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),

(iii) ~(cos 9A +cos A); 10. !{cos [27r(vk + v)t] + cos 211(vk - v)t};

11. 2a0 cos(211x/A)cos(211vt); -5 12. 12.182, 0.13534, 40.447, 4.54 X 10 ;

13. 2, -6, 0.5, 4, -1, -3, 6; 14. 0.6309, 1.4307, 2.8074; 15. 2.303; 16. 11/6, 11/3; 11/4, 1.266, 2.837, 3.9176.

CHAPTER 2

1. (i) o, (ii) 1; 2. (i) yes, (ii) no;

3. (i) 2x, (ii) 3x2 , (iii) 4x- 3, (iv) - 1/(1 + x) 2; 4. (i) 1/3' (ii) -74/5; 5. (i) 2lx2 + 8x, (ii) (2/3)x-213 , (iii) -(3/8)x-7/ 4,

. 3 2 4 3 3 (~v) 8x + 7x , (v) 30x - 60x + 20x - 21, (vi) 4x ,

(vii) 3x112 (1x + 2)(1x- 1) + x(21x + 1), (viii) 1/(x + 1) 2,

(ix) 6x(l- 3x2), (x) (x2 - 4x + 2)/(x- 2) 2, (xi) -40(1- 5x) 7, (xii) x-l/2(1- x)-312,

(xiii) -8nx(l - 2x2) 2n-l, (xiv) 3 cos 3x - 3 sin 3x,

(xv) 3 sin x cos x(sin x- cos x), (xvi) 3x2 [cos x3

(xvii) -5 cosec2 (5x + 1), (xviii) -4x cosec 2x2 cot (xix) 5 exp(5x), (xx) -2 exp(5- 2x), (xxi) 1/x,

. 3] - s~n x , 2 2x ,

284

(xxii) -tan x, (xxiii) 3x2 I (x3 + 3), (xxiv) 1n(sin x) + x cot x, (xxv) -sin x exp(cos x),

(xxvi) 1n 3 x 3x, (xxvii) 1n 2 x sec x tan x x 2 sec x

( ''') -1/2 (/) XXV~~~ X exp X ;

6. 0; 7. (i) x = 3, minimum; x = -1/3, maximum, (ii) x = 1, maximum;

x = -1, minimum, (iii) approx. x = -0.5, minimum; approx.

x = 0.7, maximum; approx. x = 2.9, minimum;

11. (i) 4/1 (1 - 16x2), (ii) a/ (a2 + x2),

(iii) 4x/{ (x2 - 1)1[(x2 - 1) 2 - (x2 + 1) 2]};

(iv) -1/ [1(9 - x 2)];

13. (i)- (6x + 7y)/(7x + 18y),

(ii) [2x - 4x(x2 + /)] / [4y(x2 + /) + 2y],

(iii) (3y - 3x2) I o/ - 3x);

14. (i) tx~x + 1)(x- 2)] 1 / 2 1~1 1 + 1 2x 2 'i + -;-:;-r ;:-=-z - x2 + 1

(x + 1) (2x + 3)

(ii) (sin x)tan x[(sec2 x)1n(sin x) + 1], - 2x 2+ 3j'

X(x2 + 1)1/2 [1 2 1 (iii) (x + 1)2/3 [X+ x2: 1 - 3(x + 1)j.

CHAPTER 3

1. (i) 0.08x5 , (ii) (4/3)x3 - (5/2)x2 + x, (iii) -2.5x-0 · 4 ,

(iv) 1n(x + 3), (v) 1n(x2 + 4), (vi) (2/3) (ax + b) 312 /a,

(vii) 1/3 exp(3x- 1), (viii) 1/16(x2 + 1) 8 ,

(ix) 1/2 exp(x2), (x) 1/3 sin 3x + 3 cos 3x, (xi) 1/4 sin 4 x, (xii) 1n(1 +tan x), (xiii) (ax+ b)/2a +[sin 2(ax + b)]/4a,

(xiv) sin x- (1/3)sin3 x, (xv) x/8 -(sin 4x)/32, (xvi) -2 cot 2x, (xvii) -(1/11) cos 11x/2 - (1/5) cos Sx/2;

2. (i) 1/4 sin-1 2x + 1/2xl(l- 4x2), (ii) 1n(tan x),

(iii) 1/2 tan - 1 [2 tan(1/2x)], (iv) 1/3 sin x 3 ,

(v) -(1/6)1n(1- 2x3), (vi)- 2 cos lx, (vii) -(1/2)1n(1 + 2 cos x), (viii) 1/2(1n x) 2 ;

