======================SCIEN_MATH=============================
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Don Sauer 10/17/09 dsauersanjose@aol.com

Andy N wrote:
> very interesting (IMHO). The square
> root of number, n, is INDEX of sum of odd numbers
that equal n.
The square root of 9 is (1+3+5=9)= 3.
THREE odd numbers. Likewise,
the square root of 25 is (1+3+5+7+9=25)= 5.
The sum of odd numbers 1... can be expressed as
n

\
/ (2n1)

1
Which expands to 1+3+5...
where n is the "index" or the target square root.
This can be simplified to:
n

\
2*( / n )  n

1
The sum in parenthesis can be simplified to n*(n+1)/2.
So the final simplification of the sum of odd numbers is:
2* n*(n+1)/2  n
Which reduces to exactly n*n!
Mean sum of test scores/number of people being tested
Median score were 50 % of people are above & 50 % below
Mode most commonly occuring test score
Average usually means "mean",
57.29578 degrees radian or 0.017453293 radians a degree.
Asphere = 4 * PI * R^2 so: 4 Pi steradians in a sphere.
(0.017453293)^2 =0.0003046174 steradians in"square degree"
1/0.0003046174 = 3282.8064 "square degrees" insteradian
4 Pi 3282.8064 = 41252.961 "square degrees" in a sphere.
60 x 60 nautical miles = one square degree earth surface.
It's not actually square; it bulges in the middle.
a+0 = a a*0 = 0 a*1 = a
a+(a) = 0 a*(1/a) = 1 a + b = b+a
a*b = b*a a*(b+c) = a*b+ a*c a*b*c = c*a*b
a*x^2 +b*x +c = 0 x =(1/2*a)( b +/sqrt(b^2 4*a*c) )
x^1 = x x^0 = 1 x^(n) = 1/x^b
x^(1/2) = sqrt(x) x^a*x^b = x^(a+b)
y = log_b(x) if x = b^y
log_b(x) = log_b(c)*log_c(x)
log_b(1) = 0 log_b(b) = 1
log_b(x*y) = log_b(x) +log_b(y)
log_b(x/y) = log_b(x)  log_b(y)
Complex Numbers
Note: Some textbooks use the letter j
to represent the imaginary part of a complex
number. I have used the more universal i throughout.
A complex number, z, is of the form: z = x + iy
or, using polar coordinates: r = [r,theta]
where x and y are real numbers and:
i  sqrt(1) i^2 = 1
The modulus of a complex number is: z = sqrt( x^2 +y^2 )
The argument of a complex number is: arg(z) = atan(y/x)
The conjugate of a complex number is: z* = x iy
When theta is measured in radians:
The exponential form of a complex number:
[r,theta] = r*exp(i*theta)
cos(z) = ( exp(i*z) + exp(i*z) )/2
sin(z) = i*( exp(i*z)  exp(i*z) )/2
exp(i*z) = cos(z) + i *sin(z)
RightAngled Triangle
Rightangled triangle where
a is the shortest side adjacent to angle ,
b is the side opposite and
c is the longest side (the hypotenuse)
\
 \
 \
a Phi\ c
 \
__ \
_____\ sin(phi) = b/c
b cos(phi) = a/c tan(phi) = b/a
Trigonometrical Identities cos^2(A) + sin^2(A) = 1
Sine Law Cosine Law
Triangle where
side a is opposite angle A,
side b is opposite angle B and
side c is opposite angle C
/\
/B \
a / \ c
/ \ a/sinA = b/sinB = c/sinC
/C A\
/__________\
b
Addition process of finding the sum of the addend and the augend.
Roman Numerals
I 1 II 2 III 3 IIII or IV 4
V 5 VI 6 VII 7 VIII 8
IX 9 X 10 XI 11 XII 12
XIII 13 XIV 14 XV 15 XVI 16
XVII 17 XVIII 18 XIX 19 XX 20
XXV 25 XXX 30 XXXV 35 XL 40
XLV 45 L 50 LX 60 LXX 70
LXXX 80 XC 90 C 100 CL 150
CC 200 CCL 250 CCC 300 CCCL 350
CCCC or CD 400 CDL 450 D 500 DC 600
DCC 700 DCCC 800 DCCCC or CM 900 M 1000
MD 1500 MM 2000 MMD 2500 MMM 3000
Prefix Symbol Factor Prefix Symbol Factor
yotta Y 10+24 deci d 101
zeta Z 1021 centi c 102
exa E 1018 milli m 103
peta P 1015 micro u 106
tera T 1012 nano n 109
giga G 109 pico p 1012
mega M 106 femto f 1015
kilo k 103 atto a 1018
hecto h 102 zepto z 1021
deca d 101 yocto y 1024
Euler's Constant C = 0.57721566
The limit of
C = sum( 1/r, 1=>r=>n ) ln(n)
as n tends towards infinity.
It can be denoted either by the symbol
(the Greek symbol gamma) or C.
e e = 2.7182818285...
e = limit( (1 + 1/m)^m , m=>infinity )
e is an irrational number
(ie. it can never be expressed as the
ratio of two integers).
The value of e isn't coincidental  the gradient
of a graph of ex at any point x equals ex,
making it extremely useful in calculus.
The letter e was first used
by the Swiss mathematician Leonhard Euler (17071783).
Eule also responsible for notations of function, f(x),
__
\
the /_ summation symbol ,
the letter pi for ratio of circumference to diameter
and i to represent the square root of 1.
totally blind in 1768 but still continued his work.
Infinity
Something larger than anything that can be quantified.
It can be regarded as being equal to
1/0. Symbol oo.
Irrational Numbers
An irrational number is any number
that cannot be expressed as the ratio of two whole
(integer) numbers. You can never exactly write down
an irrational number as a decimal
number  there are simply an infinite number of decimal
places. Examples of irrational
numbers include e and .
The term surd can be applied to irrational
roots or sums of irrational roots.
Integer Numbers
An integer number is any whole number
(a number without a fractional or decimal part),
positive and negative, including zero.
In other words the set [...,3,2,1,0,1,2,3,...]. The
__
//
symbol //_ is used to represent set of integer numbers.
Natural Numbers
A natural number is any number in the set [1,2,3,...]
or [0,1,2,3,...]  any integer number
greater than or equal to zero. It should be noted
that the inclusion of zero is by definition
only  you should specifiy if you are to include zero
in the set. The set of natural numbers is
\ 
\\ 
denoted by the symbol  \\.
Rational Numbers
A rational number is any number which
can be expressed as the ratio of two integer
numbers. For example, 1/2, 7/8 and 13/7.
It is important to remember that not all rational
numbers can be written exactly as a
decimal number  1/9 = 0.11111111... and similarly
that a decimal number such as 0.88888888...
should not be disregarded as a rational
number just because it cannot be written
exactly as a decimal number.
___
/ \
 
