Appendix A

Greek Alphabet

The physical sciences make extensive use of the Greek alphabet. Since at least parts of this alphabet may be new to some readers, and because the way the lower-case letters are drawn can vary, we list in the table A.I the upper- and lower-case characters and their names. No attempt is made to list pronunciations since it is common to mix Greek and English.

I Lower Upper Case Name I Lower Upper Case Name

0: A alpha v N nu {3 B beta ~ ~ Xl

'Y r gamma 0 0 omicron 8 ~ delta 7r II pi E,€ E epsilon p P rho ( Z zeta a E sigma TJ H eta T T tau e,i) 8 theta v Y upsilon t I iota ¢,cp <I> phi K, K kappa X X chi A A lambda 1/J 'lI psi J-L M mu w n omega

Table A.t. The Greek Alphabet

267

Appendix B

Units

The following is a printout of the ASCII file 'units'. Use it to determine the abbreviations of the constants and symbols. The user can add more constants and conversion factors with any ASCII editor.

# LIST OF UNITS AND CONSTANTS # # converts expressions automatically to MKSA (51) units # prepared by W. E. Baylis for use in his text "Theoretical Methods in the # Physical Sciences, An Introduction to Problem Solving with MAPLE V", # Birkhauser Boston 1994. # based on E. R. Cohen and B. N. Taylor. Rev Mod Phys 59. 1121 (1987) If # Metric prefixes Exa := 10 A18 Peta := 10 A15 Tera 10 A12 Giga 10 A9 Mega 10 A6 kil 0 10 A3 mi 11 i 10A(-3) micro 10A(-6) nano 10A(-9) pico 10A(-12) femto 10 A(-15) atto 10A(-18) If

269

270

#Common conversions deg_ := Pi*rad_/180 cycle_ := 2*Pi*rad_ rad_ := 1 Hz_ := cycle_/s_ hr_ := 60*min_ min_ := 60*s_ day_ :~ 24*hr_ yr_ := 365.24219*day_ mm_ := milli*m_ cm_ := O.OI*m_ km_ := kilo*m_ in_ :- 2.54*cm_ ft_ := 12*in_ yd_ := 3*fC

APPENDIX B. UNITS

:#angular measure :# relation between cycle and radian :# radian (caution: some authors use cycle_=I) :# Hertz (frequenc~)

:#tropical year

mi := 5280*ft_ :#statute mile ly_ := 94605282*Giga*m_:#light-year pc_ := 3.261633*ly_ :#parsec hectare_ :- (100*m_)h2 :# 100 ares. unit of area acre_ := 0.40468564*hectare_:#acre. 160 sQ.rods 1 := m_ h3/1000 :11 1 iter L_ := 1_ :/1 alternative liter symbol ml :- mi 11 i *1_ cc_ := ml :/1 cubic centimeter ga1_ := 3.785412*1_ :# gallon (US. liquid) gal_imp:= 4.54609*1_ :# Imperial gallon qL :- gaL/4 :/1 quart(US. liquid) PC := qL/2 :# pint(US. liquid) g_ := kg_/1000 :# gram (cgs unit of mass) Tonne_ := 1000*kg_ :# metric ton lb_ := 0.453592*kg_ :# pound mass (do not confuse with 1b_force) ton_ := 2000*lb_ :# short ton oz_ := lb_/16 :# ounce (US. fluid) N_ := kg_*m_/s_h2 :# Newton (metric unit of force) dyne_ := g_*cm_/s_h2 :# cgs unit of force 1b_force := 4.44822*N_ :# caution: 1b is used both for force and mass P_ :- g_/(cm_*s_) :# poise. unit of viscosity J_ :="N_*m_ :# Joule (metric unit of energy) erg_ := dyne_*cm_ :# cgs unit of energy ca1_ := 4.1868*J_ :# calorie (cgs heat unit) Ca1_ := ki1o*cal_ :# dieticians calorie Btu_ := 1055.056*J_ :/1 British thermal unit eV_ := 1.60217733*10h (-19)*J_:# electon Volt (energy) Hartree_:-27.21139613*eV_ :# atomic unit of energy - 2 Rydbergs of energ) W_ :- J_/s_ :11 Watt hp_ := 745.7*W_ :# horsepower