3 ( ') . (' ') 3 . 3 2 6 . , ~ S~n X - X COS X, ~~ X S~n X + X COS X - X S~n X -

6 cos x, (iii)1/2x2 1n x- 1/2x, (iv) 1/4x4 1n x- (1/16)x4 ,

(v) ex(0.4 sin 2x + 0.2 cos 2x), (vi) x tan-1 x­

(1/2)1n(1 + x2), (vii) x/b- a 1n(a + bx)/b2 ,

285

(viii) x - 2 1n(2x + 3), (ix) (113)x3 + (112)x2 + x +

1ntx- 1), (x) 1n(1- x) + 31(1- x), (xi) x + (5l3)1n(x- 2)-

(213)1n(x + 1), (xii) (112)x2 - 2x + 2 1n(x + 1) - 1n(x- 1),

(xiii) -(312)1n x + (513)1n(x- 1) - (116)1n(x + 2),

(xiv) (115) [(312)1n(x2 + 4) - tan-1 (112x) - 3 1n(l - x)],

(xv) (1;/8)tan-1 [(x + 3)118], (xvi) (11113)tan-1 (2x + 1)1113,

(xvii) cosh-1 (x - 2)112, (xviii) l(x2 + 2x - 1) +

cosh-1 (x + 1)112, (xix) 2 sin-1 (xll2) + (112)1n 1(2- x 2),

(xx) -cosh-1 [(2x + 1)1x], (xxi) -(112)x(1- x 2) 112 +

(112)sin-1 x, (xxii) -21(1- x) + (213)(1- x) 312 ; 4. (i) 2135, (ii) (16155) (1 + 1/12). (iii) -(7 18) + 1/12,

(iv) 2 + 2 1n(5/3); 6. 0; 7. (i) n,O, (ii) 0,0, (iii) n,O; 8. 918; 9. 57n;

10.

11.

12.

2 X 3512 . ' 2

(4/3)na b; 2

7f 12; 2

13. (1lp)sin pt, (ml4p )(1- cos 2pT);

14.

15.

16. 17. 18.

19.

20.

(nl6,1l2);

(a) (1/RT)1n(V21v1), (b) kl(1 - y) [v21-y - v1

1-y];

(512,0); (a) 45MI2, (b) 45M/7; (a) 2/n, (b) 112;

(8kTI7fiD) 112 ; 2

4V0 lng.

CHAPTER 4

1. (i) -2x sin(x2 + y 2), -2y sin(x2 + y2), .. 3 2 2 2 2 3 2 2 2 2

(n) (y - x y) I (x + y ) , (x - xy ) I (x + y ) ,

(iii) a/y2 , -2axly3 ; 2. (i) exp(x + y) [cos(x - y) + sin(x- y)],

(ii) exp(x + y) [sin(x - y) - cos(x - y)], (iii) 2 cos(x- y)exp(x + y), (iv) 2 sin(x- y)exp(x + y), (v) -2 cos(x- y)exp(x + y);

4. (i) [(1ly) - (z/x2)] {ln[(x/y) + (y/z) + (z/x)] + 1},

(ii) [(-x//) + (1/z)] {1n[(x/y) + (y/z) + (z/x)] + 1},

(iii) [(-y/z2) + (11x)] {1n[(x/y) + (y/z) + (z/x)] + 1};

5. 5x\ 6. -3;

286

7. (i) (1ly)dx- (xli)dy, (ii) (2xy + l)dx + (x2 + (iii) y exp(xy)dx + x exp(xy)dy;

2 3xy )dy,

10. 14.

16.

17.

18.

19.

(i) (aflau) + y(aflav), (ii) -(aflau) + x(aflav);

~~ [sin ¢ aae + cot e cos ¢ aaq,J ,

ih [ a . a l - 2n cos¢ 3e- cote s~n q, ~J;

(i) -(6x + 7y)l(7x + 18y), (ii) [2x- 4x(x2 + y2)]1

[4y(x2 + y 2 ) + 2y], (iii) (3y- 3x2)1(3y2 - 3x), . m-1 n x+y

(~v) cot x cot y, (v) -(mx y - 1n a x a )I m n-1 x+y

(nx y - 1n ax a );

(i) RIV0P, _ _!_ [ RI(V - b) 1. VO 2aiV3 - RTI(V- b) 2j'

(i) exact, (ii) inexact, (iii) inexact, (iv) exact;

4096/3, (ii) 12813, (iii) lle3 - 3e, (iv) 413, 312. •

CHAPTER 5

1. (i) 3i + j - 3k, i- 7 i +k, -8, 10i + 3j + llk, -7, -1

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,

-1 cos (21154);

2. (i) 21114, 31114, -11114; (ii) 1;./6, 2/16, 1/16; (iii) 11126, 3/126, -4/126; (iv) 3/114, -11114, 21114;

3. i + 5 j +k, 154212; 4. 14, 6 i + 2j + 10k; 5. (i) 0, 19 i + 17 j - 20k; (ii) o, -24 i - 6 j + 12k;

(iii) -4, 4i + 2j + 4k;

11. (i) (4t + 3) i + (6t2 + 4)j + 2k, 4 i + 12tj; (ii) 3 cos t i - 2 sin 2t j + 2t k; -3 sin t i - 4 cos 2t j + 2 k;

12. m[(2t 3 + 2t + 1) i - 3t2 (t + 1) j - 3(2t2 - 2t - 1)k];

13.

14.

( ") _ 2A[xi + yj + zk], (ii) 2a(xi + yj + zk), ~ 2 2 2 2

(x + y + z ) (iii) [2ax + b(y + z)] i + [2ay + b(x + z)] j + [2az + b(x + y)] k; (i) 0, 2(y- z)i + 2(z- x)j + 2(x- y)k;

(ii) 2y + 2 + xy + 2xz, xz i- (yz + z2) j + (2x - 1)k; 2

(iii) 3 + 2z- 3z , - 2yi- 6xi + (2x- 2)k.