The \_/ symbol is
\
used to represent the set of rational numbers.
Real Numbers
real number is any rational or irrational number.
The set of real numbers is denoted by the
____
 \
_/
symbol  \\.
 \\
Factorial
factorial of a number n
is the product of all integer numbers from 1 to n.
It gives the number
of different ways (permutations) of arranging n objects.
n! = n (n  1) (n  2) ... 3.2.1
For example:
2! = 2.1 = 2
R B B R
3! = 3.2.1 = 6
B G R G R B R B G
G B R R G B B R G
By definition 0!=1. This is because (n1)!=n!/n, so:
for n = 3; 2! = 3!/3 = 6/3 = 2
for n = 2; 1! = 2!/2 = 2/2 = 1
for n = 1; 0! = 1!/1 = 1/1 = 1
This makes sense
as there is only one way to arrange nothing.
My thanks to all those people who replied regarding this
exp(x) = 1 + x +(x^2/2!) +(x^3/3!)+ (x^4/4!)+ (x^5/5!)...
cos(x) = 1 (x^2/2!) + (x^4/4!)+ ...
sin(x) = + x (x^3/3!) + (x^5/5!)...
for all values of x
(1 + x)n
(1 + x)n = 1 + nx + (n(n  1) / 2!)x 2 + ...
for 1 x 1
(1 + x) 1
(1 + x) 1 = 1  x + x 2 + ... + (1)r x r + ...
for 1 x 1
(1  x) 1
(1  x) 1 = 1 + x + x 2 + ... + x r + ...
for 1 x 1
Maclaurin's Formula
f(x) = f(0) + (f'(0)/1!) + f''(0)/2!) + ... +
(f'(n  1)(0)/(n 1)!) x^(n  1) + R_n(x)
where R_n(x) = ( f'(n)(x_0/n!)x^n (Lagrange Form)
or R_n(x) = ( f(n)(x*_0/(n  1)! )*(x  x*_0 )^(n  1)
( Cauchy Form)
position r> r> = (x, y, z )
velocity v> v> = (x_dot, y_dot, z_dot )
acceleratio a> a> = (x_ddot,y_ddot,z_ddot)
s(t) = s_0 + intergal( mag(v>(t)), dt )
r>(t) = r_0> + intergal( v>(t) , dt )
v>(t) = v_0> + intergal( a>(t) , dt )
v(t) = v_0 +a*t s(t) = s_0 +v_0*t +(1/2)*a*t^2
__ to orbit e_t>
 to it e_n>
for curvature k
radius of curvature rho
e_t> = v>/mag(v>) = d(r>)/d(s)
e_n> = e_dot_t>/mag( e_dot_t> )
e_dot_t> = (v/rho)e_n> = d(r>)/d(s)
rho = mag( 1/k )
k> d(`e_t,ds) = d( d(`r, ds), ds) = mag( d(psi)/d(s)
solute Zero
___
/ \
 
