, Essential
Mathematical Methods
for the
Physical Sciences
Instructor Solution Manual
K F RILEY and M P HOBSON
———- OOO ———-
,Contents
1 Matrices and vector spaces 1
2 Vector calculus 53
3 Line, surface and volume integrals 79
4 Fourier series 108
5 Integral transforms 137
6 Higher-order ODEs 168
7 Series solutions of ODEs 206
8 Eigenfunction methods for ODEs 226
9 Special functions 247
10 Partial differential equations 267
11 Solution methods for PDEs 289
12 Calculus of variations 323
13 Integral equations 354
14 Complex variables 374
15 Applications of complex variables 389
16 Probability 412
17 Statistics 444
ii
, Preface
For reasons that are explained in the preface to Essential Mathematical Methods
for the Physical Sciences the text of the third edition of Mathematical Methods
for Physics and Engineering (MMPE) (Cambridge: Cambridge University Press,
2006) by Riley, Hobson and Bence, after a number of additions and omissions,
has been republished as two slightly overlapping texts. Essential Mathematical
Methods for the Physical Sciences (EMMPS) contains most of the more advanced
material, and specifically develops mathematical methods that can be applied
throughout the physical sciences; an augmented version of the more introductory
material, principally concerned with mathematical tools rather than methods,
is available as Foundation Mathematics for the Physical Sciences. The full text
of MMPE, including all of the more specialized and advanced topics, is still
available under its original title.
As in the third edition of MMPE, the penultimate subsection of each chap-
ter of EMMPS consists of a significant number of problems, nearly all of which
are based on topics drawn from several sections of that chapter. Also as in the
third edition, hints and outline answers are given in the final subsection, but
only to the odd-numbered problems, leaving all even-numbered problems free
to be set as unaided homework.
A separate manual containing complete solutions to the two hundred and
thirty plus odd-numbered problems in EMMPS is available to students. This web
site file contains fully-worked solutions for all the problems in EMMPS, both
the odd-numbered and the even-numbered; it is available only to registered
instructors. For each problem, the original question is reproduced and followed
by a fully-worked solution. In some cases where the original problem makes
internal reference to the main text, the questions have been reworded, usually
by including additional information; the solutions themselves do contain some
numbered references to topics and results in the main text.
In many cases the solution given here is even fuller than one that might be
expected of a good student who has understood the material. This is because
we have, where possible, aimed to make the solutions both utilitarian and the
basis for further instruction. To this end, we have included comments that are
intended to show how the plan for the solution is formulated and have provided
the justifications for particular intermediate steps (something not always done,
even by the best of students). We have also tried to write each individual
substituted formula in the form that best indicates how it was obtained, before
simplifying it at the next or a subsequent stage. Where several lines of algebraic
iii
Mathematical Methods
for the
Physical Sciences
Instructor Solution Manual
K F RILEY and M P HOBSON
———- OOO ———-
,Contents
1 Matrices and vector spaces 1
2 Vector calculus 53
3 Line, surface and volume integrals 79
4 Fourier series 108
5 Integral transforms 137
6 Higher-order ODEs 168
7 Series solutions of ODEs 206
8 Eigenfunction methods for ODEs 226
9 Special functions 247
10 Partial differential equations 267
11 Solution methods for PDEs 289
12 Calculus of variations 323
13 Integral equations 354
14 Complex variables 374
15 Applications of complex variables 389
16 Probability 412
17 Statistics 444
ii
, Preface
For reasons that are explained in the preface to Essential Mathematical Methods
for the Physical Sciences the text of the third edition of Mathematical Methods
for Physics and Engineering (MMPE) (Cambridge: Cambridge University Press,
2006) by Riley, Hobson and Bence, after a number of additions and omissions,
has been republished as two slightly overlapping texts. Essential Mathematical
Methods for the Physical Sciences (EMMPS) contains most of the more advanced
material, and specifically develops mathematical methods that can be applied
throughout the physical sciences; an augmented version of the more introductory
material, principally concerned with mathematical tools rather than methods,
is available as Foundation Mathematics for the Physical Sciences. The full text
of MMPE, including all of the more specialized and advanced topics, is still
available under its original title.
As in the third edition of MMPE, the penultimate subsection of each chap-
ter of EMMPS consists of a significant number of problems, nearly all of which
are based on topics drawn from several sections of that chapter. Also as in the
third edition, hints and outline answers are given in the final subsection, but
only to the odd-numbered problems, leaving all even-numbered problems free
to be set as unaided homework.
A separate manual containing complete solutions to the two hundred and
thirty plus odd-numbered problems in EMMPS is available to students. This web
site file contains fully-worked solutions for all the problems in EMMPS, both
the odd-numbered and the even-numbered; it is available only to registered
instructors. For each problem, the original question is reproduced and followed
by a fully-worked solution. In some cases where the original problem makes
internal reference to the main text, the questions have been reworded, usually
by including additional information; the solutions themselves do contain some
numbered references to topics and results in the main text.
In many cases the solution given here is even fuller than one that might be
expected of a good student who has understood the material. This is because
we have, where possible, aimed to make the solutions both utilitarian and the
basis for further instruction. To this end, we have included comments that are
intended to show how the plan for the solution is formulated and have provided
the justifications for particular intermediate steps (something not always done,
even by the best of students). We have also tried to write each individual
substituted formula in the form that best indicates how it was obtained, before
simplifying it at the next or a subsequent stage. Where several lines of algebraic
iii
