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Competing with High Quality Data
by Graham Shaw, Brian R. Martin
Mathematics for Physicists
Editors' preface to the Manchester Physics Series
Authors' preface
Notes and website information
‘Starred’ material
Website
Examples, problems and solutions
1 Real numbers, variables and functions
1.1 Real numbers
1.2 Real variables
1.3 Functions, graphs and co-ordinates
Problems 1
Notes
2 Some basic functions and equations
2.1 Algebraic functions
2.2 Trigonometric functions
2.3 Logarithms and exponentials
2.4 Conic sections
Problems 2
Notes
3 Differential calculus
3.1 Limits and continuity
3.2 Differentiation
3.3 General methods
3.4 Higher derivatives and stationary points
3.5 Curve sketching
Problems 3
Notes
4 Integral calculus
4.1 Indefinite integrals
4.2 Definite integrals
4.3 Change of variables and substitutions
4.4 Integration by parts
4.5 Numerical integration
4.6 Improper integrals
4.7 Applications of integration
Problems 4
Notes
5 Series and expansions
5.1 Series
5.2 Convergence of infinite series
5.3 Taylor's theorem and its applications
5.4 Series expansions
*5.5 Proof of d'Alembert's ratio test
*5.6 Alternating and other series
Problems 5
Notes
6 Complex numbers and variables
6.1 Complex numbers
6.2 Complex plane: Argand diagrams
6.3 Complex variables and series
6.4 Euler's formula
Problems 6
Notes
7 Partial differentiation
7.1 Partial derivatives
7.2 Differentials
7.3 Change of variables
7.4 Taylor series
7.5 Stationary points
*7.6 Lagrange multipliers
*7.7 Differentiation of integrals
Problems 7
Notes
8 Vectors
8.1 Scalars and vectors
8.2 Products of vectors
8.3 Applications to geometry
8.4 Differentiation and integration
Problems 8
Notes
9 Determinants, Vectors and Matrices
9.1 Determinants
9.2 Vectors in n Dimensions
9.3 Matrices and linear transformations
9.4 Square Matrices
Problems 9
Notes
10 Eigenvalues and eigenvectors
10.1 The eigenvalue equation
*10.2 Diagonalisation of matrices
Problems 10
Notes
11 Line and multiple integrals
11.1 Line integrals
11.2 Double integrals
11.3 Curvilinear co-ordinates in three dimensions
11.4 Triple or volume integrals
Problems 11
12 Vector calculus
12.1 Scalar and vector fields
12.2 Line, surface, and volume integrals
12.3 The divergence theorem
12.4 Stokes' theorem
Problems 12
Notes
13 Fourier analysis
13.1 Fourier series
13.2 Complex Fourier series
13.3 Fourier transforms
Problems 13
Notes
14 Ordinary differential equations
14.1 First-order equations
14.2 Linear ODEs with constant coefficients
*14.3 Euler's equation
Problems 14
Notes
15 Series solutions of ordinary differential equations
15.1 Series solutions
15.2 Eigenvalue equations
15.3 Legendre's equation
15.4 Bessel's equation
Problems 15
Notes
16 Partial differential equations
16.1 Some important PDEs in physics
16.2 Separation of variables: Cartesian co-ordinates
16.3 Separation of variables: polar co-ordinates
*16.4 The wave equation: d'Alembert's solution
*16.5 Euler Equations
*16.6 Boundary conditions and uniqueness
Problems 16
Notes
Answers to selected problems
Problems 1
Problems 2
Problems 3
Problems 4
Problems 5
Problems 6
Problems 7
Problems 8
Problems 9
Problems 10
Problems 11
Problems 12
Problems 13
Problems 14
Problems 15
Problems 16
Index
EULA
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Editors' preface to the Manchester Physics Series
CONTENTS
Editors' preface to the Manchester Physics Series