271

Pa_ := N_/m_A2 :4/ Pascal. the metric unit of pressure atm_ :- 101325*Pa_ :4/ atmosphere = 29.9213 in_ of mercury bar_ := 100000*Pa_ :4/ bar of pressure torr_ := atm_1760 :4/ mm_ height of mercury L := J_/C :4/ metric unit of Volt StatV_ := 299.792458*L :/1 cgs (esu) unit of potential C := A_*s_ :4/ Coulomb (metric unit of electric charge) StatC := C/2997924580 :/1 cgs (esu) unit of electric charge StatA_ := A_/2997924580 :11 cgs (esu) unit of current F_ := C/V_ :4/ Farad (metric unit of capacitance) Wb_ := V_*s_ :4/ Weber (metric unit of magnetic flux) H_ := V_*s_/A_ :4/ Henry (metric unit of inductance) ohm_ :- L/A_ :4/ metric unit of resistance T_ :- Wb_/m_A2 :/1 Tesla (metric unit of magnetic field B) gauss_ :- T_/I0000 :4/ cgs (emu) unit of magnetic induction B Oe_ := gauss_ :/1 oersted. cgs (emu) unit of magnetic field II 4/ Physical Constants mol_ :- 6.0221367*10A(-23) :4/Avogadro constant. 1 mole alpha_ := 1/137.036 :4/fine-structure constant c_ := 299792458*m_/s_ :/lspeed of light (exact) h_ := 6.6260755*10A(-34)*J_*s_:4/Planck constant hbar_ :- 1.05457266*10A(·34)*J_*s_:4/Planck constant/2 Pi e_ :- 1.60217733*10A(-19)*C_:4/elementary charge Ryd_inf:= 10973731.57/m_ :4/Rydberg constant for infinite mass mu_B :- 9.2740154*10A(-24)*J_/T_:4/Bohr magneton mu_O :- 4*Pi*10A(-7)*N_/A_A2 :4/permeability of vacuum eps_O :- I/(muO_*c_A 2) :4/permittivity of vacuum

H

G_ :- 6.6726*10A(-11)*m_A3/(kg_*s_A2):#Newtonian gravitational constant k_ := 8.31451*J_/(K_*mol_) :4/Boltzmann constant m_e :- 9.1093897*10A(-31)*kg_:4/electron mass m_p .= 1836.1527*m_e :4/proton mass m_n 1838. 68366*m_e :4/neutron mass u_ 1.66054*10A(-27)*kg_ :4/unified atomic mass unit (amu) M(C12)/12 m_sun 1.989*lOA30*kg_ :/lmass of sun m_earth:= 5.98*10A24*kg_ :/lmass of earth m_moon :- 7.34*lOA22*kg_ :4/mass of moon bohr_ :=-0.52917725*10A(-10)*m_:4/bohr radius aO (atomic unit of length) R_moon :- 1.74*Mega*m_ :4/average radius of moon R_earth:-.6.37*Mega*m_ :4/average radius of earth R_sun :- 696*Mega*m_ :4/average radius of sun r_earth:= 149.59787*Giga*m_ :4/average radius of earth's orbit about sun AU_ :- r_earth :4/Astronomical Unit r_moon :- 384.0*Mega*m_ :4/average radius of moon's orbit about earth g_acc :- 9.80665*m_/s_A2 :4/acceleration of gravity on earth's surface

Appendix C

Constants and Functions in Maple

We summarize below the names of a few common constants which Maple knows.

Constant Value

Pi 7r = 3.14159265 ... E e = exp(1) = 2.718281728459 ... I i=A gamma 'Y = 0.5772156649 ... infinity 00

The following table lists some of the common functions available in Maple. While some of these functions are in the Maple core, others are available in the Maple library. All those listed here can be invoked directly, without calling any special packages.

