Bibliography GENERAL W. J. Moore, Physical Chemistry, Longman, London (5th edn, 1972) L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics, McGraw-Hill, New York (1935) W. Kauzmann, Quantum Chemistry, Academic Press, New York (1957). P. W. Atkins, Molecular Quantum Mechanics, Clarendon Press, Oxford (1970) H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Mass. (1950) GENERAL MATHEMATICAL TEXT G. Stephenson, Mathematical Methods for Science Students, Longman, London (2nd edn,.l973) MATHEMATICAL TABLES H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, New York (4th edn, 1961) M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York (1965) CHAPTER 5 S. Simons, Vector Analysis for Mathematicians, Scientists and Engineers, Pergamon, Oxford (2nd edn, 1970) CHAPTER 6 J. A. Green, Sequences and Series, Routledge & Kegan Paul, London (1958) CHAPTER 7 W. Ledermann, Complex Numbers, Routledge & Kegan Paul, London (1960)
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Bibliography
GENERAL
W. J. Moore, Physical Chemistry, Longman, London (5th edn, 1972) L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics,
McGraw-Hill, New York (1935) W. Kauzmann, Quantum Chemistry, Academic Press, New York (1957). P. W. Atkins, Molecular Quantum Mechanics, Clarendon Press,
Oxford (1970) H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Mass.
(1950)
GENERAL MATHEMATICAL TEXT
G. Stephenson, Mathematical Methods for Science Students, Longman, London (2nd edn,.l973)
MATHEMATICAL TABLES
H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, New York (4th edn, 1961)
M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York (1965)
CHAPTER 5
S. Simons, Vector Analysis for Mathematicians, Scientists and Engineers, Pergamon, Oxford (2nd edn, 1970)
CHAPTER 6
J. A. Green, Sequences and Series, Routledge & Kegan Paul, London (1958)
CHAPTER 7
W. Ledermann, Complex Numbers, Routledge & Kegan Paul, London (1960)
282
CHAPTER 8
I. N. Sneddon, Fourier Series, Routledge & Kegan Paul, London (1961)
P. D. Robinson, Fourier and Laplace Transforms, Routledge & Kegan Paul, London (1968)
CHAPTER 10
G. Stephenson, An Introduction to Matrices, Sets and Groups, Longman, London (1965)
CHAPTER 12
G. Stephenson, An Introduction to Partial Differential Equations for Science Students, Longman, London (2nd edn, 1970)
CHAPTER 13
D. D. McCracken and W. S. Dorn, Numerical Methods and Fortran Programming, Wiley, New York (1964)
CHAPTER 14
Biometrika Tables for Statisticians, Biometrika Trustees, University College, London
R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and Medical Research, Oliver & Boyd, Edinburg~ (6th edn, 1963)
J. Murdoch and J. A. Barnes, Statistical Tables, Macmillan, London (2nd edn, 1970)
0. L. Davies and P~ L. Goldsmith, Statistical Methods in Research and Production, Oliver & Boyd, Edinburgh (4th edn, 1972)
C. J. Brooks, I. G. Betteley and S. M. Loxston, Mathematics and Statistics, Wiley, London (1966), chapters 9-16
J. Topping, Errors of Observation and Their Treatment, Chapman & Hall, London (3rd edn, 1962)
E. S. Swinbourne, Analysis of Kinetic Data, Nelson, London (1971)
Solutions to Problems
CHAPTER 1
1. (i) -~, (ii) -~,(iii) l; 2 ( ") 2 (" ") 1/2 (". ") 1 • ~ y versus x , ~~ y versus x , ~~~ n x versus t,
(iv) ln k versus u112 , (v) ln y versus ln k; 3. (i) (3 ± 141)/4, (ii) (-2 ± 131)/3, (iii) no real solutions; 4. (i) All x except x = ±1/13, (ii) x < 1 and x > 3, (iii) all
x except -3, 1, 2; 5. (i)-(iii) odd, (iv) even, (v) odd; 6. 1/12, 1/12, 1/2, 13/2, 1/2, 13; 9. (i) ~ sin 6A, (ii) ~(cos SA- cos 9A),
cos (-41105); (ii) 3i- 2j- 2k, i- 4j, 0, 4i + i + 5k, 11, nl2; (iii) 3i- 2 j + 2k,- j- 4j, 0, -4 i + j + 7k, 5, TI I 2 ; ( i v) 3 i + 3 i + 3 k' - i - j - k' 6' 0' -9' 0; ( v) 5 i - j - k ' 3 i - 3 j + 3 k' 0' 3 i T 9i + 6 k. 15' TI I 2; (vi) 4 i + 3 j - 4k, 2 i - j + 21k, 2, - i + 8 j + 5k, -3,
z = (5 + 2k- 3k2)/(1 + k3); (ii) x = 5- z, y = 3- z; 11. (i) yes, (ii) no; 12. c = 1; x = -z, y = 0; c = 2: x = 0, y = -z; 13. (i) x = -7z/11, y = 19z/11; (ii) x = -llz/10, y = 13z/10;
5. (i) X= 1, y = 2, Z = 3; (ii) X= 6/7, y = 10/7, Z = -2/7; (iii) X = 2. y = 3, Z = -4; (iv) X = 0, y = -2, Z = 1; (v) x = -48/53, y = -59/53, z = 7 /53;
6. (i) 1: x1 = x2 , x3 = 0; 1 + 12: x1 = -x3 /l2, x 2 = x 3!12, x 3 ; 1 - 12: x 1 = x3!12, x 2 = -x3!12, x3 ;
(ii) 1: x1 = -2x3 , x 2 = -2x3 ; 2: x 1 = x 2 = -x3 ;
8 " kt = (b - a)\c - a) ln(a ~ x) + (a - b)\c - b) In(b ~ x)
+ (a - c)\b - c) ln(c ~ x);
292
( ') -3x 2x ('') 3x -x 9. 1 y c c1 e + c2 e , 11 c1 e + c2 e
10.
