Mathematical Notations and Symbols xi - Ira A. Fulton …vps/ME505/IEM/00 04.pdf3. Set Notations 4. Vectors and Matrices 5. Constants and Numbers Mathematical Notations and Symbols

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

Mathematical Notations and Symbols xviii