CHAPTER 6

1. 250500;

2. n(n + l)ax/2; 3. (220 - 1) /219; 4. 55944; 5. If exp(-hv/kT) < 1, the sum is 1/[exp(hv/kT)- 1]; 7. (i) divergent, (ii) divergent, (iii) divergent,

(iv) divergent, (v) convergent, (vi) divergent, (vii) convergent, (viii) divergent;

8. (i) divergent, (ii) absolutely convergent, (iii) divergent, (iv) conditionally convergent;

9. (i) [x[ < 1, (ii) [x[ < 1, (iii) [x[ < 2, (iv) [x[ < 1;

10. (i) 3 5 X X

x + 3T + 5T + ... ,for all x;

(ii) 2 3 x + x2 + x3 + ... ,for [x[ < 1 and x -1;

(iii) 2x2 8x4 32x6 y - ----z;: + 6! + ... , for all x;

(iv) x 1 x2 3 x3 15 x4 1 + 2 - 4 2T + 8 3T - 16 4T + · · ·' for I X I < 1;

(v) 3 15 X 2 3 5 7 3

X X ~+ 1 -2x +--- ... ' 4 2: 8 5:

(vi) 3 1 3x 5 'IT X X

for 2 x-----4 5 - ... , 2 X 3 2 X X

(vii) x3 2x5 17x7 x + :l + ~ + :3:[) + ... , for [x[

for

[x[

0 <

'IT < -· 2'

[x[ <

< 1, -1

cos

2 3 11. 1, cos 8, !(3 cos 8- 1), !(5 cos 8- 3 cos 8)

1· '

X < 'IT" '

287

4 2 5 3 ~(35 cos 8-30 cos 8 + 3); k{63 cos 8-70 cos 8+15cos8).

CHAPTER 7

1. (i) 13 - 3i, 13 + 3i; (ii) 41 - 13i, 41 + 13i; (iii) -9 - 38i, -9 + 38i; (iv) 14 - lzi, 14 + lzi; (v) 5, 5; (vi) (22 + 59i)/61, (22 - 59i)/ 61; (vii) -(3/2) - (5i/2), -(3/2) + (5i/2); (viii) -(1/5)- (2/6i/5), -(1/5) + (2/6i/5); (ix) -95 + 59i, -95 - 59i;

2. (i) 5, cos-1 (4/5); (ii) 13, cos-1 (5/13); (iii) 2, TI/3;

(iv) 1, TI/2; (v) /5/2, cos-1(2/15); (vi) llo, cos-1(3/110); 3. (i) 1.938 + 0.495i, (ii) -(3/2) + (3/3/2)i, (iii) 4 + 3i,

(iv) /2 - lzi; 4. (i) 3i/2, 313 - 3i; (ii) (1/3) - (i/13), 6 cos(TI/12) +

6i sin(TI/12);

5. cos 8(4 sin 8 - 8 . 3 ) 8 cos 4 8 - 8 2 s~n 8 , cos 8 + 1,

5 sin 8 - 20 . 3 8 + 16 sin5 8, 16 cos 5 8 - 20 cos 3 s~n 8 +

5 cos 8; 11. (i) (1/12) + (i/12)' (ii) -(1/12) + (i/ 12) '

(iii) -(1/12) - (i/12), (iv) (1//z) - (i/12), (v) i, (iv) -i;

288

12. 2113 , 2113 (-(1/2) + (l3/2)i), 2113 (-(1/2) - (l3/2)i);

13. 13118 (0.989 + 0.146i), 13118 (-0.146 + 0.989i),

131/ 8 (-0.989- O.l46i), 131/ 8(0.146- 0.989i); 14. ~o = No<Xo + xl + x2 + x3 + x4 + Xs)

~1 = Nl[xo + !xl - !x2 - X3- ~x4 + ~xs + (il3/2)(x1 + x2 - x4 - x5>]

~-1 = N_l[xo + !xl- !x2- X3- !x4 + !xs - (il3/2)(x1 + x2 - x4 - x5>]

~2 = N2[xo- !xl - ~x2 + X3 - !x4 - !xs + (il3/2)(x1 - x2 + x4 - x5>]

~-2 = N_2[xo- !xl - !x2 + X3 - ~x4 - ~xs - (il3/2)(x1 - x2 + x4 - x5>]

~3 = N3<xo- xl + X2- X3 + X4- xs>·

CHAPTER 8

r even

sin rx r

4 • l [ I (112 - ..i.)sin rx- 112 11 r odd l r r 3

6" t + I f 24 2 ] cos(k11x); k=llk 11

I sinr rx]; r even

7• 1 2 ~ 1 1 J 2 + ~ I k cos(kx/a) - I k cos(kx/a) ; k=l,5,9... k=3,7,11 ...