273

274 APPENDIX C. CONSTANTS AND FUNCTIONS IN MAPLE

Function abs(x) sqrt(x) sin(x) cos (x) tan(x) cot (x) esc (x) sec (x) arccsin(x) arctan (x) exp(x) sinh (x)

cosh(x) tanh (x)

coth((x) In(x) log10(x) n! GAMMA (x)

Beta(x,y)

binomial(n,m) rand erf(x) erfc(x) Dirac(x)

Description

Ixl .jX sine of x cosine of x tangent of x = sin(x)/ cos (x) cotangent of x = 1/ tan(x) cosecant of x = 1/ sin (x) secant of x = 1/ cos (x) inverse function of sine of x inverse function of tangent of x exponential eX hyperbolic sine of x, (eX - e-X)/2 hyperbolic cosine of x, (eX + e-X)/2 hyperbolic tangent of x, sinh(x)/ cosh(x) hyperbolic cotangent of x, 1/ tanh(x) natural logarithm of x common logarithm (base 10) of x n factorial, n(n - 1)(n - 2) ... (2)(1) Gamma function, r(x) = 1000 dttx - 1 exp( -t) r(n + 1) = n! for integer n Beta function B(x, y) = f~ dttX - 1(1 - t)y-l = f(x)f(y)/f(x + y) binomial coefficient, (::.) = m!(:~m)' random number error function 2 rx dse-s2 = 1 JV2x dte- t2 / 2

,~JO ~ -v'2x complementary error function, 1- erf (x) Dirac delta function, 6(x)

Index

A abelian, 208 absolute value

of complex number, 202 absorption coefficient, 5 acceleration

gravitational, 13 accuracy, 124 additive inverse, 92 affine space, 87 algebra

center of, 248 quaternion, 248

algebraic product, 228 analytic function, 71 angular momentum, 113, 243 angular velocity, 213 anticommutative, 111 Apple Macintosh, v, 31 arcsine function, 49 area, 178 Argand, Jean Robert, 205 ASCII, 34 assignment statement, 33 assocciativity

of functional composition, 49 associative, 92 associativity

of vector addition, 91

275

astronomical unit, 4 asymptotic expansion, 79 atmosphere

of earth, 14 atmospheric pressure, 14 atomic elements, 26 atomic mass, 27 attractor, 215 Atwood machine, double, 57 average, 125 average value

of a function, 179 Avogradro's number, 14

B ballistic trajectories, 109 basis, 107

left-handed, 112 orthonormal, 107 right-handed, 112

basis elements, 231 of the Pauli algebra, 230

basis vectors, 226 binary operators, 35 binomial distribution, 138 binomial expansion, 72 bins, 132 bivector, 229, 233, 235

spacetime, 255 block and tackle, 10

276

boost of paravectors, 252

boost parameter, 252 boosts

composition of, 254 of parabivectors, 255 successive, 253

branches of a function, 210

C calorie, 24 Calorie

dietician's, 28 Cartesian coordinates, 241 center of algebra, 248 centrifugal force, 263 chaos, 218 character strings, 35 chemical isotopes, v, 26 chemical symbols, 27 circular motion, 68

uniform, 249 cliffor, 225

even part, 247 imaginary part, 247 odd part, 247 real part, 247 scalar part, 247 vector part, 247

Clifford algebra, 225, 245 Clifford, William Kingdon, 227 collinear vectors, 99 communtivity

of vector addition, 89 commutative, 208

INDEX

commutative ring with identity, 208

complementary idempots, 261 complex conjugate, 209 complex conjugation, 245 complex numbers, 201, 248 complex phase, 206 complex plane, 202 complex variable, 201 composition of functions, 51 conservative, 216 constants in Maple, 273 continuation of function, 206 converge, 135 conversion

of units, 4 conversion factors, v, 22, 269 Coriolis force, 263 cosh, 74 cosh function, 205 cosine function, 47 counter, 138 coupled differential equations, 19 coupled oscillators, 221 covariance, 249 critical damping, 191, 222 cross product, 111, 227, 235

double, 117 cubic relation, 46 curvature method, 169 cycloid motion, 241-242 cyclotron frequency, 242