Authors' preface
Notes and website information
‘Starred’ material
Website
Examples, problems and solutions
1 Real numbers, variables and functions
1.1 Real numbers
1.2 Real variables
1.3 Functions, graphs and co-ordinates
Problems 1
Notes
2 Some basic functions and equations
2.1 Algebraic functions
2.2 Trigonometric functions
2.3 Logarithms and exponentials
2.4 Conic sections
Problems 2
Notes
3 Differential calculus
3.1 Limits and continuity
3.2 Differentiation
3.3 General methods
3.4 Higher derivatives and stationary points
3.5 Curve sketching
Problems 3
Notes
4 Integral calculus
4.1 Indefinite integrals
4.2 Definite integrals
4.3 Change of variables and substitutions
4.4 Integration by parts
4.5 Numerical integration
4.6 Improper integrals
4.7 Applications of integration
Problems 4
Notes
5 Series and expansions
5.1 Series
5.2 Convergence of infinite series
5.3 Taylor's theorem and its applications
5.4 Series expansions
*5.5 Proof of d'Alembert's ratio test
*5.6 Alternating and other series
Problems 5
Notes
6 Complex numbers and variables
6.1 Complex numbers
6.2 Complex plane: Argand diagrams
6.3 Complex variables and series
6.4 Euler's formula
Problems 6
Notes
7 Partial differentiation
7.1 Partial derivatives
7.2 Differentials
7.3 Change of variables
7.4 Taylor series
7.5 Stationary points
*7.6 Lagrange multipliers
*7.7 Differentiation of integrals
Problems 7
Notes
8 Vectors
8.1 Scalars and vectors
8.2 Products of vectors
8.3 Applications to geometry
8.4 Differentiation and integration
Problems 8
Notes
9 Determinants, Vectors and Matrices
9.1 Determinants
9.2 Vectors in
n
Dimensions
9.3 Matrices and linear transformations
9.4 Square Matrices
Problems 9
Notes
10 Eigenvalues and eigenvectors
10.1 The eigenvalue equation
*10.2 Diagonalisation of matrices
Problems 10
Notes
11 Line and multiple integrals
11.1 Line integrals
11.2 Double integrals
11.3 Curvilinear co-ordinates in three dimensions
11.4 Triple or volume integrals
Problems 11
12 Vector calculus
12.1 Scalar and vector fields
12.2 Line, surface, and volume integrals
12.3 The divergence theorem
12.4 Stokes' theorem
Problems 12
Notes
13 Fourier analysis
13.1 Fourier series
13.2 Complex Fourier series
13.3 Fourier transforms
Problems 13
Notes
14 Ordinary differential equations
14.1 First-order equations
14.2 Linear ODEs with constant coefficients
*14.3 Euler's equation
Problems 14
Notes
15 Series solutions of ordinary differential equations
15.1 Series solutions
15.2 Eigenvalue equations
15.3 Legendre's equation
15.4 Bessel's equation
Problems 15
Notes
16 Partial differential equations
16.1 Some important PDEs in physics
16.2 Separation of variables: Cartesian co-ordinates
16.3 Separation of variables: polar co-ordinates
*16.4 The wave equation: d'Alembert's solution
*16.5 Euler Equations
*16.6 Boundary conditions and uniqueness
Problems 16
Notes
Answers to selected problems
Problems 1
Problems 2
Problems 3
Problems 4
Problems 5
Problems 6
Problems 7
Problems 8
Problems 9
Problems 10
Problems 11
Problems 12
Problems 13
Problems 14
Problems 15
Problems 16
Index
EULA
List of Tables
Chapter 3
Table 3.1
Chapter 4
Table 4.1
Table 4.2
Chapter 5
Table 5.1
Chapter 12
Table 12.1
Table 12.2
Chapter 14
Table 14.1
Chapter 15
Table 15.1
List of Illustrations
Chapter 1
Figure 1.1
Graph of the function
f
(
x
) =
x
3
− 3
x
2
− 6
x
+ 8.
Figure 1.2
Graphs of the functions
f
(
x
) = 3
x
2
− 15 (dashed line) and
f
(
x
) =
x
3
+ 4
x
(solid line).