8 · ..!. + \ 22 ( l) (r/2)+1 L - cos rx. 11 r even 11(r - 1)

CHAPTER 9

1. (i) -12, (ii) -59, (iii) 176, (iv) 732; 4. (i) 2, 5, 17, (ii) 7.0867, 0.4567 ± l.8218i, (iii) 0.6852,

-8.4111, 5. 7259; 6. (i) X= i, y = 2, Z = 3; (ii) X= 6/7, y = 10/7, Z = -2/7;

(iii) X= 2, y = 3, Z = -4; (iv) X= -5, y = 7, Z = 2; 7. (i) x = (1/3) -A, y =A, z = 1/3; (ii) x = 3/8, y = 3/8;

z = 1/2; (iii) no solution; 8. (i) No solution for k = 1, 2, there are no values of k for

which there is more than one solution; 9. Determinant of coefficients of x, y, z is zero; k = 2,

x = (12 - 7z)/ll, y = (2 + 19z)/ll;

289

10. (i) X= (2- 3k- 5k2)/(l + k3), y = (3 + 5k + 2k2)/(1 + k3),

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;

(iii) no solution.

CHAPTER 10

1. (i)

[ ~ : ~]' ~-~ -: ~]' 5 5 3 -5 3 3

[13

24

11 3 -5 ' [11 15 16

28 25 5 31 20

11 6 8 2 15

[- ~ ~ ~] ' [ : - ~ 2 5 3 -1 4

5]' [11 1 15

0 16 3~ :]· 3~[-~ 1 ~ 20 15 12 -4

-1~]' 10

;3[ ~~ -~ ~~]' 21~2[ 4:~ -11 13 7 -117

-193 -196

37 -140

49 273

(ii)

[ ~ ~ -:]· [~ -~ =~]' [~: 1 5 1 1 1 3 10

: _:]· [:~ : 20 4 11 -3

[ 5 6 1] ' 13 1 0] ' [21 2 0 3 1 5 2 6

-3 1 2 0 2 -1 -8

3: ~~ ' -9~~-~~ 6 0 18

2]' 13 -23

-12

-13

-13

- 2~~-921 -~ -:]· 21~6[ ~: :: -:~~]; -6 14 -199 129 -54

[: : :]· [-: _: -~ ' ~~~ :~ 1~ ' ~~~ ~: 1~ 1 3 4 1 -1 6 4 16 -3 13 5 -3

(iii)

[ ~ : ~]' [~ ~ ~' ~~~ ~: 1:]·- 2~6[-~:

2 1 5 0 2 -1 7 17 -3 4

1] ' 13

6 -46

-33 3

- 2~[-921 -~ _:]· 56~6~-~:: -~~: ~:~]; -6 14 -122 -262 -672

290

(iv) 1 + 13 1 + 13 0 ' 1 - 13 13 - 1 0 • [ 0

1 0 --2- 2 --2- --2--1 0 0

1 + 13 1 + 13 0 1 - 13 1 - 13 0 0 0 1 ---2- --2- --2- --2-

0 0 2 0 0 0

H 1 0 • [ 1/2 -13/2 l [/3/2

-1/2 0

0 0 13/2 1/2 0 1/2 13/2 0

0 1 0 0 1 0 0 1

[~ -1

0 • [ 1/2 -13/2 0 • [13/2 -1/2 0

0 0 13/2 1/2 0 1/2 13/2 0

0 1 0 0 1 0 0 1

[~ -1

~r 0

0

2. (i) [: ~ : • 4; (ii) [:; 1: :~]· 36;

(iii) [-21

3

-1 -1 , 6; (iv) [-2 4 2 , 0; 2 2 -1 2 1

-3 -3 0 0 0

-0i), Hermitean; (ii) [

4

12

- i 2 + i

- 3i 1 - 2i

+ i 3 - 4i

3. (i) (~ 5 + 2il ' 7 - 3i

4 + i

not Hermitean

(iii) [~ -i ~]' l. 0 -]_

0 i 0

Hermitean; (iv) 2 - i

4

-3i

1 + 3il' 3i 3

4. (i) [1/3 2/3

2/3

2/3 -2/3], 1/3 2/3

-2/3 -1/3

(ii) [13/2 -1/2

0

not Hermitean;

1/2 0]; both are 1312 0 orth~gona1

matr1.ces; 0 1

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 ;

3: x 1 = 4x3 ; x 2 = x3 /2, x3 ;

(iii) 0: x1 = 13x3, x2 = O, x3 ; 2: x1 = !x2, x2, x3 = (l3/2)x2; -2: x1 = -!x2 , x2, x3 = -(l3/2)x2 ; (iv) 2: x1 = 0, x2 = ix3 ; 12: x1 = l2x3, x2 = -ix3, x3; -12: x1 = -12x3 , x2 = -ix3 , x3 .

