======================SCIEN_MATH=============================================== PI = 4*(1-1/3+1/5+etc) ln = 2.30258509log10 lnx = (x-1)/x+ ((x-1)/x)^2/2 +etc e^x = 1+x+x^2/2!+etc sinx = x-x^3/3!+x^5/5!+etc cosx = 1-x^2/2!+x^4/4!+etc 1/(x-1) = 1+x+x^2+x^3+etc Note 0! = 1! =1 ------------------------------------------------------------------------------- e^jx=cosx-jsinx ------------------------------------------------------------------------------- Quadratic Equation Have a*x^2 + b*x + c = 0 find X X = (-b +/-sqrt(b^2-4*a*c))/2a ------------------------------------------------------------------------------- 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!*(n-r)!) \r/ ------------------------------------------------------------------------------- permutation permutations order of combination happens also counts always r! larger than combinations M truncated N (M)n = M!/(M-n)! ------------------------------------------------------------------------------- Euler Eq exp(jx) =cos(x) +j*sin(x) 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", ------------------------------------------------------------------------------- Addition process of finding the sum of the addend and the augend. ------------------------------------------------------------------------------- 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. ------------------------------------------------------------------------------- 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. ------------------------------------------------------------------------------- Alegebra Rules 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) ) Power Rules x^1 = x x^0 = 1 x^(-n) = 1/x^b x^(1/2) = sqrt(x) x^a*x^b = x^(a+b) Log Rules 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) ------------------------------------------------------------------------------- Right-Angled Tria Right-angled 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 ------------------------------------------------------------------------------- Trig 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 ------------------------------------------------------------------------------- differentiation differentiate( a )by(dx) = 0 differentiate( a*x )by(dx) = a differentiate( 1/x )by(dx) = -1/x^2 differentiate( x^n )by(dx) = n*x^(n-1) differentiate( abs(x) )by(dx) = x/abs(s) = sgn(x) differentiate( x^(1/2) )by(dx) = (1/2)*x^(-1/2) differentiate( c^x )by(dx) = c^x*ln(c) differentiate( exp(x) )by(dx) = exp(x) differentiate( exp(a*x) )by(dx) = a*exp(x) differentiate( ln(x) )by(dx) = 1/x differentiate( log_c(x) )by(dx) = 1/(x*ln(c)) = log_c(e)/x differentiate( sin(x) )by(dx) = cos(x) differentiate( cos(x) )by(dx) = -sin(x) differentiate( sin(a*x) )by(dx) = a*cos(x) differentiate( f(x)*g(x) )by(dx) = f*differentiate(g(x))by(dx) + g*differentiate(f(x))by(dx) differentiate( f(x)+g(x) )by(dx) = differentiate(f(x))by(dx) + differentiate(g(x))by(dx) differentiate( exp(u) )by(dx) = exp(u)*differentiate( u )by(dx) differentiate( b^u )by(dx) = b^u*ln(b)*differentiate( u )by(dx) ------------------------------------------------------------------------------- integration integrate( a )by(dx) = a*x +constant integrate( x^n )by(dx) = x^(n+1)/(n+1) +constant integrate( 1/x )by(dx) = ln(abs(x)) +constant integrate( a^x )by(dx) = a^x/ln(a) +constant integrate( exp(a*x) )by(dx) = exp(a*x)/a +constant integrate( ln(x) )by(dx) = x*ln(x) -x +constant integrate( sin(x) )by(dx) = -cos(x) +constant integrate( cos(x) )by(dx) = +sin(x) +constant integrate( tan(x) )by(dx) = -ln(abs(cos(x))) +constant ------------------------------------------------------------------------------- General Differential 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 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^(n-1) 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) ======================MATRIX===================================== MATRIX order = m x n roll x column 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..... Add [A]+[B] = [B]+[A] but [A]*[B] >< [B]*[A] 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 | 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 ) Inverse matrix Invers[A] =[A]^-1 [A]*Invers([A]) = [I] 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]) Transpose Matrix Transp([A]) = [A]^T | a_1_1 a_1_2 | [A]= | | | a_2_1 a_2_2 | | a_1_1 a_2_1 | Transp[A]= | | | a_1_2 a_2_2 | transp([A]*[B]) = transp([B])*transp([A]) Minor a matrix minor( row,col,[A]) | a1 b1 c1| [A]= | a2 b2 c2| | a3 b3 c3| | . b1 c1| |b1 c1| minor of a2 => | . . . | => |b3 c3| | . b3 c3| cofactor cofactor(row,col,[A]) = ( -1^(row+col) )*minor( row,col,[A]) 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) ------------------------------------------------------------------------------- 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 10-1 zeta Z 1021 centi c 10-2 exa E 1018 milli m 10-3 peta P 1015 micro u 10-6 tera T 1012 nano n 10-9 giga G 109 pico p 10-12 mega M 106 femto f 10-15 kilo k 103 atto a 10-18 hecto h 102 zepto z 10-21 deca d 101 yocto y 10-24 ------------------------------------------------------------------------------- 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 = 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 (1707-1783). Eule also responsible for notations of function, f(x), ------------------------------------------------------------------------------- __ \ summation 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 (n-1)!