Figure 1.3
Cartesian co-ordinate system for the points
A
(
x
1
,
y
1
) and
B
(
x
2
,
y
2
).
Figure 1.4
Construction to deduce the equation of a circle in Cartesian co-ordinates.
Figure 1.5
The linear function
y
=
mx
+
c
with parameters: (a)
m
= 1 and
c
= −2, 0, 2 and (b)
m
= 1, 2, 3 and
c
= 2.
Figure 1.6
The functions: (a)
y
=
x
3
+ 2
x
2
− 5
x
; and (b)
y
= 2/(
x
2
− 1). The blue dashed lines show the tangents at the points (2, 6) and ± 1, for curves (a) and (b), respectively.
Figure 1.7
Construction to show that the product of the gradients of perpendicular lines is −1.
Figure 1.8
Use of inequalities to define regions of the
xy
-plane.
Figure 1.9
A right-handed three-dimensional Cartesian co-ordinate system.
Figure 1.10
Figure 1.11
Chapter 2
Figure 2.1
Plot of the function
y
=
f
(
x
) =
x
3
− 3
x
2
− 4
x
+ 7.
Figure 2.2
Angle, arc and sector.
Figure 2.3
Plane polar co-ordinates (
r
, θ).
Figure 2.4
Polar angles θ, for constant
r
; diagrams (a) and (b) represent the same point
P
, and diagrams (c) and (d) represent the same point
P
′.
Figure 2.5
A particle/point moving counter-clockwise in a circular trajectory with constant radius
r.
Figure 2.6
The circular functions sin θ and cos θ as functions of θ.
Figure 2.7
The construction used to establish the result (2.31).
Figure 2.8
The circular functions (a) tan θ and (b)
as functions of θ.
Figure 2.9
The circular functions (a) cosec θ and (b)
as functions of θ.
Figure 2.10
Inverse circular functions: (a)
, (b)
and (c)
as functions of
x
.
Figure 2.11
Construction to prove the identity (2.36a).
Figure 2.12
Definition of the angles and lengths of the sides for the sine and cosine rules. The angles are labelled by the same letters
A
,
B
,
C
as the vertices, while the opposing sides are labelled by the corresponding lower case letters
a
,
b
,
c
, respectively.
Figure 2.13
Construction to prove the sine and cosine rules.
Figure 2.14
Figure 2.15
Plots of log
n
(
x
) for
n
= 2, 3 and 10.
Figure 2.16
The exponential function
a
x
for
a
= 1/2, 1, 3/2 and 2.
Figure 2.17
Construction to show that the exponential function is proportional to its gradient.
Figure 2.18
The functions
e
x
,
e
−
x
andln
x
.
Figure 2.19
The functions sinh
x
and cosh
x
and their relation to the exponential functions. Note that cosh
x
and sinh
x
become equal at large positive values of
x
.
Figure 2.20
The function tanh
x
.
Figure 2.21
Geometrical interpretation of conic sections.
Figure 2.22
Derivation of the polar equation for a conic section.
Figure 2.23
The standard forms for: (a) the parabola (
e
= 1); (b) the ellipse (
e
< 1); and (c) the hyperbola (
e
> 1). Only one focus and directrix is shown for the ellipse and hyperbola.
Figure 2.24
Chapter 3
Figure 3.1
The function
f
(
x
) =
x
/|
x
| in the vicinity of
x
= 0.
Figure 3.2
Geometrical interpretation of the derivative d
f
/d
x
.
Figure 3.3
Construction to find the limit of sin θ/θ as θ → 0.
Figure 3.4
The behaviour of a function (solid line) and its derivative (dashed line) in the vicinity of a stationary point
x
=
x
0
, together with the gradient at
x
0
(dotted line), for (a) a minimum, (b) a maximum and (c, d) points of inflection.
Figure 3.5
The functions
f
(
x
) =
x
2
,
x
3
and
x
4
.