CHAPTER 11

291

1. (i) y = sin-1 (cx), (ii) y = c(x2 + 1), (iii) y = c(1 + x) - 1,

(iv) 1n y = -(x2/2y2) + c, (v) (y- x) = c(y + x) 3 , ( . ) ( 2) 2 ( .. ) 6 3 4 5 V1 y exp X - X = c, V11 X y + X y = C,

(viii) y = lx4 + c/x, (ix) y = (x + c)exp(-x2),

(x) y = -(5/sin x)exp(cos x) + c/sin x,

(xi) y = !x3 + cx3 exp(1/x2);

2. (i) 8n0 , (ii) ~ x 104;

3. kt = 1/(a- x) - 1/a, 1/ak; -1 -1 4. ABa = 0.0578 days , ALa= 0.0693 hours ; 314.7 units,

17 units, 11 units; 5. (i) i = exp(-Rt/L)(i0 - E0/R) + E0/R,

(ii)

6. 1 rcx+2)l t = 5k1 lnL2(1 - x)j;

7. 1 t = 2 2 X

2kl[Ck1a/k2) + (k1 /4k2 )]

I[Ck1a/k2) + (k12/4k/)J + (k1/2k2) + x

[ 2 2 J I (k1a/k2) + (k1 /4k2 ) - (k1/2k2) - x

l[(k1 a/k2) + (k12 /4k/)J + (k/2k2) ]· - ln 2 2 '

l[(k1a/k2) + (k1 /4k2 )] - (k1/2k2)

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,

(iv) (x2/3) + (2/9)x- (25/27),

(v) (x5/15) - (x4/9) + (13/27)x3 - (13/27)x2 - (x/3), (vi) (5/2)e3x, (vii) !x2 e-2x, (viii) 3 e-x;

. 2x -x 2x (1) y = c1 e + c2 e + (1/3)x e ,

( .. ) -x/5 ( 11 e c1 cos 4x + c2 sin 4x) + 2x - 1,

( .. ') 5x 111. c1 e + c 2 e-x- (3/26)cos x- (l/13)sin x,

(iv) e-2x(c1 cos 2x + c2 sin 2x) - sin 3x - 3 cos 3x,

X X (v) c1 e + c2x e + ! cos x - ! sin x;

14. q = c1 exp{[- 2~ + zij(R2 - 4t]]t}

+ c2 exp{ [- ~ - 2i}(R2 - 4t]] t}

E0L[(q/CL) - 1l] + sin at

12 [(q/CL) - a2J 2 + R2a2

E0Ra ~----"-~-:::---::--: cos at. L 2 [ (q/CL) - a2J 2 + R2a2

CHAPTER 12

1. u(x,t) I c exp(-r2 ~2kt/~2 )sin(r~x/~); r=1 r

00

2. 1/J(x,y) = I c sin(nx/a)sinh[n(b - y)/a]; r=1 r

2 2 2 3 E_ a R(r) +E. dR(r) = k2· _1_ d e = -k2

' R dr2 R dr ' 8(8) dS2

4. (i) 1/J(x,y,z) = 3~2 sin(nx~x/a)sin(ny~y/a)sin(nz~z/a), a

(ii) ljJ(x,y,z) = ~Sb) sin(n ~x/a)sin(n ~y/b)sin(n ~z/c). rlaDCl X y Z

CHAPTER 13

1. (i) 1.414, -1.414; (ii) 0.311, (iii) 2.798 radians, (iv)- 2.427, (v) 0.708, (vi) 3.146;

2. (i) 0.8795, 0.8814, (ii) 0.4471, 0.4458, (iii) 2.5413, 2.5456, (iv) 1.8539, 1.8536, (v) 0.9068, 0.9097, (vi) 1.0970, 1.0948;

3. 0.8362, 0.8862, exact result 0.8862; 4. Numerical solutions:

(i) Euler method predictor-corrector method

(ii) Euler method predictor-corrector method

y(l) 0.5 0.5

y(O.l) 0.090 0.090

(iii) y(l) Euler method 1.927 predictor-corrector method 2.029

(iv) y(l) Euler method 0.655 predictor-corrector method 0.678

Analytical solutions:

y(2) -3.0 -3.0

y(0.2) 0.163 0.163

y(2) 33.453 34.146

y(2) 0.214 0.248

(i) y = x - (3x2/2) + 1, (ii) y = x ex,

y(3) -9.5 -9.5

y(0.3) 0.222 0.222

y(3) 247.554 250.136

y(3) -0.422 -0.415

y(4) -19.0 -19.0

y(0.4) 0.267 0.268

y(4) -0.669 -0.699

(iii) y = 2(x2 - x)ex + 0.391, (iv) y = !(cos x +sin x); 5. (i) (a) X = y = Z = U = 1, (b) X = 14.42, y = -7 .09,

Z = -2.45, U = 3.07; (ii) X = 1, y = 2, Z 3, W = 0;

6. (i) 0, no inverse; (ii) 831, 1 -98 44 47 56 831 155 100 409 -326

84 81 -159 -48

-4 -83 -32 121

CHAPTER 14

293

y(5) -31.5 -31.5

y(0.5) 0. 303 o. 304

y(5) -0.301 -0.341

1. (i) 15.59, 15.6, 0.0289, 0.17, 0.0538; (ii) 20.99, 21.0, 0.2344, 0.4842, 0.0625; (iii) 44.80, 45, 299.72, 17.31, 0.972;

2. (i) 30.85%, (ii) 6.68%, (iii) 0.621%; 3. x ± 0.675a;

2 4. (i) y = 1.0236x + 0.8818, am 2 0.001247, ab = 0.01091;

2 2 (ii) y = 101.09x - 3.293, am 0.8123, ab 31.2736;

5. Regression of y on x gives y = 5.018lx + 0.0460; regression of x on y gives y = 5.0276x - 0.0484;

6. y = 0.998 2 + 0.0495x + 0.9987x ;

7. l (~ "' 8) 5.04, sample from normal distribution;

x 2 (~ = 18) = 38.79, significant deviation from normality; 8. u = 3.68: very highly significant increase in purity,

81.64 ± 1.67; t = 2.67: significant increase in purity 95% confidence limits 81.64 ± 2.66.