=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 ------------------------------------------------------------------------------- Euler Eq exp(jx) =cos(x) +j*sin(x) 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) = ( fï(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 Á 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 Different 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^(n-1) 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 co-ordinates by the difference in x co-ordinates. 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 right-angles to the tangent at that point: \ / normal _\/\ / \\/ | |\tangent ------------------------------------------------------------------------------- Parametric Equations Parametric equations express co-ordinates 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 square of the hypotenuse of a right-angled 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!*(n-r)!) \r/ ------------------------------------------------------------------------------- permutation permutations order of combination happens also counts always r! larger than combinations M truncated N (M)n = M!/(M-n)! ------------------------------------------------------------------------------- Euler Eq exp(jx) =cos(x) +j*sin(x) PI = 4*(1-1/3+1/5+etc) ln = 2.30258509log10 lnx = (x-1)/x+ ((x-1)/x)^2/2 +etc e^x = 1+x+x^2/2!+etc sinx = x-x^3/3!+x^5/5!+etc cosx = 1-x^2/2!+x^4/4!+etc 1/(x-1) = 1+x+x^2+x^3+etc Note 0! = 1! =1 ------------------------------------------------------------------------------- Head + Tails (H+T)^N = sum of all n terms Cn*H^n*T^(n-1) Cn =N!/(n!*(N-n)!) ------------------------------------------------------------------------------- e^jx=cosx-jsinx x=-b/2*a +/- sqrt(b^2-4*a*c)/2*a ------------------------------------------------------------------------------- area triangle = (s*(s-a)*(s-b)*(s-c))^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!/(m-n)! Combinations P(m/n)=n!*C(m/n) ------------------------------------------------------------------------------- POWERPOLY FIT y=b0 +b1x +b2x^2+etc for n terms/n equations áy=nb0+b1áx+b2áX^2+etc áyx=b0áx+b1áx^2 +b2áx^3 +etc áyx^m=b0áx^m+b1áx^(m+1) +b2áx^(m+2) +etc use Cramer to solve for b terms ------------------------------------------------------------------------------- ECLIPSE SPHERE x^2/a^2+y^2/b^2=1 S =4*PI*R^2 Foci=ae V = (4*PI*R^3)/3 e={(a^2-b^2)/a}^1/2 ------------------------------------------------------------------------------- PARABOLA x^2=4yp HYPERBOLA p=foci x^2/a^2-y^2/b^2=1 ------------------------------------------------------------------------------- CURVE FIT fy(t)=b*fx(t)+a b=(náfxfy-áxáy)/(ná(fx)^2-(áfx)^2) a=áfy/n-b*áfx/n r^2=SDxy^2/(SDx*SDy) SDxy^2=(áfxfy-áfxáfy/n)^2 SDx=á(fx^2)-(áfx)^2/n SDY=á(fy^2)-(áfy)^2/n coeffcient of determination= a probablity term= b*(Mxy-MxMy)/(MSy-My^2) 1) y= bx +a fy= y fx= x 2) y= A*exp^bx lny= lnA +bx fy=lny fx=x a= lnA 3) y= a +b*lnx fy= y fx= lnx 4) y= a*x^b lny = b*lnx +lnA ------------------------------------------------------------------------------- POWERPOLY FIT y=b0 +b1x +b2x^2+etc for n terms/n equations áy=nb0+b1áx+b2áX^2+etc áyx=b0áx+b1áx^2 +b2áx^3 +etc áyx^m=b0áx^m+b1áx^(m+1) +b2áx^(m+2) +etc use Cramer to solve for b terms ------------------------------------------------------------------------------- Permutation N draws of M objects P(m/n)=m!/(m-n)! Combinations P(m/n)=n!*C(m/n) ------------------------------------------------------------------------------- Means Mx =á f(x)/n Sx={(náx^2-áxáx)/n(n-1)}^1/2 Sy={(náy^2-áyáy)/n(n-1)}^1/2 Syx={(náxy-áyáx)/n(n-1)}^1/2 Covariance Sxy =(náxy-áyáx)/n(n-1) Corelation R = Sxy^2/(Sy*Sx) ------------------------------------------------------------------------------- NORMAL DISTR €(x)= {1/sdÌ2ü} exp^(-(x-µ/2sd)^2) Q(x) =¼ €(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) ------------------------------------------------------------------------------- probablity 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 x-1 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^2-4*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 (x-x1)/a = (y-y1)/b = (z-z1)/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^2-b^2)/a <1 | |a \ / \__ __/ ---------- ------------------------------------------------------------------------------- Given 3 sides of any triangle a,b,c Area = sqrt(s*(s-a)*(s-b)*(s-c)) 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^(n-1)*q*n!/(n-1)! ....p^(n-k)*q^k*n!/(k!*(n-1)!) ------------------------------------------------------------------------------- NOTE 1! =0! =1 ------------------------------------------------------------------------------- if p = win and q=lose Prob_of_k_sucesses = combination( n take k)*p^k*q(n-k) combination( n take k) = n!/( k!*(n-k)!) 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 = 1-F(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) = (x-1)/x + ( (x-1)/x )^2/2 + ( (x-1)/x )^3/3... standard dev =sqrt( ( ( (sum(x^2) -sum(x)^2)/n )/(n-1) ) ------------------------------------------------------------------------------- __ __ __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, [] . ------------------------------------------------------------------------------- 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 -- \ / (2n-1) -- 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!