11. 12.
13.
-2x -2x (iii) c1 cos 2x + c2 sin 2x, (iv) c1x e + c2 e
(v) e2x(c1 cos 3x + c2 sin 3x), (vi) e-x(c1 cos 2x + c2 sin 2x);
x = 0.04 cos(l3t);
(i) x = 0.04 e-t/2048 cos(l3t), (ii) x = -0.04 e-3t + 0.08 e-t; (i) y = -0.2(cos 3x + 3 sin 3x), (ii) y = !x(-3 cos 2x +sin 2x), (iii) y = -(4/50)sin 2x - (3/50) cos 2x,
of 11 derivative of 30 integral of 53 Taylor and Maclaurin series
expansions of 150
Maclaurin series 146-51 Maclaurin series for binomial
expansion 148 Maclaurin .series for cosine
functions 149 Maclaurin series for
exponential functions 149 Maclaurin series for
logarithmic functions 150 Maclaurin series for sine
functions 149 magnitude of a vector 116, 120 matrix 194 ff.
complex conjugate of 202 determinant of 200 diagonal of 199 eigenvalues and
eigenvectors of 206 element of 194 equality of 195 Hermitean 202 Hermitean adjoint 202 inverse 203 linear transformation of
208 multiplication of
by a constant null 199 orthogonal 205 reflection 210 rotation 210 singular 200 square 199 symmetric 202 transpose 200 unit 200 unitary 205
matrix addition 194
195-7 195
300
matrix algebra 194-7 matrix solution of simultaneous
equations 205 maxima 34-6 Maxwell's relations 110 mean 261 mean deviation 263 mean value of a function 78-82 median 262 ml.m.ma 34-6 mode 263 modulus of a complex number
155 modulus of a real number 32 moment of a couple 126 moment of a force 125 moment of inertia 75-8 moment of momentum 132 multiple qngle formulae 10,
160 multiple integral 82-7 multiplication of complex
numbers 154 multiplication of complex
numbers in Argand diagram 158
multiplication of matrices 195-7
multiplication of matrix by a constant 195
multiplication of real numbers 153
'nabla' operator 134 Newton's method for solution of
non-linear equations 242-4 noise 259 normal distribution 265
2 x -test for 277 normal distribution in sampling
266 normal equations 269 normalisation 168, 170 normalisation of a frequency
distribution 262 null matrix 199 null vector 116 numerical evaluation of a
determinant 256 numerical evaluation of inverse
of a matrix 256 numerical integration 244-9
Gaussian quadrature in 248
Newton-Cotes formulae for 248
Simpson's rule for 246 trapezoidal rule for 245
numerical metho.ds 242 ff.
order of a differential equation 214
ordinary differential equations 213 ff.
degree of 214 first order 214-20
exact 216 homogeneous 215 linear 218 simple 214 variables separable
parabola 3 parallelepiped, volume of 127 parallelogram law for vector
addition 116 partial derivative, definition
of 90-2 geometrical interpretation
of 92 higher 93
partial differentiation 90 ff. change of variable in
94 ff. partial fractions 56-9 partial sum of a series 140 particular integral 61
particular solution 214, 221 parts, integration by 52-5 Pauli principle 186 periodic function 170 pivoting 256 point function 132 polar co-ordinates, spherical
101 polar co-ordinates in Argand
diagram 157 polar co-ordinates in two
dimensions 95 polarisability 229 polynomial 4
Legendre 150 population 266 postmultiply 199 potential 69 potential of a dipole in a
field 123 precision 259 predictor-corrector method 253
predictor formula 254 premultiply 199 principle axes 78 principle of least squares 267 probability 80 probability distribution curve
261 probability distribution
function 80 probability in significance
tests 273 probability of an observation
261 product cross 123
derivative of 26 dot 120 scalar 120, 166, 198
of functions 167 triple 126 of vectors in n
dimensions 167 vector 123
triple 128 product of complex numbers 154 product of matrices 195-7 progression, arithmetic 139
geometric 140
quadratic function 3 quadrature 244-9
Gaussian 248
301
Newton-Cotes formulae for 248
Simpson's rule for 246 trapezoidal rule for 245
quotient, derivative of 26 quotient of complex numbers 155
radian 6 radius of convergence range 263 rate of change 23, 33
140 ratio, common rational number 153 real axis 156
145
153 real numbers, algebra of real part of a complex number
154 reduced equation 221 reduced mass 73, 76 reduction formula 54 reflection, matrix
representation of 210 regression, curvilinear 272
line of 270 regression analysis 269 regression coefficient 269 relative frequency 260, 262 remainder in Taylor and
Maclaurin series 146 repeated integral 83 resolution of a vector 118,
homogeneous 188-91 inconsistent 188 inhomogeneous 187 matrix notation of 198 matrix solution of 205 numerical solution of 254 secular 189
sine function, definition of 6 derivative of 28 integral of 47 Maclaurin series for 149
singular matrix 200
Slater determinant 185 slope at turning points 34-6 slope of a straight line 2 slope of a curve 23, 34 solid of revolution 69 solution, to differential
equations, general 214 particular 214, 221
solution to non-linear equations 242-4
specific heats 97 spherical polar co-ordinates
101 use of, in Schr5dinger
equation 238 spherical top 78 square matrix 199 square wave function 170 standard deviation 264 standard error of the mean 266 standard normal distribution