mathematical notations and symbols xi - ira a. fulton …vps/me505/iem/00 04.pdf3. set notations 4....
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Mathematical Notations and Symbols
xi
MMAATTHHEEMMAATTIICCAALL NNOOTTAATTIIOONNSS AANNDD SSYYMMBBOOLLSS
Contents: 1. Symbols
2. Functions 3. Set Notations 4. Vectors and Matrices 5. Constants and Numbers
Mathematical Notations and Symbols xii
SYMBOLS { } 1,2,3,...= set of all natural numbers
0 { } 0,1,2,3,...=
{ } ..., 2,1,0,1,2,...= − set of integers
m m ,nn
= ∈ ∈
set of all rational numbers
{ } 23.4367...= ± set of all real numbers (infinite decimals)
{ } a ib a,b= + ∈ set of all complex numbers, where I is an imaginary unit, 2i 1= −
2 x if x 0x x
x if x 0≥
= = − < absolute value of real number x∈
i imaginary unit, 2i 1= −
z a ib= + complex number
z a ib= − complex conjugate
( )Re z a= real part
( )Im z b= imaginary part
( ) ( )arg z , Arg z argument, principle argument
2 2z zz a b r= = + = absolute value of complex number z∈ , modulus
∞ infinity, l.u.b. of
−∞ minus infinity, g.l.b. of
N
kk 0
a=∑ 0 1 N a a ... a= + + + sigma-notation for summation
M
ii 1
c=∏ 1 2 M c c c= ⋅ ⋅⋅ ⋅ product notation for product of M numbers
n! 1 2 n= ⋅ ⋅⋅ ⋅ factorial
klim→∞
limit
max maximum
min minimum
sup suprenum (lowest upper bound)
inf infinum (greatest lower bound )
g.l.b. greatest lower bound (infinum)
l.u.b. lowest upper bound (suprenum)
Mathematical Notations and Symbols
xiii
0ε > , 0δ > small positive real numbers
B 0> big positive real number
= equal (used in equations and in assignments)
≡ identically equal (used in definitions)
≈ approximately equal (used for numerical representation of real numbers, 2 1.42≈ )
≠ not equal
∈ belongs, element of
proportional, similar
< less than
≤ less or equal
> greater than
≥ greater or equal
significantly less
significantly greater
⊥ orthogonal to
parallel
→ approaches, goes to
⇒ then, follows, therefore, implies, it is necessarily,
⇐ it is sufficient
⇔ if and only if, it is sufficient and necessarily, equivalent
and so on until
∀ for all
∃ there exists
DNE does not exist
∨ and
∧ or
square root n nth root % percent
x∆ increment, difference between two values dx differential, infinitesimally small increment
nk
( )
n!k ! n k !
=−
binomial coefficients
( )na b+ n
n k k
k 0
na b
k−
=
=
∑ Newton’s Binomial Theorem
Mathematical Notations and Symbols xiv
FUNCTIONS ( )f x , ( )y f x= function
( )1f x− inverse function ( )1f f x x− = and ( )1f f x x− =
f g ( ) f g x= composition
( )x clim f x→
limit
dfdx
, y′ , x , xf derivative, first derivative, ordinary derivative
2
2
d fdx
, y′′ , x , xxf second derivative
k
k
d fdx
, ( )ky kth order derivative
fx∂∂
partial derivative
( )F x antiderivative of the function ( )f x : ( ) ( )F x f x′ =
( )f x dx c+∫ indefinite integral with an arbitrary constant of integration c
( )b
a
f x dx∫ definite integral
( ) b
aF x , ( ) b
aF x bracket notation for definite integration, ( )
b
a
f x dx∫ ( ) b
aF x= ( ) ( )F b F a= −
( )x S
max f x∈
maximum of the function ( )f x on set S
( )x Smin f x∈
minimum of the function ( )f x on set S
( )xΓ gamma function
blog x ln xlnb
= logarithm with the base b
ln x natural logarithm, logarithm with the base e ( )1sin x− , ( )arcsin x inverse sine function, arcus sine, etc
( )erf x error function
( )erfc x complimentary error function, ( ) ( )erfc x 1 erf x= −
( )1 x 0
sgn x1 x 0
>= − <
sign function
( ) ( )f x p f p+ = periodic function with the period p 0>
f , ( )abs f absolute value, f if f 0
ff if f 0
≥= − <
⋅ norm
i imaginary unit, 2i 1= −
Re real part Im imaginary part ∇ Nabla operator
∆ , 2∇ Laplace operator
Mathematical Notations and Symbols
xv
SET NOTATIONS a,b,c,...,x, y,z elements of sets
A,B,...,U ,V ,W sets
x A∈ element x belongs to set A
y B∉ element y does not belong to set B
∅ empty set
A B= equality of sets
A B∪ , n
ii 1
X=
union
A B∩ , n
ii 1
X=
intersection
A| B subtraction
cA compliment
( )a,b { }x a x b= ∈ < < open interval
[ )a,b { }x a x b= ∈ ≤ < semi-open interval
[ ]a,b { }x a x b= ∈ ≤ ≤ closed interval
( )a,∞ { }x x a= ∈ > semi-infinite open interval
VECTORS AND MATRICES a , a vectors
1
2
n
xx
x
=
x
,
1
2
n
xx
x
=
x
column-vector
1 0 00 , 1 , 00 0 1
= = =
i j k standard basis for 3
1 2 n
1 0 00 1 0
, ,...,
0 0 1
= = =
e e e
standard basis for n
2 2 21 2 nx x ... x= + + +x norm
Mathematical Notations and Symbols xvi
( )1 2 ma ,a ,...,a=a row-vector
( )
T1
2T1 2 n
n
xx
x ,x ,...,x
x
= =
x
transpose
11 12 1n
21 22 2nm n
m1 m2 mn
a a aa a a
a a a
×
= =
A A
m n× matrix
V ,U ,W vector spaces, subspaces
( )dim V dimension of vector space
{ }1 2 nspan , ,...,u u u span of vectors 1 2 n, ,...,u u u
mn set of all real m n× matrices
( )im f image of map f
( )ker f kernel of map f
( )rank f rank of map f
1 0 00 1 0
0 0 1
=
I
unit matrix, identity matrix
11 1n
m1 mn
a adet
a a=A
determinant
adjA adjoint
ijA minor
ijC cofactor
RREF row reduced echelon form
11 22 nnTr a a ... a= + + +A trace, the sum of the main diagonal entries
( )ϕ r scalar field
( )a r vector field
gradϕ ϕ= ∇ gradient
diva = ∇ ⋅a divergence
curl = ∇×a curl (rotor)
λ=Ax x eigenvalue problem
λ eigenvalue
x eigenvector
Mathematical Notations and Symbols
xvii
CONSTANTS AND NUMBERS π 3.1415927...=
e 2.7182818...=
γ 0.5772157...=
2 1.4142136...=
3 1.7320508...=
g 9.80616= 2
ms
acceleration of gravity
c 2.997925e 8= + ms
speed of light
h 6.626196e 34= − [ ]J s⋅ Planck’s constant
σ 5.670e 8= − 2 4
Wm K ⋅
Stefan-Boltzmann constant
R⊕ 6 371 000= [ ]m Earth’s radius
D
1.39e+9= [ ]m diameter of the Sun
S 1.496e+11= [ ]m distance between the Sun and the Earth
( 1 astronomical unit)
cS 1353= 2
Wm
Solar constant
N 6.024e23= moleculesmol
Avagadro’s number
P 101325= [ ]Pa , 2
Nm
standard atmospheric pressure