Mathematics reference
Notation
18Ma1
MathRef
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:

  1. Italics represent scalars;
  2. Bold italics represent vectors;
  3. Bold represent unit vectors or tensors;
  4. 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.
a--b
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.
ab
means a to the power b.
a1/2
means a to the power one-half, 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 x-axis basis vector
j
means the unit y-axis basis vector
k
means the unit z-axis 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.
v2
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.
MT
means the transpose of M.
MH
means the Hermitian transpose of M.
tr M
means the trace of M.
det M
means the determinant of M.
Mn
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 exclusive-or 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 not-and Q").
P ~| Q
means the negative inclusive disjunction of P and Q ("P not-or Q").
P ~^ Q
means the negative exclusive disjunction of P and Q ("P not-exclusive-or 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 en-dash, 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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