Figure 3.6
The function
y
= (1 +
x
)/(1 −
x
), showing the asymptotes
x
= 1 and
y
= −1 (dashed lines), the root
x
= −1(•) and three sample values at
x
= 0, 2, 3 (
). There are no stationary points.
Figure 3.7
The function (3.48) showing the roots, stationary points (•) at
x
= −1, 2 and four sample values at
x
= −2, 0, 1, 3, (
).
Figure 3.8
The functions exp ( −
x
2
) and
x
exp ( −
x
2
), showing the roots, stationary points (·) and sample points (
) used, together with the asymptotic behaviour to define the shapes of the curves.
Figure 3.9
Chapter 4
Figure 4.1
Integral of the function
f
(
x
) between the limits
a
and
b
.
Figure 4.2
Integral of the function sin
x
between the limits 0 and π.
Figure 4.3
Figure 4.4
Construction to derive Eq. (4.11).
Figure 4.5
Infinite integrals over
f
(
x
) and
g
(
x
) (see text).
Figure 4.6
Figure 4.7
Construction used to obtain (4.52)
Figure 4.8
The surface of revolution generated by
y
=
f
(
x
) for
a
≤
x
≤
b
and the interval
x
to (
x
+ d
x
).
Figure 4.9
Figure 4.10
Definition of the moment of inertia (4.57) for a point particle.
Figure 4.11
Figure 4.12
Chapter 6
Figure 6.1
Argand diagram.
Figure 6.2
Addition of two complex numbers.
Figure 6.3
The cube roots of unity.
Chapter 7
Figure 7.1
The original and rotated co-ordinate systems (7.30).
Figure 7.2
A two-dimensional surface
f
(
x
,
y
) showing a maximum (denoted by Max) and a minimum (denoted by Min).
Figure 7.3
A two-dimensional surface
f
(
x
,
y
) showing an example of one type of saddle point.
Figure 7.4
The ellipse (7.47a) and the four stationary points (
+
) given by (7.48b).
Chapter 8
Figure 8.1
Graphical representation of vectors.
Figure 8.2
Addition of vectors.
Figure 8.3
Addition and subtraction of vectors.
Figure 8.4
Constructions to show that
s
=
a
+ (
b
+
c
) and
s
= (
a
+
b
) +
c
are identical. Note that the vertex
P
is not necessarily in the plane defined by the other three vertices.
Figure 8.5
The vectors
a
, 2
a
and −2
a
.
Figure 8.6
Construction to prove the parallelogram relation.
Figure 8.7
Decomposition into Cartesian components.
Figure 8.8
Definition of a position vector.
Figure 8.9
Direction ratios and direction cosines.
Figure 8.10
Construction to prove the result (8.6a).
Figure 8.11
Right-hand screw rule for vector products.
Figure 8.12
Angular velocity.
Figure 8.13
Diagram used to calculate the torque about a point
O
due to a force acting at a point
P
or a point
P
′ lying on the same line of action.
Figure 8.14
Figure 8.15
Geometrical interpretation of a triple scalar product.
Figure 8.16
Torque as a triple scalar product.
Figure 8.17
Definitions of the angles on the sphere.
Figure 8.18
Construction of equation for a straight line.
Figure 8.19
Shortest distance between (a) a point and a line, (b) two lines.
Figure 8.20
A point
R
in the plane defined by the points
A, B, C,
having position vectors
a
,
b
,
c
,
as given in (8.42). In the figure we have chosen the point
R
so that
s
and
s
′ are both less than 1.
Figure 8.21
Shortest distance between (a) the origin and a plane; and (b) an arbitrary point
P
and a plane.
Figure 8.22
Intersection of two planes.
Chapter 9
Figure 9.1
The rotation of the two-dimensional vector (9.47) through an angle θ.
Chapter 10
Figure 10.1
An example of coupled motion, showing the coupling of two weights via a spring.
Figure 10.2
The normal modes of the system shown in Figure 10.1.
Figure 10.3
An ellipsoid, showing the principal axes
x
′,
y
′,
z
′ and the lengths of the semi-axes
a, b, c.
