Miscellaneous

Riemann Hypothesis

zeta(s) = 1/1^s + 1/2^s + 1/3^s + ... (s = a + it) all 0's of zeta(s) in strip 0<=a<=1 lie on central line a=1/2

Twin Primes occur infinitely

Twin primes are primes that are 2 integers apart. Examples include 5 & 7, 17 & 19, 101 & 103

Goldbach's Postulate

Every even # > 2 can be expressed as the sum of 2 primes.
4=2+2, 6=3+3, 8=3+5, 10=5+5, 12=5+7, .. , 100=3+97, ...

Euclid's Parallel Postulate

Through a point, not on a line, there exists exactly 1 line parallel to the given line. (Then there's those non-Euclidean people...)

 (k=1..) 1/kⁿ = ?

Although others have found that this expression equals PI² / 6 when n=2, PI⁴ / 90 when n = 4 and similar solutions for all possible even values of n, no one has discovered an exactvalue when n is an odd integer (3, 5, 7, ...) (note: when n=1, the sum does not converge, but it does has relations to thegamma constant).

 Vector Notation: The lower case letters a-h, l-z denote scalars. Uppercase bold A-Z denote vectors. Lowercase bold i, j, k denote unit vectors. <a, b>denotes a vector with components a and b. <x₁, .., x_n>denotes vector with n components of which are x₁, x₂, x₃, ..,x_n. |R| denotes the magnitude of the vector R.

|<a, b>| = magnitude of vector = (a ²+ b ²)

|<x₁, .., x_n>| = (x₁²+ .. + x_n²)

<a, b> + <c, d> = <a+c, b+d>

<x₁, .., x_n> + <y₁, .., y_n>= < x₁+y₁, .., x_n+y_n>

k <a, b> = <ka, kb>

k <x₁, .., x_n> = <k x₁, .., k x₂>

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<a, b> <c, d> = ac + bd

<x₁, .., x_n> <y₁, ..,y_n> = x₁ y₁ + .. + x_n y_n>

R  S= |R| |S| cos ( = angle between them)

R  S= S  R

(a R)  (bS) = (ab) R  S

R  (S + T)= R  S+ R  T

R  R = |R| ²

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|R x S| = |R| |S| sin ( = angle between both vectors). Direction of R xS is perpendicular to A & B and according to the right hand rule.

        | i  j  k |

R x S = | r1 r2 r3 | = / |r2 r3|   |r3 r1|   |r1 r2| \

        | s1 s2 s3 |   \ |s2 s3| , |s3 s1| , |s1 s2| /

S x R = - R x S

(a R) x S = R x (a S) = a (Rx S)

R x (S + T) = R x S + Rx T

R x R = 0

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If a, b, c = angles between the unit vectors i, j,k and R Then the direction cosines are set by:

    COs a = (R  i) / |R|; COs b = (R  j) / |R|; COs c = (R  k) / |R|

|R x S| = Area of parrallagram with sides Rand S.

Component of R in the direction of S = |R|COs  = (R  S) / |S|(scalar result)

Projection of R in the direction of S = |R|COs  = (R  S) S/ |S| ² (vector result)

Prelude: A vector, as defined below, is a specific mathematical structure. It has numerous physical and geometric applications, which result mainly from its ability to represent magnitude and direction simultaneously. Wind, for example, had both a speed and a direction and, hence, is conveniently expressed as a vector. The same can be said of moving objects and forces. The location of a points on a cartesian coordinate plane is usually expressed as an ordered pair (x, y), which is a specific example of a vector. Being a vector, (x, y) has a a certain distance (magnitude) from and angle (direction) relative to the origin (0, 0). Vectors are quite useful in simplifying problems from three-dimensional geometry.
Definition:A scalar, generally speaking, is another name for "real number."
Definition: A vector of dimension n is an ordered collection of n elements, which are called components.
Notation: We often represent a vector by some letter, just as we use a letter to denote a scalar (real number) in algebra. In typewritten work, a vector is usually given a bold letter, such as A, to distinguish it from a scalar quantity, such as A. In handwritten work, writing bold letters is difficult, so we typically just place a right-handed arrow over the letter to denote a vector. An n-dimensional vector A has n elements denoted as A1, A2, ..., An. Symbolically, this can be written in multiple ways:

A = <A1, A2, ..., An>
A = (A1, A2, ..., An)

Example: (2,-5), (-1, 0, 2), (4.5), and (PI, a, b, 2/3) are all examples of vectors of dimension 2, 3, 1, and 4 respectively. The first vector has components 2 and -5.
Note: Alternately, an "unordered" collection of n elements {A1, A2, ..., An} is called a "set."
Definition: Two vectors are equal if their corresponding components are equal.
Example: If A = (-2, 1) and B = (-2, 1), then A = B since -2 = -2 and 1 = 1. However, (5, 3) not_equal (3, 5) because even though they have the same components, 3 and 5, the component do not occur in the same order. Contrast this with sets, where {5, 3} = {3, 5}.
Definition: The magnitude of a vector A of dimension n, denoted |A|, is defined as

|A| = sqrt(A1^2 + A2^2 + ... + An^2)

