle {\mathcal {B}}} , ... (the lower-case script face is rarely used because of the possible confusion with the standard face), German fraktur a {\displaystyle {\mathfrak {a}}} , A {\displaystyle {\mathfrak {A}}} , b {\displaystyle {\mathfrak {b}}} , B {\displaystyle {\mathfrak {B}}} , ..., and blackboard bold N {\displaystyle \mathbb {N} } , Z {\displaystyle \mathbb {Z} } , Q {\displaystyle \mathbb {Q} } , R {\displaystyle \mathbb {R} } , C {\displaystyle \mathbb {C} } , H {\displaystyle \mathbb {H} } , F q {\displaystyle \mathbb {F} _{q}} (other letters are rarely used in this face, or their use is unconventional). It is commonplace to use alphabets, fonts and typefaces to group symbols by type (for example, boldface is often used for vectors and uppercase for matrices). The use of specific Latin and Greek letters as symbols for denoting mathematical objects is not described in this article. For such uses, see Variable § Conventional variable names and List of mathematical constants. However, some symbols that are described here have the same shape as the letter from which they are derived, such as ∏
{\displaystyle \textstyle \prod {}} and ∑
{\displaystyle \textstyle \sum {}} . These letters alone are not sufficient for the needs of mathematicians, and many other symbols are used. Some take their origin in punctuation marks and diacritics traditionally used in typography; others by deforming letter forms, as in the cases of ∈ {\displaystyle \in } and ∀ {\displaystyle \forall } . Others, such as + and =, were specially designed for mathematics.
Layout of this article Normally, entries of a glossary are structured by topics and sorted alphabetically. This is not possible here, as there is no natural order on symbols, and many symbols are used in different parts of mathematics with different meanings, often completely unrelated. Therefore, some arbitrary choices had to be made, which are summarized below. The article is split into sections that are sorted by an increasing level of technicality. That is, the first sections contain the symbols that are encountered in most mathematical texts, and that are supposed to be known even by beginners. On the other hand, the last sections contain symbols that are specific to some area of mathematics and are ignored outside these areas. However, the long section on brackets has been placed near to the end, although most of its entries are elementary: this makes it easier to search for a symbol entry by scrolling. Most symbols have multiple meanings that are generally distinguished either by the area of mathematics where they are used or by their syntax, that is, by their position inside a formula and the nature of the other parts of the formula that are close to them. As readers may not be aware of the area of mathematics to which the symbol that they are looking for is related, the different meanings of a symbol are grouped in the section corresponding to their most common meaning. When the meaning depends on the syntax, a symbol may have different entries depending on the syntax. For summarizing the syntax in the entry name, the symbol ◻ {\displaystyle \Box } is used for representing the neighboring parts of a formula that contains the symbol. See § Brackets for examples of use. Most symbols have two printed versions. They can be displayed as Unicode characters, or in LaTeX format. With the Unicode version, using search engines and copy-pasting are easier. On the other hand, the LaTeX rendering is often much better (more aesthetic), and is generally considered a standard in mathematics. Therefore, in this article, the Unicode version of the symbols is used (when possible) for labelling their entry, and the LaTeX version is used in their description. So, for finding how to type a symbol in LaTeX, it suffices to look at the source of the article. For most symbols, the entry name is the corresponding Unicode symbol. So, for searching the entry of a symbol, it suffices to type or copy the Unicode symbol into the search textbox. Similarly, when possible, the entry name of a symbol is also an anchor, which allows linking easily from another Wikipedia article. When an entry name contains special characters such as [, ], and |, there is also an anchor, but one has to look at the article source to know it. Finally, when there is an article on the symbol itself (not its mathematical meaning), it is linked to in the entry name.
Arithmetic operators
+ (plus sign) 1. Denotes addition and is read as plus; for example, 3 + 2. 2. Denotes that a number is positive and is read as plus. Redundant, but sometimes used for emphasizing that a number is positive, specially when other numbers in the context are or may be negative; for example, +2. 3. Sometimes used instead of ⊔ {\displaystyle \sqcup } for a disjoint union of sets.