\ / /  \
_\ /_ \__/ Omega
_____, _______
/    
,/    Pi
___,
/ 
\_ tau T
=! is identically equal to, defined as
>< does not equal
=about= is approximately equal to
> is greater than
>= is greater than or equal to
>> is much greater than
< is less than
=< is less than or equal to
<< is much less than
+
 plus or minus, error margin
: is to, ratio, such that
. .
. . as
.
. . therefore
. .
. because
\__/
\/ for all
___
//
//
//__ integer set
\ 
\\ 
 \\ natural set
___
/ \
 
\_/ rational set
\
___
/ \

\__/ complex set
____
 \
_/
 \\ real set
 \\
__
 Not All
__
__
__ there exists
{ } set
< > mean
__
/
V (square) root of
* denotes an operation
/
/__ angle
__
== congruent

 parallel
__ perpendicular
___
/ \
  intersection
 
\___/ union
___
/
\___ a subset
_ /_
/ /
\/___ is not a subset
/
____
/____
\____ belong to
_ /_
/_/__
\/___ does not belong to
/
/
(/) empty set
/
,\' cardinality
^
/_\ finite difference or increment
colon
semicolon
% per cent
' first derivative, feet, arcminutes
" decond derivative, inches, arcseconds
degrees
~ difference
... ellipsis
<=> is equivalent to
=> implies
! factorial
oo infinity
/
 integral
/
> maps into, approaches the limit
___
\
/__ the sum of the terms indicated (sigma)
__
 the product of the terms indicated (pi)
oc is proportional to
__
\/ vector differential
Fraction
fraction number divided by another number.
Denoted by a slash / or a bar .
Two numbers, a and b can be shown thus: a/b.
a is called the denominator and b
is called the numerator.
General Differentials
d( k)/d(x) = 0 constant not change with repect to x
d(k*x)/d(x) = k slope does change with repect to x
d( u*v)/d(x) = u*d(v)/d(x) + v*d(u)/d(x)
u is like a constant when observering d(v)/d(x)
d( y)/d(x) = (d( y)/d(t))*(d( t)/d(x) )
d( x^n)/d(x) = n*x^(n1)
d( ln(x))/d(x) = 1/x x > 0
d( exp(x))/d(x) = exp(x)
d( a^x)/d(x) = ln(a)*a^x
d( sin(x))/d(x) = cos(x)
d( cos(x))/d(x) = sin(x)
Gradient
gradient oflinear graph of y against x
can be found by choosing any two points on
the line and dividing the difference in y coordinates
by the difference in x coordinates.
dy/dx is used to represent the gradient of a line:
 /
 /
 /  dy
 /__
 / dx
/______
gradient at point of curve which is locally straight
(the graph appears to be a straight line when magnified)
is found by considering the gradient of the tangent that
point.
process of finding dy/dx for a given function y of x
is called differentiation.
Int( x^n , d(x) oo => +oo ) = ( 1/(n+1) )*x^(n+1) +C
n >< 1
Int( 1/x , d(x) oo => +oo ) = ln(x) +C
Int( u*d(v)/d(x), d(x) oo => +oo ) =
u*v  Int( v*d(u)/d(x), d(x) oo => +oo )
Int( sin(a*x) , d(x) oo => +oo ) = cos(a*x)/a +C
Int( cos(a*x) , d(x) oo => +oo ) = sin(a*x)/a +C
Subtraction is the inverse operation of addition
and has symbol  (minus). In the expression
a  b = x, a is called the minuend,
b is called the subtrahend and x is called the
difference.
Normal
normal at point on graph is line at rightangles
to the tangent at that point:
\ / normal
_\/\
/ \\/
 \tangent
Parametric Equations
Parametric equations express coordinates
of points on a surface or curve in terms of other
variables (or parameters)
which can be regarded as individual variables.
Perimeter
perimeteris the sum of length of each of vertices.
For a circle, the
perimeter is more often called the circumference.
Pythagoras's Theorem
The Pythagoras's theorem simply states
square of the hypotenuse of a rightangled triangle is
equal to the sum of the squares on the other two sides.
theorem extended to three dimensions,
^
/\
 * P(x,y,z) r^2 = x^2 + y^2 + z2