D data analysis, 121 de Moivre's theorem, 207 decay rate, 40

INDEX

definite integral, 178 degrees of freedom, 136, 221 density

of earth, 13 of nickel, 13 of water, 13

density function, 186 derivative, 49 determinant, 115, 117 differential equation, 176 differential equations

coupled, 191 differentiation, 52 dimensional analysis, 5 Dirac algebra

real, 250 directory tree, 19 diskettes, 20

3.5-inch, 21 5.25-inch, 20 formatting, 21

displacement, 86 distribution, 123

moments of, 133 distributions, 132

Gaussian, 134 square deviation of, 134 standard deviation of, 134 variance of, 134

distributive, 91 division by complex numbers, 208 domain of function, 45 dot product, 99, 235

of Pauli cliffors, 255 of vectors, 227

doubling time, 53 dual, 230

E earth, 12

atmosphere of, 14 density of, 13 gravitational field, 65 orbit about sun, 12

eccentricity, 193 eigenfrequency, 221 eigensolutions, 221 eigenvalues

of matrix, 115 Einstein, Albert, 13, 98

277

general theory of relativity, 98

principle of equivalence, 98 special theory of relativity,

249 summation convention, 107

Einstein's breath, 13 electromagnetic field, 256 electron configuration, 27 equation

linear, 186 equation of motion, 94 equations in Maple, 33 equilibrium

stable, 69, 218 unstable, 218

equilibrium position, 67 error function, 142 Euler, Leonhard, 205 Euler relation, 205 even cliffors, 247 even function, 73 event

not A, 144 events

278

mutually exclusive, 144 exp, 45 experiments, 135 exponential decay, 249 exponential function, 45, 49

of vector, 238 exponentiation, 39 expressions, 33 extrapolation, 147

F fictitious force, 97 field, 209, 245, 248 field lines, 65 file allocation table, 21 first-order Lagrange interpolation,

150 fixed point, 191 floppy disks, 20 floppy drives, 19 force

centrifugal, 263 Coriolis, 263

fractal, 211 fractal program, v Fractint, v, 211 frame of reference, 95

laboratory, 241, 252 rest, 252 rotating, 241

fuel consumption, 24 functions, 45

analytic, 71 composition of, 48 even, 73 inverses of, 49 odd, 73

of a vector, 238 of Pauli cliffors, 262 products of, 48 sums of, 48

functions in Maple, 273

G galaxies, 254 Galileo,66

INDEX

Gauss-Newton method, 169 Gauss-type integration, 185 Gaussian distribution, 134, 141 Geiger counter, 138 geometric algebra, 225, 233, 245

of spacetime, 250 Gibbs, J. Willard, 226 global minimum, 170 grade automorphism, 247 grades

of the Pauli algebra, 247 gradient, 167 gradient method, 168 gradient operator

paravector, 255 Grassmann, Herman, 227 gravitational acceleration, 3 gravitational constant G, 13 gravitational field of earth, 65 Greek letters, 267 group, 208

H half-life, 40 Hamilton, William Rowan, 227 hard drives, 19 Heaviside, Oliver, 226 helicity, 234

INDEX

help command, 38 Hermite interpolation, 154 Hermitean conjugate, 246 Hestenes, David, 227 histogram, 132 Hodge dual, 230 hyperbolic cosine, 74 hyperbolic sine, 74 hyperbolic trig functions, 205 hypersurface, 174

I idempotent elements, 261 imaginary i, 230, 245 imaginary axis, 202 imaginary number, 201

power of, 204 imaginary term, 199 indefinite integral, 175 inertial frames, 95 inner product

of vectors, 227 integration

numerical, 180 interpolation, 147 inverse

additive, 37 multiplicative, 37 of a vector, 238 of Pauli cliffor, 250, 262 of matrix, 115

inversion, 243 involution, 245 isotopes, 26

J Julia set, 211

K Kepler's second law, 244 Kepler's third law, 12 Kronecker delta, 107

L laboratory frame, 252 Lagrange interpolation, 152 laser light

279

reflections of, 243 least-squares method, 148 Levenberg-Marquardt algorithm,

170 Levi-Civita symbol, 112, 227 lifetime

of isotopes, 27 light-seconds, 4 light-year, 26 limits, 78 linear approximation, 62 linear combination, 91 linear dependence, 46

of vectors, 99 linear equation, 186 linear interpolation, 150 linear operations, 175 linear regression, 148 linear vector space, 92 linear fits, 165 In (natural logarithm), 49 logical operators, 37 Lorentz factor, 253 Lorentz scalar, 251 Lorentz transformation

of 4-vector, 251 of parabivectors, 255 of paravectors, 251

280

Lorentz transformations, 225 Lorentz-force equation, 242

M magnetic monopoles, 256 Mandelbrot

Benoit, 211 Mandelbrot set, 211 Maple, iv, 16

assignment statement, 33, 41 case sensitivity of, 40 command prompt, 19 constants in, 273 CPU time in, 43 end session, 38 equations, 33 expressions, 33 functions in, 273 introduction to, 31 list, 76 natural integer, 35 operators, 35 parsing, 35 plots in, 49 punctuation mark, 35 set, 76 strings, 35 student version, 20 variable names, 33 worksheets, vi, 22, 31