− (minus sign) 1. Denotes subtraction and is read as minus; for example, 3 − 2. 2. Denotes the additive inverse and is read as minus, the negative of, or the opposite of; for example, −2. 3. Also used in place of \ for denoting the set-theoretic complement; see \ in § Set theory.
× (multiplication sign) 1. In elementary arithmetic, denotes multiplication, and is read as times; for example, 3 × 2. 2. In geometry and linear algebra, denotes the cross product. 3. In set theory and category theory, denotes the Cartesian product and the direct product. See also × in § Set theory.
· (dot) 1. Denotes multiplication and is read as times; for example, 3 ⋅ 2. 2. In geometry and linear algebra, denotes the dot product. 3. Placeholder used for replacing an indeterminate element. For example, saying "the absolute value is denoted by | · |" is perhaps clearer than saying that it is denoted as | |.
± (plus–minus sign) 1. Denotes alternatives of a plus sign or a minus sign. 2. Denotes the range of values that a measured quantity may have; for example, 10 ± 2 denotes an unknown value that lies between 8 and 12.
∓ (minus–plus sign) Used paired with ±, denotes the opposite sign; that is, + if ± is −, and − if ± is +.
÷ (division sign) Although widely used for denoting division in Anglophone countries, it is no longer in common use in mathematics and its use is "not recommended". In some countries, it can indicate subtraction.
: (colon) 1. Denotes the ratio of two quantities. 2. In some countries, may denote division. 3. In set-builder notation, it is used as a separator meaning "such that"; see {□ : □}.
/ (slash) 1. Denotes division and is read as divided by or over. Often replaced by a horizontal bar. For example, 3 / 2 or 3 2 {\displaystyle {\tfrac {3}{2}}} . 2. Denotes a quotient structure. For example, quotient set, quotient group, quotient category, etc. 3. In number theory and field theory, F / E {\displaystyle F/E} denotes a field extension, where F is an extension field of the field E. 4. In probability theory, denotes a conditional probability. For example, P ( A / B ) {\displaystyle P(A/B)} denotes the probability of A, given that B occurs. Usually denoted P ( A ∣ B ) {\displaystyle P(A\mid B)} : see "|".
√ (square-root symbol) Denotes square root and is read as the square root of. Rarely used in modern mathematics without a horizontal bar delimiting the width of its argument (see the next item). For example, √2.
√ (radical symbol) 1. Denotes square root and is read as the square root of. For example, 3 + 2 {\displaystyle {\sqrt {3+2}}} . 2. With an integer greater than 2 as a left superscript, denotes an nth root. For example, 3 7 {\displaystyle {\sqrt[{7}]{3}}} denotes the 7th root of 3.
Equality, equivalence and similarity
= (equals sign) 1. Denotes equality. 2. Used for naming a mathematical object in a sentence like "let x = E {\displaystyle x=E} ", where E is an expression. See also ≝, ≜ or :=.
≝, ≜, := Any of these is sometimes used for naming a mathematical object. Thus, x ≜ E {\displaystyle x\triangleq E} , x = d e f E {\displaystyle x\mathrel {\stackrel {\scriptscriptstyle \mathrm {def} }{=}} E} , x := E {\displaystyle x\mathrel {:=} E} and E =: x {\displaystyle E\mathrel {=:} x} are each an abbreviation of the phrase "let x = E {\displaystyle x=E} ", where E {\displaystyle E} is an expression and x {\displaystyle x} is a variable.
≠ (not-equal sign) Denotes inequality and means "not equal".
≈ (approximately equals sign) The most common symbol for denoting approximate equality. For example, π ≈ 3.14159 {\displaystyle \pi \approx 3.14159} .
~ (tilde) 1. Between two numbers, either it is used instead of ≈ to mean "approximatively equal", or it means "has the same order of magnitude as". 2. Denotes the asymptotic equivalence of two functions or sequences. 3. Often used for denoting other types of similarity, for example, matrix similarity or similarity of geometric shapes. 4. Standard notation for an equivalence relation. 5. In probability and statistics, may specify the probability distribution of a random variable. For example, X ∼ N ( 0 , 1 ) {\displaystyle X\sim N(0,1)} means that the distribution of the random variable X is standard normal. 6. Notation for proportionality. See also ∝ for a less ambiguous symbol.