_________\
/ /
/
/_
Radius of Curvature
radius of curvature, p, of line or plane
radius of a circle or sphere that would fit into curve:
____
___/ ^ \___

 p
Sector
A sector is shape created by two radii of a circle
or ellipse and the arc connecting them:
When is measured in radians, length of arc l is
____ l
/ A \
\ /
\theta/
\ / r
\ /
V
theta in radians , the lenght of arc l
l = r*theta
and the area A is found by:
A = (1/2)*r^2*theta
CUBIC SPLINES convenient way of handling smooth and graceful curves
with a sparse data
influence point #1
x_1,y_1 o
/ x_2,y_2 influence
o_ point #2
/ ______ 
__/ __
_/ o
/ x_3,y_3
o x_0,y_0 final point
initial points
x = A*t^3 + B*t^2 + C*t +D
y = E*t^3 + F*t^2 + G*t +H
A = x_3  3*x_2 + 3*x_1  1*x_0
B = 3*x_2  6*x_1 + 3*x_0
C = + 3*x_1  3*x_0
D = + 1*x_0
E = y_3  3*y_2 + 3*y_1  1*y_0
F = 3*y_2  6*y_1 + 3*y_0
G = + 3*y_1  3*y_0
H = + 1*y_0
x_0 = D
x_1 = D + C/3
x_2 = D + C*2/3 + B/3
x_3 = D + C + B + A
y_0 = H
y_1 = H + G/3
y_2 = H + G*2/3 + F/3
y_3 = H + G + F + E
PRIME NUMBERS 1 2 3 5 7 11 13 17 19 23
29 31 37 41 43 47 53 59 61 67
71 73 79 83 89 97 101 103 107 109
combinations /n\ n draw r n!/(r!*(nr)!)
\r/
permutation permutations order of combination happens also counts
always r! larger than combinations
M truncated N (M)n = M!/(Mn)!
Euler Eq exp(jx) =cos(x) +j*sin(x)
PI=4*(11/3+1/5+etc)
ln=2.30258509log10
lnx=(x1)/x+
((x1)/x)^2/2 +etc
e^x=1+x+x^2/2!+etc
sinx=xx^3/3!+x^5/5!+etc
cosx=1x^2/2!+x^4/4!+etc
1/(x1)=1+x+x^2+x^3+etc
Note 0! = 1! =1
(H+T)^N = sum of all n
terms Cn*H^n*T^(n1)
Cn =N!/(n!*(Nn)!)
e^jx=cosxjsinx
x=b/2*a +/ sqrt(b^24*a*c)/2*a
area triangle= (s*(sa)*(sb)*(sc))^1/2
where s=(a+b+c)/2
integer square
1+3=4 1+3+5=9 etc...
Permutation
N draws of M objects
P(m/n)=m!/(mn)!
Combinations
P(m/n)=n!*C(m/n)
Means=Mx= f(x)/n
Sx={(nx^2xx)/n(n1)}^1/2
Sy={(ny^2yy)/n(n1)}^1/2
Syx={(nxyyx)/n(n1)}^1/2
Covariance=
Sxy=(nxyyx)/n(n1)
Corelation R= Sxy^2/(Sy*Sx)
NORMAL DISTR (x)=
{1/sd^212} exp^((x/2sd)^2)
Q(x) = (x)x for 0infinity )
irrational number <>= ratio of two integers
was also responsible for
the notations of function, f(x),
the letter pi
i to represent the square root of 1
__
\
the /_ summation symbol
blind in 1768 but still continued his work..
C or gamma = 0.57721566 Euler's Constant
limit( sum( 1/r, r = 1=>n) ln(n)) n>oo