Index

abscissa 1 absolute convergence 144, 145 absolute value of a complex

number 156 acceleration as a derivative

33 vector representation 131

accuracy 259 Adams-Bashforth predictor 254 Adams-Moulton corrector 254 addition of complex numbers

154 addition of matrices 194 addition of real numbers 154 addition of vectors 117 adjoint, Hermitean 202 adjoint matrix 203 algebra of complex numbers

154-6 algebra, matrix 194-7

vector 116-18 angle between vectors 121, 124 angular momentum operators 102 angular momentum vector 132 angular velocity 131 antisymmetry of a wavefunction

186 arc, length of 70 area, calculation by integration

65-8 representation of by a

vector 125 argand diagram 156 argand diagram in cartesian

co-ordinates 156 argand diagram in polar

co-ordinates 157 argand plane 156 argument of a complex number

157 argument of a function 1 arithmetric progression 139

associative law for matrix addition 195

associative law for matrix multiplication 197

associative law for product of vectors and scalars 118

associative law for real numbers 153

asymmetric top 78 asymptote 37 atomic orbitals, complex form

162 auxiliary equation 221 axes, principal 78

right-handed rectangular cartesian 118

back substitution 255 Bessel's correction 266 binomial expansion 148 bisection method 242 Bloch sum 163 boundary conditions 235

cartesian axes, right-handed rectangular 118

cartesian co-ordinates 1 Argand diagram in 156 three-dimensional harmonic

oscillator in 238 centre of mass 71-5 chain rule for differentiation

27 change of variable in partial

differentiation, first derivative 94-7

second derivative 97-103 2

X -test 277 class 260

limit 260 width 260

cofactor of a determinant 186 cofactors, matrix of 203 column vector 197 common difference 139 common ratio 140 commutation in addition and

multiplication of real numbers 153

commutation in matrix addition 195

commutation in multiplication of a vector by a scalar 118

commutation in partial differentiation 94

commutation in scalar product 120

commutation in vector addition 117

commutation in vector product 124

compact methods 257 comparison test 142 complementary function 221 completeness of set of vectors

or functions 168 completing the square 57-8 complex conjugate 155

matrix 202 complex numbers 153 ff.

absolute value of 156 addition 154 algebra 154 complex conjugate 155 definition 153 division 154 modulus 155 multiplication 154 quotient 154 subtraction 154

components of a vector 118, 168

conditional convergence 144 confidence limits 275, 277 conformable matrices 195 contin\lity 21

equation of 135 convergence, absolute 144, 145

conditional 144, 145 interval of 145 radius of 145

convergence of improper integrals 63

convergence of power series 145

295

convergence of series 140-5 tests for 141-5

coplanarity of vectors 127 corrector formula 254 cosine, direction 119 cosine formula 122 cosine function, definition of

6 derivative of 29 integral of 47 Maclaurin series for 149

Cramer's rule 188 critical constants 38

damping 225 cross product 123 curl of a vector field 135 curve, slope of 23, 34-6

tracing a 37

D-operator method· 226 d'Alembert's ratio test 142 damped harmonic motion 224 damping, critical 225 definite integral, applications

to area 65-8 centre of mass 71-5 length of arc 70 mean value 78-82 moment of inertia 75-8 volumes of solids of

revolution 69 work done by a force 68

definition of 61 double 82-6 multiple 82-7 properties of 61-4 repeated 83 triple 75, 84, 86-7

definite integral as a summation 64

definite integral or even and odd functions 62

degree of a differential equation 214

degree of freedom in statistics 274

'del' operator 134 de Moivre's theorem 159, 162

application of to multiple angle formulae 160

application of to roots of complex numbers 160

dependent variable 1 derivative, chain rule 27

296

definition of 22 directional 133 higher, of a single

variable 32 of many variables 93

partial 90 ff. vector 129-32

derivative of exponential function 29

derivative of generalised exponential function 30

derivative of hyperbolic function 31

derivative of implicit functions 41, 106, 107

derivative of inverse trigonometric functions 31

derivative of logarithmic function 30

derivative of powers of x 24 derivative of product 26 derivative of quotient 26 derivative of sum 25 derivative of trigonometric

functions 28 determinants 177 ff.