Maple commands :,34 ;,34 array, 115 assign, 81, 155 assume, 260 asympt, 79

collect, 155 conjugate, 209

INDEX

convert( , polynom), 77, 79 crossprod, 115 D, 52 degree, 80 describe, 126 diff, 52, 129 Digits, 42 done, 38 dotprod, 115 dsolve(,numeric), 188, 193 dsol ve, 176, 193 E, 54 erf, 142 evalc, 207 evalf, 42, 76, 179 example, 44 exp, 40 expand, 81 for ... to ... do, 152 GAMMA, 40 gamma, 40 Help, 38 I, 207 1m, 207 innerprod, 115 int, 175 interp, 152 inverse, 115 linalg [grad], 171 linsol ve, 115 map, 76,127 mean, 127 nops, 80 NULL, 194 op, 80

INDEX

package [command] , 126 phaseportrait, 218 plot,49 plot 3d, 128 quit, 38 RandNormal, 142 Re, 207 regression, 162 rhs, 189 sdev, 136 select, 80 seq, 76, 150 series, 77, 179 serr, sin, 43 solve, 40, 57, 149 spline, 158 standarddeviation, 136, 137 stats, 126 stop, 38 subs, 190, 194 sum, 76 union, 37 while, 201 with(linalg), 114 with(package), 126 with(Palg);, 258 Help, 32 odeplot, 188

Maple tokens, 35 \,36 ",35 *,37 -,37 @, 51 I, 37 !,36

**, 37 +,37 7,36 #, 36 and, 37 intersect, 37 or, 37 caret, 37

281

escape characters, 36 parentheses, 37 white-space characters, 36

mass and weight, 3 mathematics packages, ii matrices

Pauli, 231 matrix

trace of, 232 matrix representation, 225, 232 Maxwell's equations, 256 mean, 125 mean square deviation, 125 measurements

analysis of, 121 methane, 106 method of steepest descent, 168 metric prefixes, 24 Minkowski metric, 250 mirrors, 243 modulus

of complex number, 206 molecules

methane, 106 number in liter of air, 14

moments of a distribution, 133

mouse, 16 MS-Windows, v

282

multilinear product, 117

N nabla symbol, 167 naturru abundance

of isotopes, 27 naturru modes, 221 net force, 94 Newton

second law of, 33 Newton's method, 168 Newton's second law, 94 noncommutative, 111 nonlinear systems, 216 normal distribution, 141 notation

component-free, 225, 249 covariant, 249

nuclear spin of isotopes, 27

nullary operators, 35 N umerov method, 187

o observer, 108 odd cliffors, 247 odd function, 73 operators in Maple, 35 orbitru plane, 192 order-of-magnitude estimate, 12 orthogonal, 101 orthogonal elements, 261 oscillator

damped,189 nonlinear, 216

oscillators, 249 coupled, 221

p Pade approximants, 75 parabivector, 255 parallelepiped

volume of, 244 parameter space, 167 parametric plot, 190 paravector space, 231 paravectors, 225, 249 parsec, 28 partial derivative, 129 Pauli algebra, 225, 233

INDEX

general element of, 233 subspaces, 231

Pauli cliffors, 233 functions of, 262 inverses of, 250, 262 scalar (dot) product of, 255 square roots of, 262

Pauli spin matrices, 231 Pauli-algebra package, 258 pendulum, 66, 216 perigee, 193 perihelion, 193 periodic functions, 47 periodic table, 26 perpendicular, 101 phase portrait, 217 phase-space plot, 190 phase portrait, 190 phase-space trajectories, 190, 217 physical consants, v physical constants, 22 plane

representation of, 235 rotations in, 239

plane wave, 256

INDEX

planetary motion, 191 plotting in Maple, 49 points, 86 Poiseuille's law, 8 Poisson distribution, 139 population, 135 population distribution function,