≡ (triple bar) 1. Denotes an identity; that is, an equality that is true whichever values are given to the variables occurring in it. 2. In number theory, and more specifically in modular arithmetic, denotes the congruence modulo an integer. 3. May denote a logical equivalence.
≅ 1. May denote an isomorphism between two mathematical structures, and is read as "is isomorphic to". 2. In geometry, may denote the congruence of two geometric shapes (that is the equality up to a displacement), and is read "is congruent to".
Comparison
< (less-than sign) 1. Strict inequality between two numbers; means and is read as "less than". 2. Commonly used for denoting any strict order. 3. Between two groups, may mean that the first one is a proper subgroup of the second one.
> (greater-than sign) 1. Strict inequality between two numbers; means and is read as "greater than". 2. Commonly used for denoting any strict order. 3. Between two groups, may mean that the second one is a proper subgroup of the first one.
≤ 1. Means "less than or equal to". That is, whatever A and B are, A ≤ B is equivalent to A < B or A = B. 2. Between two groups, may mean that the first one is a subgroup of the second one.
≥ 1. Means "greater than or equal to". That is, whatever A and B are, A ≥ B is equivalent to A > B or A = B. 2. Between two groups, may mean that the second one is a subgroup of the first one.
≪ and ≫ {\displaystyle \ll {\text{ and }}\gg } 1. Means "much less than" and "much greater than". Generally, much is not formally defined, but means that the lesser quantity can be neglected with respect to the other. This is generally the case when the lesser quantity is smaller than the other by one or several orders of magnitude. 2. In measure theory, μ ≪ ν {\displaystyle \mu \ll \nu } means that the measure μ {\displaystyle \mu } is absolutely continuous with respect to the measure ν {\displaystyle \nu } .
≦ and ≧ {\displaystyle \leqq {\text{ and }}\geqq }
Rarely used symbols, generally synonyms of ≤ and ≥, respectively.
≺ and ≻ {\displaystyle \prec {\text{ and }}\succ } Often used for denoting an order or, more generally, a preorder, when it would be confusing or not convenient to use < and >.
Set theory
∅ (null sign) Denotes the empty set, and is more often written ∅ {\displaystyle \emptyset } . Using set-builder notation, it may also be denoted { } {\displaystyle \{\}} .
# (number sign) 1. Number of elements: #
S {\displaystyle \#{}S} may denote the cardinality of the set S. An alternative notation is | S | {\displaystyle \vert S\vert } ; see | ◻ | {\displaystyle \vert \square \vert } . 2. Primorial: n
# {\displaystyle n{}\#} denotes the product of the prime numbers that are not greater than n. 3. In topology, M # N {\displaystyle M\#N} denotes the connected sum of two manifolds or two knots.
∈ Denotes set membership, and is read "is in", "belongs to", or "is a member of". That is, x ∈ S {\displaystyle x\in S} means that x is an element of the set S.
∉ Means "is not in". That is, x ∉ S {\displaystyle x\notin S} means ¬ ( x ∈ S ) {\displaystyle \neg (x\in S)} .
⊂ Denotes set inclusion. However two slightly different definitions are common. 1. A ⊂ B {\displaystyle A\subset B} may mean that A is a subset of B, and is possibly equal to B; that is, every element of A belongs to B; expressed as a formula, ∀ x , x ∈ A ⇒ x ∈ B {\displaystyle \forall x,\,x\in A\Rightarrow x\in B} . 2. A ⊂ B {\displaystyle A\subset B} may mean that A is a proper subset of B, that is the two sets are different, and every element of A belongs to B; expressed as a formula, A ≠ B ∧ ∀
x , x ∈ A ⇒ x ∈ B {\displaystyle A\neq B\land \forall {}x,\,x\in A\Rightarrow x\in B} .
⊆
A ⊆ B {\displaystyle A\subseteq B} means that A is a subset of B. Used for emphasizing that equality is possible, or when A ⊂ B {\displaystyle A\subset B} means that A {\displaystyle A} is a proper subset of B {\displaystyle B} .
⊊
A ⊊ B {\displaystyle A\subsetneq B} means that A is a proper subset of B. Used for emphasizing that A ≠ B {\displaystyle A\neq B} , or when A ⊂ B {\displaystyle A\subset B} does not imply that A {\displaystyle A} is a proper subset of B {\displaystyle B} .