general order 180 minor of 180, 186 numerical evaluation of

256 properties of 181-6 scalar triple product as

127 second-order 177 secular 189 Slater 185 third-order 178-80 transposed 182 upper triangle 256 vector product as 125 zero 183, 185

deviation, mean 263 standard 264

diagonal matrix 199 difference, common 139 difference between complex

numbers 154 difference between matrices

195 difference between vectors

117 differential 39

exact 104 line integral of 111

total 103 differential equations, first

order ordinary 214-20 exact 216 homogeneous 215 linear 218 simple 214 variables separable 214

ordinary 213 ff. partial 213, 234 ff

separation of variables 234

second order ordinary 221-30

homogeneous 221-6 inhomogeneous 226-30

differential vector operator 133

differentiation, definition of 24

logarithmic 41 partial 90 ff. rules for 24-32

differentiation of exponential function 29

differentiation of function of a function 27

differentiation of generalised exponential function 30

differentiation of hyperbolic functions 31

differentiation of implicit functions of many variables 107-8

differentiation of implicit functions of single variable 41, 106

differentiation of inverse trigonometric functions 31

differentiation of logarithmic function 30

differentiation of powers of x 24

differentiation of product 26 differentiation of quotient 26 differentiation of sum 25 differentiation of trigonometric

functions 28 differentiation of vectors 129 dipole moment, average

component of 81 calculation of 74 induced 230 potential in a. field 123

direc.tion cosine 119, 122 directional derivative 133 Dirichlet's conditions 170 distributive law in matrix

multiplication 197 distributive law for real

numbers 153 distributive law for scalar

product 120 distributive law for vector

addition 118 distributive law for vector

product 124 divergence of improper

integrals 63 divergence of series 140-5 divergence of a vector field

134 division of complex numbers

155 division of complex numbers

Argand diagram 158 dot product 120 double integral 82-6

in

eigenfunction of a matrix 206 eigenfunction of an operator

30 eigenvalue of a matrix 206 eigenvalue of an operator 30 envelope 260 equality of complex numbers

154 equality of matrices 195 equality of vectors 116, 166 equipotential surfaces 133 error analysis 259 ff. error function 265 errors 259 Euler reciprocity relation 104 Euler's formula 162, 223 Euler's method for solution of

differential equations 250 improved 251

exact differential 104 line integral of 111

exact differential in differential equations 216-19

exact differential equation 216

expansion coefficient 169 expansion in power

297

series 146-51 binomial expansion 148 cosine function 149 exponential series 149 sine function 149

expansion in terms of orthogonal functions 169

expansion of a fluid, coefficient of 36

expectation value 81 exponent 10 exponential function,

definition of 10 derivative of 29

generalised 30 Maclaurin expansion of 149 standard integral of 47

exponential function of a complex argument 161

field, scalar and vector 132 flux 123 flux density 135 flux in terms of divergence

134 force, moment of 125

vector notation for work done by 122

work done by variable 68 force constant 224 forced oscillation 229 Fourier component 170 Fourier expansion in solution

of partial differential equations 236

Fourier integral theorem 175 Fourier series 169-74 Fourier transform 174 frequency distribution 259-64 frequency distribution curve

260 frequency of an observation

260 relative 80, 260, 262

function, cosecant 8 cosine 6 cotangent 8 definition of 1 even 5

definite integral of 62 explicit 4 exponential 10 hyperbolic 12

298

implicit 4 differentiation of 41,

106-8 inverse trigonometric 14 limit of 19-21 linear 2 logarithmic 11 many valued 4 odd 5

definite integral of 62

orthogonal 166 ff. periodic 170 point 132 polynomial 4 quadratic 3 relation of with vector

166 saw-tooth 170 secant 8 sine 6 single-valued 3 square wave 170 step 171 tangent 6 transcendental 5 trigonometric 6 undefined 4

function of a function, differentiation of 27

function of many variables 4 fundamental theorem of integral

calculus 64

Gaussian distribution 265 Gaussian elimination 254 Gaussian law of error 265 Gaussian quadrature 248 general solution of a

differential equation 214, 221

geometrical progression 140 gradient of a scalar field 132 graph 1 Gregory's series 151

Hamming's method 254 harmonic motion, damped 224

simple 224 harmonic oscillator, three­

dimensional 237 higher derivatives of many

variables 93 higher derivatives of single

variable 32

higher derivatives with change of variable 97-103

histogram 260 homogeneous first-order

differential equation 215 homogeneous linear equations

188 homogeneous second-order linear

differential equation 221-6 homogeneous secular equations

189 Ruckel molecular orbital theory

189, 207, 243 hyperbolic function, definition

of 12 derivatives of 31 standard integrals for 47

identity 56 imaginary axis 156 imaginary part 154 inconsistent simultaneous

equations 188 increment for a single variable

38 increment for many variables

103 independence of a set of

functions or vectors 168 independent variable 1 inequalities 16 inflection, point of 34-6 inhomogeneous second order

differential equations 226-30

for exponential function 227

for polynomial function 227

for sine and cosine functions 228

inhomogeneous simultaneous equations, determinantal solution of 177-81, 187-8

matrix solution of 205-6 integer 153 integral, definite 61-4

double 82-6 improper 63-4 indefinite 45 line 110-12 multiple 82-7 particular 60 repeated 83 standard 46-7

surface 134 triple 84, 86-7

integrand 45 trigonometric 51

integrating factor 106 integrating factor in

differential equations 218 integration, definition of 45

methods of 46-61 numerical 244-9

Gaussian 248 Newton-Cotes formula

248 Simpson's rule 246 trapezoidal rule 245

integration by completing the square 57

integration by partial fractions 56-9

integration by parts 52-5 integration by substitution

47-51 algebraic 51, 60 elementary 47-9 trigonometric 49-51

integration of irrational algebraic fractions 59

integration of rational algebraic fractions 55-9

intercept 2 interquartile range 263 interval of convergence 145 inverse matrix 203

