Mathematics reference
Notation


A unified mathematical notation used throughout these pages.


Typeface.

Typeface is meant to reflect the types of the entities involved in
expressions.
The conventions used on these pages are:
 Italics represent scalars;
 Bold italics represent vectors;
 Bold represent unit vectors or tensors;
 Upright text represents Booleans or matrices.

Scalars.

 oo

means infinity.
 =a

means exactly a (usually by definition).
 ~a

means approximately a.
 <a

means less than a.
 >a

means greater than a.
 ab

means somewhere between a and b.
 a...b

means varies between a and b.
 +a

means plus or minus a;
that is, a or a.
 a
[footnote 1]

means negative a.
 a + b

means a plus b.
 a  b [1]

means a minus b.
 a + b

means a plus b and a
minus b.
 a + b

means a minus b and a
plus b.
 a b

means a times b.
 a/b

means a divided by b.
 a^{b}

means a to the power b.
 a^{1/2}

means a to the power onehalf, or
the square root of a.
 a == b

means a is defined as b.
 a = b

means a is equal to b.
 a != b

means a is not equal to b.
 a ~= b

means a is approximately equal to b.
 a < b

means a is less than b.
 a <= b

means a is less than or equal to b.
 a << b

means a is much less than b.
 a > b

means a is greater than b.
 a >= b

means a is greater than or equal to b.
 a >> b

means a is much greater than b.
 a o= b

means a is proportional to b;
that is, there exists some k such that
a = k b.
 a > b

means a approaches b.
 f(a)

means f is a function of a.
 f[b]

means that, for f(a), f is evaluated at
a = b.
 da/db

means the derivative of a with respect to
b.
 @a/@b

means the partial derivative of a with respect to
b.

Vectors.

 o

means the zero vector.
 i

means the unit xaxis basis vector
 j

means the unit yaxis basis vector
 k

means the unit zaxis basis vector
 v [1]

means negative v; that is, (1) v.
 u + v

means u plus v.
 u  v [1]

means u minus v.
 u + v

means u plus v and
u minus v.
 u + v

means u minus v and
u plus v.
 u dot v

means the vector dot product of u and v.
 u cross v

means the vector cross product of u and v.
 r v

means the scalar vector product of r and v.
 v/r

means the scalar vector quotent of v and r;
that is, (1/r) v.
 norm v
 v

means the norm (or magnitude) of v.
 v^{2}

means the norm square of v; that is,
v dot v or v^{2}.
 u == v

means u is defined as v.
 u = v

means u is equal to v.
 u != v

means u is not equal to v.
 u = v

means u and v are parallel vectors;
that is, there exists some k such that u = k v.
 u /= v

means u and v are perpendicular vectors;
that is, u dot v = 0.
 unit v
[footnote 2]

means a unit vector in the direction of v; that is,
the vector v == unit v == v/v.
 F(a)

means F is a function of a.
 F[b]

means that, for F(a), F is
evaluated at a = b.
 dF/dr

means the derivative of F with respect to
r.
 @F/@r

means the partial derivative of F with respect to
r.
 grad F

means the gradient of F ("del F").
 div F

means the divergence of F ("del dot F").
 curl F

means the curl of F ("del cross F").

Matrices.

 I

means the identity matrix.
 M

means negative M; that is, (1) M.
 M + N

means M plus N.
 M  N

means M minus N.
 r M

means r times M.
 M N

means M times N.
 v M

means v times N.
 M v

means M times v.
 M^{1}

means the inverse of M.
 M^{T}

means the transpose of M.
 M^{H}

means the Hermitian transpose of M.
 tr M

means the trace of M.
 det M

means the determinant of M.
 M^{n}

means M to the power n; that is, M multiplied by itself n  1 times.


Booleans.

 ~P
[footnote 3]

means the logical negation of P ("not P").
 P & Q

means the conjunction of P and Q
("P and Q").
 P  Q

means the inclusive disjunction of P and Q
("P or Q").
 P ^ Q

means the exclusive disjunction of P and Q
("P exclusiveor Q").
 P : Q

means P implies (or entails) Q
("if P, then Q"; "Q if P").
 P :: Q

means P is equivalent to Q
("if P, then and only then Q"; "Q if and only if P").
 P ~& Q

means the negative conjunction of P and Q
("P notand Q").
 P ~ Q

means the negative inclusive disjunction of P and Q
("P notor Q").
 P ~^ Q

means the negative exclusive disjunction of P and Q
("P notexclusiveor Q").
 P ~: Q

means P does not imply (or entail) Q
("it is not the case that if P, then Q"; "it is not the case that Q if P").
 P ~:: Q

means P is negatively equivalent to Q
("it is not the case that if P, then and only then Q"; "it is not the case that Q if and only if P").

Footnotes.


1.
The  operators should typographically be an endash, although
many Web browsers display it as a hyphen.

footnote 1

2.
As the example shows, this convention is often implicit. Referring to a unit
vector (say, x) with the same name as a normal vector (x)
without previous introduction implies that the former is taken to the unitized
vector of the latter; i.e., x == unit x.

footnote 2

3.
The numeric operators and the Boolean operator ~ are both unary
operators, but as there is no such thing as "approximately" in Boolean
arithmetic, there is no conflict.

footnote 3

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