⊃, ⊇, ⊋ Denote the converse relation of ⊂ {\displaystyle \subset } , ⊆ {\displaystyle \subseteq } , and ⊊ {\displaystyle \subsetneq } respectively. For example, B ⊃ A {\displaystyle B\supset A} is equivalent to A ⊂ B {\displaystyle A\subset B} .
∪ Denotes set-theoretic union, that is, A ∪ B {\displaystyle A\cup B} is the set formed by the elements of A and B together. That is, A ∪ B = { x ∣ ( x ∈ A ) ∨ ( x ∈ B ) } {\displaystyle A\cup B=\{x\mid (x\in A)\lor (x\in B)\}} .
∩ Denotes set-theoretic intersection, that is, A ∩ B {\displaystyle A\cap B} is the set formed by the elements of both A and B. That is, A ∩ B = { x ∣ ( x ∈ A ) ∧ ( x ∈ B ) } {\displaystyle A\cap B=\{x\mid (x\in A)\land (x\in B)\}} .
∖ (backslash) Set difference; that is, A ∖ B {\displaystyle A\setminus B} is the set formed by the elements of A that are not in B. Sometimes, A − B {\displaystyle A-B} is used instead; see − in § Arithmetic operators.
⊖ or △ {\displaystyle \triangle }
Symmetric difference: that is, A ⊖ B {\displaystyle A\ominus B} or A △ B {\displaystyle \ A\operatorname {\triangle } B\ } is the set formed by the elements that belong to exactly one of the two sets A and B.
∁ {\displaystyle \complement }
1. With a subscript, denotes a set complement: that is, if B ⊆ A {\displaystyle B\subseteq A} , then ∁ A B = A ∖ B {\displaystyle \complement _{A}B=A\setminus B} . 2. Without a subscript, denotes the absolute complement; that is, ∁ A = ∁ U A {\displaystyle \complement A=\complement _{U}A} , where U is a set which contains all possible sets currently under consideration, implicitly defined by the context. This set U is sometimes called the universe of discourse. 3. Used as a superscript on a set symbol, denotes the complement of that set; that is, A ∁ = U ∖ A , {\displaystyle \ A^{\complement }=U\ \backslash \ A\ ,} where U is the universal set, as in definition 2.
× (multiplication sign) See also × in § Arithmetic operators. 1. Denotes the Cartesian product of two sets. That is, A × B {\displaystyle A\times B} is the set formed by all pairs of an element of A and an element of B. 2. Denotes the direct product of two mathematical structures of the same type, which is the Cartesian product of the underlying sets, equipped with a structure of the same type. For example, direct product of rings, direct product of topological spaces. 3. In category theory, denotes the direct product (often called simply product) of two objects, which is a generalization of the preceding concepts of product.
⊔ {\displaystyle \sqcup }
Denotes the disjoint union. That is, if A and B are sets then A ⊔ B = ( A × { i A } ) ∪ ( B × { i B } ) {\displaystyle A\sqcup B=\left(A\times \{i_{A}\}\right)\cup \left(B\times \{i_{B}\}\right)} is a set of pairs where iA and iB are distinct indices discriminating the members of A and B in A ⊔ B {\displaystyle A\sqcup B} .
⨆ or ∐ {\displaystyle \bigsqcup {\text{ or }}\coprod }
1. Used for the disjoint union of a family of sets, such as in ⨆ i ∈ I A i . {\textstyle \bigsqcup _{i\in I}A_{i}.}
2. Denotes the coproduct of mathematical structures or of objects in a category.
← or → {\displaystyle \leftarrow {\text{ or }}\rightarrow }
Means "sample from". Denotes Random Sampling (probabilistic or otherwise), such as in a ← A or { 0 , 1 } ℓ → x {\displaystyle ~~a\leftarrow A~~{\text{or}}~~\{0,1\}^{\ell }\rightarrow x} . The symbol applies some specified (typically implied) Probability Distribution to a set when randomly selecting (i.e., picking) a single value from that same set. Commonly used in the field of statistics.