numerical evaluation of 256

inverse trigonometric functions, definitions of 14

derivatives of 31 Gregory's series for

inverse tangent 151 integrals of 53 standard integrals of 47

irrational number 153

Kroenecker delta 169

Laplace transform 226 least squares analysis 267-73

linear functions 268-72 non-linear functions 272

least squares principle 267 Legendre polynomials 150 Legendre polynomials in

Gaussian quadrature 249

Legendre transformation 108 limits of integration 61 line integral 110-12 line of regression 270 linear transformation 208 logarithm, common 11

definition of 11 natural 12

299

logarithmic differentiation 41 logarithmic function, definition

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

214 numerical solution of

249-54 by Euler's method 250 by improved Euler

method 251 by predictor-corrector

method 253 by Runge~Kutta method

250 by Taylor series

solution 250 order of 214 second order, linear

220-30 homogeneous 221~6 inhomogeneous 226-30

orthogonal functions 166 ff. orthogonal matrix 205 orthogonal vectors 121, 168 orthonormal functions 169

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,

168 resultant vector 116 risk 275 roots of complex numbers 160 rotation, matrix representation

of 209 row vector 197 Runge-Kutta method 250-3

fourth order 252 relation of, to Simpson's

rule 253

sample 266 scalar 116 scalar field 132

gradient of 132-4 scalar product 120, 166, 198 scalar product of functions

167 scalar product of n-dimensional

vectors 167 scaling of a matrix or

determinant 256

302

Schr5dinger equation 237-40 Schr5dinger equation for

hydrogen atom 238 Schr5dinger equation for three­

dimensional harmonic oscillator 237

separation constant 235, 238, 239

series 139 ff. binomial 148 convergence of 140-5 convergent 141 cosine 149 divergent 141 exponential 149 Fourier 169 ff. Gregory's 151 infinite 141 ff. Maclaurin 146-51 power 145 simple 139 sine 149 Taylor 146-51

series of alternating terms, convergence of 143

series of positive terms, convergence of 142

significance level 273 significance level tests 273-8

l 277 Student's t- 276 u- 274

simple harmonic motion 224 Simpson's rule 246

comparison of with trapezoidal rule 248

relation of to Runge-Kutta method 253

simultaneous equations, Cramer's rule for 188

determinantal solution of 177-81

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

tables 275 statistics 259 ff. step function 21, 171 straight-line graph 2 string, vibrating 234-7 student's t-test 276 substitution, integration by

47-51 algebraic 51, 60 elementary 47-9 in definite integrals 62 trigonometric 49-51

subtraction of complex numbers 154

subtraction of matrices 195 subtraction of real numbers 153 subtraction of vectors 117 successive first order

reactions, differential equation for 219

summation by integration 64 surface integral 134 symmetric matrix 202 symmetric top 78

t-test 276 tangent function, definition of

6 derivative of 29 integral of 51

Taylor series 146-51 Legendre polynomials for

150

logarithmic function for 150

Taylor series in Newton's method 243

Taylor series in numerical solution of ordinary differential equations 250

Taylor series in Simpson's rule 246

test ratio 143 tests for convergence of series

141-5 comparison 142 d'Alembert's ratio 145

tests for convergence of series of positive terms 142

tests for convergence of series of alternating sign 143

total differential 103 transcendental function 5 transform, Fourier 174

Laplace 226 transformation, linear 208 transpose of a determinant 182 transpose of a matrix 200 trapezoidal rule 245

comparison of, with Simpson's rule 248

trigonometric functions, definitions of 6

differentiation of 28 Maclaurin series expansion of 149 relationships between 9 standard integrals of 47 use of, in integration

49-51 trigonometric functions in

integrands 51 triple integral 75, 84, 86 triple product, scalar 126

vector 128 trivial solution to linear

equations 189 turning points 34-6

u-test 274 undetermined coefficients in

solution of differential equations 226

unit matrix 200 unit vector 116 unitary matrix 205 upper triangle determinant 256

variable, dependent and independent 1

variance 264 variance of regression

coefficient 270 vector addition 117 vector field 132

curl of 135 divergence of 134

vector operator 132 ff. vector prodvct 123 vector subtraction 117 vector triple product 128 vectors 116 ff.

column 197 components of 118 coplanarity of 127 derivatives of 129-32 direction cosines 119 magnitude of 116 null 116 products of 120

303

relation of, with functions 166

resolution of 118 row 197 unit 116, 118

vectors in n dimensions 166 velocity as a derivative 33 velocity of a vector 130 vibrating string 234

wave equation 234-7 wavefunction, antisymmetric

186 determinantal form of

185 weight in calculation of mean

80 weight in Fourier expansion

170 weight ~n Gaussian quadrature

249 weight of observation 268 work done by a force, vector

notation 122 work done by a gas 40 work done by a variable force

68 working mean 262

xn, derivative of 25 integral of 46