132 position in variate space, 132 potential

paravector, 255 power, 24 power series, 77 power-law relations, 48 power-series expansion, 71 power-series products, 74 Precession

Thomas, 254 precision, 123 principle of equivalence, 98 principle of maximum likelihood,

166 priority of operations, 37 problem solving

principles of, 8 product

distributive over addition, 228 product of terms, 131 propagation

of errors, 129 proper time, 252 pseudo-random numbers, 142 pseudoforce, 97 pseudoscalar, 233 pseudovector, 233, 235 punctuation in Maple, 35

Q quadratic relation, 46 quaternions, 227, 248

R radians, 43 radioactive decay, 39 random errors, 123 random variable, 124 range of function, 45 reciprocal basis, 118 reflection, 242 reflections

multiple, 243 reflector

corner-cube, 243 regression, 162 relative position, 94 representation, 108

coordinate-free, 249 matrix, 225

representation matrix, 232

rest frame, 252 restoring force, 67 reverberation time, 5 reversal, 246 ring, 208 rms deviation, 125

283

root mean square deviation, 125 root of a function, 200 rotating frames, 241 rotation

in plane, 239 in three dimensions, 240 of parabivectors, 255 of paravectors, 252

284

plane of, 239 rotation-dilations, 248 Runge-Kutta method, 187

S Sabine formula, 6 saddle points, 168, 218 saddle shaped surface, 202 sample, 124, 127, 135 sample mean, 127 scalar, 234

Lorentz, 251 scalar product

of Pauli cliffors, 255 of vectors, 227

scalars, 87 second-order differential equation,

176 self-similarity, 211 separatrix, 218 sequence, 76 series expansions, 77 significant digits, 124 simple harmonic motion, 68, 249 simplex method, 170 Simpson's rule, 181 sine function, 47 sinh, 74 sinh function, 205 six-vector, 255 slices, 128 smooth curves, 150 smooth functions, 156 spacetime, 249 spacetime algebra, 250 spacetime bivector, 255 spacetime position, 252

spacetime rotations, 226 spatial inversion, 246 spatial reversal, 246 special relativity, 249 spline, 148 splines, 156

natural, 158 square deviation

of distribution, 134 square root

of Pauli cliffor, 262 stable, 188 standard deviation, 125

of distribution, 134 step size, 180 Stirling's formula, 82 Stone Soup Group, 212 strings in Maple, 35 SU(2), 240 sun, 12

INDEX

earth's orbit around, 12 systematic errors, 123

T Taylor expansion, 71 tension, 11 tetrahedral molecule, 106 Thomas precession, 254 time

proper, 252 tokens, 35 tonne, 3 torque, 113 total force, 94 trace

of matrix, 115 trackball, 16

INDEX

transformation inversion, 243 Lorentz, 251 reflection, 242 restricted Lorentz, 251 rotation, 233 spatial inversion, 233

transformations, Lorentz of parabivectors, 255

translations, 87 trapezoidal rule, 180 triple cross product, 236 trivector, 229, 233, 244

U unary operators, 35 uniform circular motion, 242, 249 uniform gravitational field

motion in, 93 unit conversion, 4 unit imaginary, 230 unit imaginary i, 245 units, v units file, 269 Unix, v, 31 uranium isotopes

V

U-235, 39, 56 U-238,56

variable names, 33 variance, 125

of distribution, 134 variate, 124 vector algebra, 225 vector difference, 89 vector notation, 226

vectors, 85 6-vector, 255

285

algebraic products of, 228 axial, 234, 239 basis, 107, 226 collinear, 99 dilation of, 88 linearly independent, 99 position vector, 87 scalar (dot) product of, 227 scaled,88 sum of, 88 time derivative of, 93 unit, 87 unit normal to a plane, 105 which span a space, 107 zero, 89

velocity paravector, 252

velocity vector, 93 volume

of parallelepiped, 244

W weighted sum, 165 Wessel, Caspar, 205 Wessel-Argand diagram, 205, 240 Windows, v

double clicking in, 17 exiting, 17 File Manager, 18 group icons, 16 Help menu, 21 Main window, 18 Program Manager, 16 scroll bar, 19 tutorial, 16

286

worksheets, 2, 22

X X-windows, v, 31 Xfract, v

Z zero divisors, 245

INDEX