Figure 10.4
(a) Prolate spheroid resulting when the lengths of the semi-axes satisfy
a
=
b
<
c
. (b) Oblate spheroid resulting when the lengths of the semi-axes satisfy
a
=
b
>
c
.
Figure 10.5
(a) Hyperboloid of one sheet, (b) hyperboloid of two sheets.
Figure 10.6
Figure 10.7
Figure 10.8
Figure 10.9
Chapter 11
Figure 11.1
Integration path for (a) a single-valued function and (b) a two-valued function.
Figure 11.2
Figure 11.3
A simple closed plane curve.
Figure 11.4
Figure 11.5
Constructions for defining a double integral.
Figure 11.6
Figure 11.7
Figure 11.8
Figure used in the derivation of Green's theorem in the plane.
Figure 11.9
Two co-ordinate systems.
Figure 11.10
Curvilinear co-ordinates in a plane, showing lines of constant
u
1
and
u
2
, spaced by δ
u
1
and δ
u
2
, respectively. The area
S
to be integrated over is the interior of the closed loop and the shaded region is one of the areas δ
S
rs
.
Figure 11.11
Construction to define the area δ
S
rs
in the limit that δ
u
1
and δ
u
2
become infinitesimally small.
Figure 11.12
(a) Cylindrical polar co-ordinates ρ, φ,
z
, and the associated unit vectors
e
ρ
,
e
φ
,
e
z
. The vector
e
ρ
is in the direction of the radius vector ρ;
e
φ
is in the
xy
–plane, tangential to the circle through
P
, and in the direction of increasing φ;
e
z
is in the
z-
direction. The three vectors
e
ρ
,
e
φ
,
e
z
are mutually orthogonal. (b) Spherical polar co-ordinates
r
, θ, φ, and the associated unit vectors
e
r
,
e
θ
,
e
φ
. The vector
e
r
is in the direction of the radius vector
r
;
e
φ
is in the
xy
–plane, tangential to the circle through
P
, and in the direction of increasing φ;
e
θ
is at right angles to
e
r
in the direction of increasing θ. The three vectors
e
r
,
e
θ
,
e
φ
are mutually orthogonal.
Figure 11.13
The volume element in (a) cylindrical polar co-ordinates; (b) spherical polar co-ordinates.
Figure 11.14
Chapter 12
Figure 12.1
The motion of a fluid around a smooth solid. The coloured lines are the flow lines and the arrows show the direction of the vector field, in this case the velocity
v
(
r
).
Figure 12.2
Diagram for the derivation of the gradient of a scalar field ψ(
r
).
PR
is normal to the equipotential surface ψ
R
at the point of intersection
P
.
Figure 12.3
Gradient and directional derivative, where the vector
u
indicates the direction of
defined in the text.
Figure 12.4
Definition of a space curve for the line integral of a scalar product. The path is defined in terms of a parameter
s
, so that
r
=
r
(
s
).
Figure 12.5
Figure 12.6
Examples of open surfaces that are (a) two-sided (b) one-sided and (c) the use of a projection of a two-sided surface onto a plane to define the direction of
.
Figure 12.7
Diagram used to illustrate the evaluation of surface integrals by the projection method.
Figure 12.8
A segment through the region Ω lying parallel to the
x-
axis and with constant infinitesimal cross section d
y
d
z
, used in the derivation of the divergence theorem.
Figure 12.9
Construction to derive the divergence in orthogonal curvilinear co-ordinates.
Figure 12.10
The spherical surface
S
used to calculate the electric field due to a spherical charge distribution of radius
R
for
r
>
R
.
Figure 12.11
The infinitesimal cylinder used in Example 12.14, where
S
is the surface of the conductor and
E
=
0
within the conductor.
Figure 12.12
Flow of a fluid. The coloured lines are the flow lines; the arrows show the direction of the vector field
V
. Their lengths show the relative magnitudes of
V
.
Figure 12.13
Construction to derive Stokes' theorem.
