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,Essential Mathematical Methods
for the Physical Sciences
Student Solution Manual
K. F. RILEY
University of Cambridge
M. P. HOBSON
University of Cambridge
, Contents
Preface page vii
1 Matrices and vector spaces 1
2 Vector calculus 27
3 Line, surface and volume integrals 41
4 Fourier series 56
5 Integral transforms 72
6 Higher-order ordinary differential equations 85
7 Series solutions of ordinary differential equations 105
8 Eigenfunction methods for differential equations 116
9 Special functions 128
10 Partial differential equations 138
11 Solution methods for PDEs 149
12 Calculus of variations 166
13 Integral equations 182
14 Complex variables 192
15 Applications of complex variables 200
16 Probability 214
17 Statistics 231
v
, 1 Matrices and vector spaces
1.1 Which of the following statements about linear vector spaces are true? Where a statement is false,
give a counter-example to demonstrate this.
(a) Non-singular N × N matrices form a vector space of dimension N 2 .
(b) Singular N × N matrices form a vector space of dimension N 2 .
(c) Complex numbers form a vector space of dimension 2.
(d) Polynomial functions of x form an infinite-dimensional
vector space.
(e) Series {a0 , a1 , a2 , . . . , aN } for which N
n=0 |an | = 1 form an N-dimensional vector space.
2
(f) Absolutely convergent series form an infinite-dimensional vector space.
(g) Convergent series with terms of alternating sign form an infinite-dimensional vector space.
We first remind ourselves that for a set of entities to form a vector space, they must
pass five tests: (i) closure under commutative and associative addition; (ii) closure under
multiplication by a scalar; (iii) the existence of a null vector in the set; (iv) multiplication
by unity leaves any vector unchanged; (v) each vector has a corresponding negative vector.
(a) False. The matrix 0N , the N × N null matrix, required by (iii) is not non-singular
and is therefore not in the set.
1 0 0 0
(b) Consider the sum of and . The sum is the unit matrix which is not
0 0 0 1
singular and so the set is not closed; this violates requirement (i). The statement is false.
(c) The space is closed under addition and multiplication by a scalar; multiplication
by unity leaves a complex number unchanged; there is a null vector (= 0 + i0) and a
negative complex number for each vector. All the necessary conditions are satisfied and
the statement is true.
(d) As in the previous case, all the conditions are satisfied and the statement is true.2
(e) This statement is false. To see why, consider bn = an + an for which N n=0 |bn | =
4 = 1, i.e. the set is not closed (violating (i)), or note that there is no zero vector with unit
norm (violating (iii)).
(f) True. Note that an absolutely convergent series remains absolutely convergent when
the signs of all of its terms are reversed.
(g) False. Consider the two series defined by
n n
a0 = 12 , an = 2 − 12 for n ≥ 1; bn = − − 12 for n ≥ 0.
The series that is the sum of {an } and {bn } does not have alternating signs and so closure
(required by (i)) does not hold.
1
,Essential Mathematical Methods
for the Physical Sciences
Student Solution Manual
K. F. RILEY
University of Cambridge
M. P. HOBSON
University of Cambridge
, Contents
Preface page vii
1 Matrices and vector spaces 1
2 Vector calculus 27
3 Line, surface and volume integrals 41
4 Fourier series 56
5 Integral transforms 72
6 Higher-order ordinary differential equations 85
7 Series solutions of ordinary differential equations 105
8 Eigenfunction methods for differential equations 116
9 Special functions 128
10 Partial differential equations 138
11 Solution methods for PDEs 149
12 Calculus of variations 166
13 Integral equations 182
14 Complex variables 192
15 Applications of complex variables 200
16 Probability 214
17 Statistics 231
v
, 1 Matrices and vector spaces
1.1 Which of the following statements about linear vector spaces are true? Where a statement is false,
give a counter-example to demonstrate this.
(a) Non-singular N × N matrices form a vector space of dimension N 2 .
(b) Singular N × N matrices form a vector space of dimension N 2 .
(c) Complex numbers form a vector space of dimension 2.
(d) Polynomial functions of x form an infinite-dimensional
vector space.
(e) Series {a0 , a1 , a2 , . . . , aN } for which N
n=0 |an | = 1 form an N-dimensional vector space.
2
(f) Absolutely convergent series form an infinite-dimensional vector space.
(g) Convergent series with terms of alternating sign form an infinite-dimensional vector space.
We first remind ourselves that for a set of entities to form a vector space, they must
pass five tests: (i) closure under commutative and associative addition; (ii) closure under
multiplication by a scalar; (iii) the existence of a null vector in the set; (iv) multiplication
by unity leaves any vector unchanged; (v) each vector has a corresponding negative vector.
(a) False. The matrix 0N , the N × N null matrix, required by (iii) is not non-singular
and is therefore not in the set.
1 0 0 0
(b) Consider the sum of and . The sum is the unit matrix which is not
0 0 0 1
singular and so the set is not closed; this violates requirement (i). The statement is false.
(c) The space is closed under addition and multiplication by a scalar; multiplication
by unity leaves a complex number unchanged; there is a null vector (= 0 + i0) and a
negative complex number for each vector. All the necessary conditions are satisfied and
the statement is true.
(d) As in the previous case, all the conditions are satisfied and the statement is true.2
(e) This statement is false. To see why, consider bn = an + an for which N n=0 |bn | =
4 = 1, i.e. the set is not closed (violating (i)), or note that there is no zero vector with unit
norm (violating (iii)).
(f) True. Note that an absolutely convergent series remains absolutely convergent when
the signs of all of its terms are reversed.
(g) False. Consider the two series defined by
n n
a0 = 12 , an = 2 − 12 for n ≥ 1; bn = − − 12 for n ≥ 0.
The series that is the sum of {an } and {bn } does not have alternating signs and so closure
(required by (i)) does not hold.
1