← $ or → $ {\displaystyle {\stackrel {\$}{\leftarrow }}{\text{ or }}{\stackrel {\$}{\rightarrow }}}
Denotes Uniform Random Sampling, such as in a ← $ A or { 0 , 1 } ℓ → $ x {\displaystyle ~~a{\stackrel {\$}{\leftarrow }}A~~{\text{or}}~~\{0,1\}^{\ell }~{\stackrel {\$}{\rightarrow }}~x} . Sometimes used in statistics and cryptography, the symbol applies the Uniform Probability Distribution to a set when randomly selecting (i.e., picking) a single value from that same set. If the $ part above the arrow is replaced with the name of a variable, such as ← s {\displaystyle {\stackrel {s}{\leftarrow }}} , the variable above the arrow corresponds to the probability distribution that should be used when sampling a single value from the set.
よ (Hiragana Yo) or h Denotes the Yoneda embedding in category theory.
Basic logic Several logical symbols are widely used in all mathematics, and are listed here. For symbols that are used only in mathematical logic, or are rarely used, see List of logic symbols.
¬ (not sign) Denotes negation, and is read as "not". If E is a logical predicate, ¬ E {\displaystyle \neg E} is the predicate that evaluates to true if and only if E evaluates to false. For clarity, it is often replaced by the word "not".
∨ (descending wedge) 1. Denotes logical disjunction, and is read as "or". If E and F are logical predicates, E ∨ F {\displaystyle E\lor F} is true if either E, F, or both are true. It is often replaced by the word "or". 2. In lattice theory, denotes the join or least upper bound operation. 3. In topology, denotes the wedge sum of two pointed spaces.
∧ (wedge) 1. Denotes logical conjunction, and is read as "and". If E and F are logical predicates, E ∧ F {\displaystyle E\land F} is true if E and F are both true. It is often replaced by the word "and" or the symbol "&". 2. In lattice theory, denotes the meet or greatest lower bound operation. 3. In multilinear algebra, geometry, and multivariable calculus, denotes the exterior product (a.k.a. wedge product).
⊻ Denotes exclusive disjunction. If E and F are logical predicates, E ⊻ F {\displaystyle E\veebar F} denotes the exclusive or. Notations E XOR F and E ⊕ F {\displaystyle E\oplus F} are also commonly used; see ⊕.
∀ (turned A) 1. Denotes universal quantification and is read as "for all". If E is a logical predicate, ∀ x E {\displaystyle \forall x\;E} means that E is true for all possible values of the variable x. 2. Often used in plain text as an abbreviation of "for all" or "for every".
∃ 1. Denotes existential quantification and is read "there exists ... such that". If E is a logical predicate, ∃ x E {\displaystyle \exists x\;E} means that there exists at least one value of x for which E is true. 2. Often used in plain text as an abbreviation of "there exists".
∃! Denotes uniqueness quantification, that is, ∃ ! x P {\displaystyle \exists !x\;P} means "there exists exactly one x such that P (is true)". In other words,
∃ ! x P ( x ) {\displaystyle \exists !x\;P(x)} is an abbreviation of ∃ x ( P ( x ) ∧ ¬ ∃ y ( P ( y ) ∧ y ≠ x ) ) {\displaystyle \exists x\,(P(x)\wedge \neg \exists y\,(P(y)\wedge y\neq x))} .
⇒ 1. Denotes material conditional, and is read as "implies". If P and Q are logical predicates, P ⇒ Q means that if P is true, then Q is also true. Thus, P ⇒ Q is logically equivalent with Q ∨ ¬ P {\displaystyle Q\lor \neg P} . 2. Often used in plain text as an abbreviation of "implies".
⇔ 1. Denotes logical equivalence, and is read "is equivalent to" or "if and only if". If P and Q are logical predicates, P ⇔ Q {\displaystyle P\Leftrightarrow Q} is thus an abbreviation of ( P ⇒ Q ) ∧ ( Q ⇒ P ) {\displaystyle (P\Rightarrow Q)\land (Q\Rightarrow P)} , or of ( P ∧ Q ) ∨ ( ¬ P ∧ ¬ Q ) {\displaystyle (P\land Q)\lor (\neg P\land \neg Q)} . 2. Often used in plain text as an abbreviation of "if and only if".
