======================SCIEN_MATH============================= NEW CHEATSHEETS http://www.idea2ic.com/CheatSheet_2/CHEATSHEETS_2.html These are my personal cheatsheets designed to make access to detailed information much easier to find. They are being put on the web mainly because for now it is easy to do. The new rev of cheatsheets are the ones being continually upgraded. 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 -- \ / (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! 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) Right-Angled Triangle 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 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 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 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), __ \ 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 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^(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 The Pythagoras's theorem simply states 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 (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) Means=Mx= f(x)/n Sx={(nx^2-xx)/n(n-1)}^1/2 Sy={(ny^2-yy)/n(n-1)}^1/2 Syx={(nxy-yx)/n(n-1)}^1/2 Covariance= Sxy=(nxy-yx)/n(n-1) 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 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, [] .