Geometrically speaking, magnitude is synonymous with "length," "distance", or "speed." In the two-dimensional case, the point represented by the vector A = (A1, A2) has a distance from the origin (0, 0) of sqrt(A1^2 + A2^2) according to the pythagorean theorem. In the three-dimension case, the point represented by the vector A = (A1, A2, A3) has a distance from the origin of sqrt(A1^2 + A2^2 + A3^2) according to the three-dimensional form of the Pythagorean theorem (A box with sides a, b, and c has a diagonal of length sqrt(a2+b2+c2) ). With vectors of dimension n greater than three, our geometric intuition fails, but the algebraic definition remains.
Definition: The sum of two vectors A = (A1, A2, ..., An) and B = (B1, B2, ..., Bn) is defined as

A + B = (A1 + B1, A2 + B2, ..., An + Bn)

Note: Addition of vectors is only defined if both vectors have the same dimension.
Example:

(2, -3) + (0, 1) = (2+0, -3+1) = (2, -2).
(0.1, 2) + (-1, PI) = (0.1 + -1, 2 + PI) = (-0.9, 2+PI)

Justification: Physical and geometric applications warrant such a definition. IF a train travels East at 5 meters/second relative to the ground, which will be denoted in vector notation as VT = (0, 5), and a person on the train walks South at 1 meter/second relative to the train, which will be denoted as VP = (-1, 0), THEN the direction and speed that the person is traveling relative to the ground is represented by the vector VG = VT + VP = (0, 5) + (-1, 0) = (0 + -1, 5 + 0) = (-1, 5). This vector has a magnitude of |VG| = sqrt((-1)^2 + 5^2) = sqrt(6) = 2.449..., which means that the person is traveling at about 2.449 meters/second relative to the ground and the net direction is mostly East but slightly South.
Definition: The scalar product of a scalar k by a vector A = (A1, A2, ..., An) is defined as

kA = (kA1, kA2, ..., kAn)

Example:

2(5, -4) = (2*5, 2*-4) = (10, -8)
-3(1, 2) = (-3*1, -3*2) = (-3, -6)
0(3, 1) = (0*3, 0*1) = (0, 0)
1(2, 3) = (1*2, 1*3) = (2, 3)

Note: In general, 0A = (0, 0, ..., 0) and 1A = A, just as in the algebra of scalars. The vector of any dimension n with all zero elements (0, 0, ..., 0) is called the zero vector and is denoted 0.

Basic Operations

i = (-1)i ² = -1 
1 / i = -i 
i ^4k = 1; i ^(4k+1) = i; i ^(4k+2) = -1; i ^(4k+3) = -i (k = integer)
( i ) = (1/2)+ (1/2) i 

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Complex Definitions of Functions and Operations

(a + bi) + (c + di) = (a+c) + (b + d) i
(a + BI) (c + DI) = ac + adi + bci + bdi ² = (ac - bd) + (ad +bc) i
1/(a + BI) = a/(a ² + b ²) - b/(a ² + b ²) i
(a + BI) / (c + DI) = (ac + BD)/(c ² + d ²) + (BC - ad)/(c ² +d ²) i
a² + b² = (a + BI) (a - BI)   (sum of squares)
e ^(i ) = cos + i sin  
n ^{(a + BI)} = (COs(b ln n) + i sin(b ln n))n ^a
if z = r(COs + i sin ) then z ⁿ = r ⁿ ( COs n+ i sin n )(DeMoivre's Theorem)
if w = r(COs + i sin );n=integer, then there are n complex nth roots (z) of w for k=0,1,..n-1:
z(k) = r ^(1/n) [ COs( (+ 2(PI)k)/n ) + i sin( (+ 2(PI)k)/n ) ]
if z = r (COs + i sin ) then ln(z) = ln r + i 
sin(a + BI) = sin(a)cosh(b) + COs(a)sinh(b) i
COs(a + BI) = COs(a)cosh(b) - sin(a)sinh(b) i
tan(a + BI) = ( tan(a) + i tanh(b) ) / ( 1 - i tan(a) tanh(b))
= ( sech ²(b)tan(a) + sec ²(a)tanh(b) i ) / (1 + tan ²(a)tanh ²(b))

Trigonometric Graphs

(Math | Trig | Graphs)

 
 

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Algebraic Graphs (Math | Miscellaneous | Algebra)       Conic Sections (see also Conic Sections) Point x^2 + y^2 = 0 Circle x^2 + y^2 = r^2 Ellipse x^2 / a^2 + y^2 / b^2= 1 Ellipse x^2 / b^2 + y^2 / a^2= 1 Hyperbola x^2 / a^2 - y^2 / b^2= 1 Parabola 4px = y^2 Parabola 4py = x^2 Hyperbola y^2 / a^2 - x^2 / b^2= 1 For any of the above with a center at (j, k) instead of (0,0), replace each x term with (x-j) and each y term with (y-k) to get the desired equation.