⊤ (tee) 1. ⊤ {\displaystyle \top } denotes the logical predicate always true. 2. Denotes also the truth value true. 3. Sometimes denotes the top element of a bounded lattice (previous meanings are specific examples).
⊥ (up tack) 1. ⊥ {\displaystyle \bot } denotes the logical predicate always false. 2. Denotes also the truth value false. 3. Sometimes denotes the bottom element of a bounded lattice (previous meanings are specific examples). 4. In cryptography often denotes an error in place of a regular value. 5. For the use as a superscript, see □⊥. 6. For the similar symbol, see ⊥ {\displaystyle \perp } .
Blackboard bold The blackboard bold typeface is widely used for denoting the basic number systems. These systems are often also denoted by the corresponding uppercase bold letter. A clear advantage of blackboard bold is that these symbols cannot be confused with anything else. This allows using them in any area of mathematics, without having to recall their definition. For example, if one encounters R {\displaystyle \mathbb {R} } in combinatorics, one should immediately know that this denotes the real numbers, although combinatorics does not study the real numbers (but it uses them for many proofs).
N {\displaystyle \mathbb {N} }
Denotes the set of natural numbers { 1 , 2 , … } , {\displaystyle \{1,2,\ldots \},} or sometimes { 0 , 1 , 2 , … } . {\displaystyle \{0,1,2,\ldots \}.} When the distinction is important and readers might assume either definition, N 1 {\displaystyle \mathbb {N} _{1}} and N 0 {\displaystyle \mathbb {N} _{0}} are used, respectively, to denote one of them unambiguously. Notation N {\displaystyle \mathbf {N} } is also commonly used.
Z {\displaystyle \mathbb {Z} }
Denotes the set of integers { … , − 2 , − 1 , 0 , 1 , 2 , … } . {\displaystyle \{\ldots ,-2,-1,0,1,2,\ldots \}.} It is often denoted also by Z . {\displaystyle \mathbf {Z} .}
Z p {\displaystyle \mathbb {Z} _{p}}
1. Denotes the set of p-adic integers, where p is a prime number. 2. Sometimes, Z n {\displaystyle \mathbb {Z} _{n}} denotes the integers modulo n, where n is an integer greater than 0. The notation Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } is also used, and is less ambiguous.
Q {\displaystyle \mathbb {Q} }
Denotes the set of rational numbers (fractions of two integers). It is often denoted also by Q . {\displaystyle \mathbf {Q} .}
Q p {\displaystyle \mathbb {Q} _{p}}
Denotes the set of p-adic numbers, where p is a prime number.
R {\displaystyle \mathbb {R} }
Denotes the set of real numbers. It is often denoted also by R . {\displaystyle \mathbf {R} .}
C {\displaystyle \mathbb {C} }
Denotes the set of complex numbers. It is often denoted also by C . {\displaystyle \mathbf {C} .}
H {\displaystyle \mathbb {H} }
Denotes the set of quaternions. It is often denoted also by H . {\displaystyle \mathbf {H} .}
F q {\displaystyle \mathbb {F} _{q}}
Denotes the finite field with q elements, where q is a prime power (including prime numbers). It is denoted also by GF(q).
O {\displaystyle \mathbb {O} }
Denotes the set of octonions. It is often denoted also by O . {\displaystyle \mathbf {O} .}
S {\displaystyle \mathbb {S} }
Denotes the set of sedenions. It is often denoted also by S . {\displaystyle \mathbf {S} .}
T {\displaystyle \mathbb {T} }
Denotes the set of trigintaduonions. It is often denoted also by T . {\displaystyle \mathbf {T} .}
Calculus
□′ Lagrange's notation for the derivative: If f is a function of a single variable, f ′ {\displaystyle f'} , read as "f prime", is the derivative of f with respect to this variable. The second derivative is the derivative of f ′ {\displaystyle f'} , and is denoted f ″ {\displaystyle f''} .
◻ ˙ {\displaystyle {\dot {\Box }}}
Newton's notation, most commonly used for the derivative with respect to time. If x is a variable depending on time, then x ˙ {\displaystyle {\dot {x}}} , read as "x dot", is its derivative with respect to time. In particular, if x is the position of a moving point, then x ˙ {\displaystyle {\dot {x}}} is its v