Ergodic system which ensemble averages equal time averages

_
LOGIC A   B AND A and B
A _ B OR A or B
_
A _ B SUBSET A is subset of B
(/) Zero a null set
_
A NOT a compliment
S space (all) P(S) =1
E event P(E) >=0
If A is a subset of B P(B/A) = 1
If B is a subset of A P(B/A) >= P(B)
A or _A = S
A and _A = null
A and S = A
A and B = B and A
_A or _B = NOT (A and B)
_A and _B = NOT (A or B)
P(E) >=0 E = event
P(A xor B) = P(A) + P(B) P(A and B)
P(B/A) = probable B given A = P(A and B)/P(A) = P(B/A)
If A and B are independent P(A and B) =P(A)*P(B)
If probablity =1/x odds are x1 to 1 ..
P(A) =1/11 odds 10 to 1
Quadratic Equation Have a*x^2 + b*x + c = 0
find X X = (b +/sqrt(b^24*a*c))/2a

Demorgans Rule
Not(A and B) = A' or B'
Not(A or B) = A' and B'

____
/ \ Circle circumference = 2*PI*radius
/ \ Area = PI*radius^2
 _____\
 r / Sphere Area = 4*PI*radius^2
\ / Volume = (4/3)*PI*radius^3
\__ __/


/\ triangle Area = base*height/2
/ \ pyramide volume = Base_Area*height/3
/____\ cone Area = PI*Radius*length
volume = PI*height*radius^2/3

____
  square Area = height*wide
  perimeter = 2*(height + wide)
____ box Area = 6sides
volume = height*width*length

entropy =K*ln(Number_Of_Ways)

delta_p*delta_x >= h/(2*PI)

LINES (xx1)/a = (yy1)/b = (zz1)/c

PLANES ^ z
/\
 PLANE
\ X/A +y/B +z/C +D = 0
/ \
 \ P = D/sqrt(A^2+B^2+C^2+)
/ ___\______\ (dist orig to plane)
/ _ / y
// _
/ 
/_
x

ELLIPSE b
____ x^2/a^2 + y^2/b^2 = 1
/  \
/  \ foci = a*e
 X ______________ e = sqrt(a^2b^2)/a <1
 a
\ /
\__ __/


Given 3 sides of any triangle a,b,c
Area = sqrt(s*(sa)*(sb)*(sc)) s = (a+b+c)/2

MATRIX Conventions
1) order = m x n roll x column
2) multipy = roll times column
 > :  X  A_1_1 = A_row_col =sum(row1* column1)
 *: =  
V 
 >  :  X A_1_2 = A_row_col = sum(row1* column2)
 * :=  
 V 
ETC.....
3) [A]+[B] = [B]+[A] but [A]*[B] >< [B]*[A]
3) Identity [A]*[I] = [A]
[A]*[I]=[A] = a_1_1 a_1_2   1 0   a_1_1 a_1_2 
 * = 
 a_2_1 a_2_2   0 1   a_2_1 a21_2 
4) Cramer_Rule
y_1 = A1*x_1 +A2*x_2 x_1 =  y_1 A2  /
y_2 = A3*x_3 +A4*x_4  y_2 A4 / ( A1*A4  A2*A3 )
5) Inverse of a matrix Invers[A] =[A]^1
[A]*Invers([A]) = [I]
6) To make Invers([A])
1) replace each element by cofactor A_j_k
2) then transpos([A])
3) then divide by Det([A])
Invers([A]) =Adjoint([A])/Det([A])
7) Transpose a Matrix Transp([A]) = [A]^T
 a_1_1 a_1_2   a_1_1 a_2_1 
[A]=   Transp[A]=  
 a_2_1 a_2_2   a_1_2 a_2_2 
8) transp([A]*[B]) = transp([B])*transp([A])
9) Minor of a matrix minor( row,col,[A])
a1 b1 c1  . b1 c1 b1 c1
[A]= a2 b2 c2 minor of a2 => . . .  => b3 c3
a3 b3 c3  . b3 c3
10) cofactor(row,col,[A])
= ( 1^(row+col) )*minor( row,col,[A])
11) Determinate Det([A]) for 3X3 or 2x2
 a1 b1 c1
Det([A])=  a2 b2 c2 = (a1*b2*c3 +b1*c2*a3+c1*a2*b3) 
 a3 b3 c3 (a1*c2*b2 +b1*a2*c3+c1*b2*a3)