Figure 12.14
Construction to derive Stokes' theorem in curvilinear co-ordinates.
Figure 12.15
The circuit
C
used to derive (12.79), where the current
I
at the centre is directed out of the page.
Chapter 13
Figure 13.1
An arbitrary periodic function.
Figure 13.2
Figure 13.3
Figure 13.4
A discontinuous function.
Figure 13.5
The function
f
(
x
) given by Eq. (13.14).
Figure 13.6
The function (13.14) (solid black line), together with its Fourier representations derived from (13.15). The short-dashed blue line is the constant plus the sine term; the long-dashed blue line also includes the first cosine term; and the solid blue line includes the first two cosine terms.
Figure 13.7
The function
f
(
x
) given by (13.16).
Figure 13.8
The function (13.16) (black lines), together with its Fourier representations (coloured lines) derived from (13.17) with 4 and 11 terms (including the constant), respectively.
Figure 13.9
Possible continuations of the non-periodic function shown in (a). These have the following symmetries about
x
= 0: (b) none, (c) odd and (d) even.
Figure 13.10
The function
f
(
x
) =
x
2
for 0 ≤
x
≤ 2, together with an extension to the interval − 2 ≤
x
≤ 2, so that the resulting function is even about
x
= 0.
Figure 13.11
The functions defined by (a) Eq. (13.24) and (b) Eq. (13.25).
Figure 13.12
The real part of the wave form described by the function (13.42).
Figure 13.13
The functions: (a)
f
(
x
) = exp ( − |
x
|) and (b) its Fourier transform 1/[π(1 +
k
2
)]; and (c) the function
f
(
x
) = π (|
x
| < 4), 0 (|
x
| ≥ 4) and (d) its Fourier transform sin (4
k
)/
k
.
Figure 13.14
The functions
f
(
x
) and
g
(
k
) of Equations (13.56a) and (13.56b).
Figure 13.15
The real part of
E
(
t
) and the envelope formed by the exponential decay terms (left figure) and the intensity spectrum |
g
(ω)|
2
(right figure).
Figure 13.16
Possible forms of
f
n
(
x
), which approximate more closely to δ(
x
) as
n
→ ∞: (a) from (13.59); (b) from (13.60).
Figure 13.17
The Fourier transforms of (a)
and (b)
e
iqx
together with (c) their convolution, which is the Fourier transform of their product (13.72).
Figure 13.18
(a) The ‘top hat’ function
T
(
x
); (b) the convolution
f
(
x
).
Figure 13.19
Chapter 14
Figure 14.1
Motion
x
(
t
) of a damped harmonic oscillator: (a) light damping γ ≪ 2ω
0
; (b) critical damping γ = 2ω
0
(solid line), and heavy damping γ > 2ω
0
(dashed line). The units of
t
and
x
(
t
) are arbitrary.
Figure 14.2
Order of integration in (14.47c).
Chapter 15
Figure 15.1
Legendre polynomials: (a)
P
l
(
x
),
l
= 0, 1, 2, 3, and (b)
P
10
(
x
).
Figure 15.2
Construction for the multipole expansion.
Figure 15.3
The gamma function Γ(
x
).
Figure 15.4
Bessel functions
J
n
(
x
):
n
= 0 (solid),
n
= 1 (short dash),
n
= 2 (long dash-short dash),
n
= 3 (long dash).
Chapter 16
Figure 16.1
Figure 16.2
Diagrams used in the derivation of the density-of-states formula (16.30), where only the (
k
x
,
k
y
) plane is shown. The diagrams are not to the same scale, and in practical application the spacing between the points in the left-hand diagram is normally extremely small compared to typical
k
-values.
Figure 16.3
A travelling wave corresponding to the solution (16.81) shown at two arbitrary times
t
1
and
t
2
>
t
1
for an arbitrary choice of
u
(
x
, 0), where
.
Figure 16.4
The error function (16.100a) and the complementary error function (16.100b).
Guide
Cover
Table of Contents
Preface
Pages
xi
xiii
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