crest factor =Vpk/Vrms

inf
/
sqrt(PI)/4 =  x^2*exp(x^2/2)/*delta_x
/0

binomial distrition q+p =1
= (p+q)^n = P^n +p^(n1)*q*n!/(n1)!
....p^(nk)*q^k*n!/(k!*(n1)!)

NOTE 1! =0! =1

if p = win and q=lose
Prob_of_k_sucesses = combination( n take k)*p^k*q(nk)
combination( n take k) = n!/( k!*(nk)!)
SD = sqrt(n*p*q)
ave =n*p

guassian P(x) =exp(x^2/2)/sqrt(2*PI)
X
/
F(x) =  exp(x^2/2)/sqrt(2*PI)*delta_x
/inf
ERF error fuction = 1F(x)

( sin(x)(x) )^2 +( (cos(x) )^2 =1
exp(x) = 1 + x + x^2/2! + x^3/3! ....
cos(x) = 1  x^2/2! ....
sin(x) = + x  x^3/3! ....
ln(x) = (x1)/x + ( (x1)/x )^2/2 + ( (x1)/x )^3/3...
standard dev =sqrt( ( ( (sum(x^2) sum(x)^2)/n )/(n1) )
__ __ __2
Laplacian V = \/ dot (\/ V) = \/ V = rho/e

__
DIV D =\/ dot D> =dDx/dx +dDy/dy +dDz/dz = RHO
D> =displacement =Electric field+polarization=eo*E>+P>
[ charge changes field ]

__
DIV B = \/ dot B> = dBx/dx + dBy/dy + dBz/dz = ZERO
[ No unipoles ]

__
CURL H= \/xH>i> j> k> 
 
=d/dx d/dy d/dz= J> + delta_D/delta_t
 
Hx Hy Hz 
[ IN = H ]

__
CURL E = \/ x E> i> j> k> 
 
= d/dx d/dy d/dz =delta_B/delta_t
 
[ V=dB/dt ] Ex Ey Ez 
__
GRAD V =\/V =dV/dx_i>+dV/dy_j>+dV/dz_k>scalar to vector
__
DIV V =\/ dot V>= dVx/dx +dVy/dy +dVz/dz vector to scalar
V(x,y,z) = Vx_i> + Vy_j> + Vz_k>
__
CURL V = \/ x V>  i> j> k> 
 
=  d/dx d/dy d/dz 
 
 Vx Vy Vz 
__ __ __2
Laplacian = \/ dot (\/ V) = \/ V
__ __
DIV CURL = \/ dot (\/ x V) = ZERO
__ __ __ __ __ __2
CURL CURL = \/ x (\/ x V) = \/ (\/ dot \/) + \/ V

Q(x) = +0.31938153*T T = 1/(1+.2316419*x) X>= 0
0.35653782*T^2
+1.781477937*T^3
1.8212559787*T^4
+1.330274429*T^5


X
XX Q(x)
XXX
______ _..XXXXx..________
<15.8%>
V <2.2750%>
xXXXx  <0.1350%>
XXXXXXX V  <0.0032%>
XXXXXXXXX V 
XXXXXXXXXXX V
__..xXXXXXXXXXXXXXx..__
        
4 3 2 1 0 1 2 3 4

Vector
A vector is a set of numbers [] , ...,
[tex2html_wrap_inline9001] that transform as
[]
This makes vector a Tensor of Rank 1.
Vectors are invariant under
translation, and reverse sign upon inversion.
A vector is uniquely specified
by giving its Divergence and Curl within a region a
and its normal component
over the boundary, a result known as Helmholtz's Theorem
79). A vector from a point A to a point B is denoted
[] , and a vector v may be denoted
[] , or more commonly, [] .