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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
A concise and up-to-date introduction to mathematical methods for students in the physical sciences
Mathematical Methods in Physics, Engineering and Chemistry offers an introduction to the most important methods of theoretical physics. Written by two physics professors with years of experience, the text puts the focus on the essential math topics that the majority of physical science students require in the course of their studies. This concise text also contains worked examples that clearly illustrate the mathematical concepts presented and shows how they apply to physical problems.
This targeted text covers a range of topics including linear algebra, partial differential equations, power series, Sturm-Liouville theory, Fourier series, special functions, complex analysis, the Green’s function method, integral equations, and tensor analysis. This important text:
- Provides a streamlined approach to the subject by putting the focus on the mathematical topics that physical science students really need
- Offers a text that is different from the often-found definition-theorem-proof scheme
- Includes more than 150 worked examples that help with an understanding of the problems presented
- Presents a guide with more than 200 exercises with different degrees of difficulty
Written for advanced undergraduate and graduate students of physics, materials science, and engineering, Mathematical Methods in Physics, Engineering and Chemistry includes the essential methods of theoretical physics. The text is streamlined to provide only the most important mathematical concepts that apply to physical problems.
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Table of Contents
Cover
Preface
1 Vectors and linear operators
1.1 The linearity of physical phenomena
1.2 Vector spaces
1.3 Inner products and orthogonality
1.4 Operators and matrices
1.5 Eigenvectors and their role in representing operators
1.6 Hilbert space: Infinite‐dimensional vector space
1.5 Exercises
2 Sturm–Liouville theory
2.1 Second‐order differential equations
2.2 Sturm–Liouville systems
2.3 The Sturm–Liouville eigenproblem
2.4 The Dirac delta function
2.5 Completeness
2.6 Recap
Summary
Exercises
3 Partial differential equations
3.1 A survey of partial differential equations
3.2 Separation of variables and the Helmholtz equation
3.3 The paraxial approximation
3.4 The three types of linear PDEs
3.5 Outlook
Summary
Exercises
4 Fourier analysis
4.1 Fourier series
4.2 The exponential form of Fourier series
4.3 General intervals
4.4 Parseval's theorem
4.5 Back to the delta function
4.6 Fourier transform
4.7 Convolution integral
Summary
Exercises
5 Series solutions of ordinary differential equations
5.1 The Frobenius method
5.2 Wronskian method for obtaining a second solution
5.3 Bessel and Neumann functions
5.4 Legendre polynomials
Summary
Exercises
6 Spherical harmonics
6.1 Properties of the Legendre polynomials,
6.2 Associated Legendre functions,
6.3 Spherical harmonic functions,
6.4 Addition theorem for
6.5 Laplace equation in spherical coordinates
Summary
Exercises
7 Bessel functions
7.1 Small‐argument and asymptotic forms
7.2 Properties of the Bessel functions,
7.3 Orthogonality
7.4 Bessel series
7.5 The Fourier‐Bessel transform
7.6 Spherical Bessel functions
7.7 Expansion of plane waves in spherical harmonics
Summary
Exercises
8 Complex analysis
8.1 Complex functions
8.2 Analytic functions: differentiable in a region
8.3 Contour integrals
8.4 Integrating analytic functions
8.5 Cauchy integral formulas
8.6 Taylor and Laurent series
8.7 Singularities and residues
8.8 Definite integrals
8.9 Meromorphic functions
8.10 Approximation of integrals
8.11 The analytic signal
8.12 The Laplace transform
Summary
Exercises
9 Inhomogeneous differential equations
9.1 The method of Green functions
9.2 Poisson equation
9.3 Helmholtz equation
9.4 Diffusion equation
9.5 Wave equation
9.6 The Kirchhoff integral theorem
Summary
Exercises
10 Integral equations
10.1 Introduction
10.2 Classification of integral equations
10.3 Neumann series
10.4 Integral transform methods
10.5 Separable kernels
10.6 Self‐adjoint kernels
10.7 Numerical approaches
Summary
Exercises
11 Tensor analysis
11.1 Once over lightly: A quick intro to tensors
11.2 Transformation properties
11.3 Contraction and the quotient theorem
11.4 The metric tensor
11.5 Raising and lowering indices
11.6 Geometric properties of covariant vectors
11.7 Relative tensors
11.8 Tensors as operators
11.9 Symmetric and antisymmetric tensors
11.10 The Levi‐Civita tensor
11.11 Pseudotensors
11.12 Covariant differentiation of tensors
Summary
Exercises
A Vector calculusVector calculus
A.1 Scalar fields
A.2 Vector fields
A.3 Integration
A.4 Important integral theorems in vector calculus
A.5 Coordinate systems
B Power seriesPower series
C The gamma function, Γ(
x
)
Recursion relation
Limit formula
Reflection formula
Digamma function
D Boundary conditions for Partial Differential Equations
Summary
References
Index
End User License Agreement
List of Tables
Chapter 3
Table 3.1 Generic form of PDEs for scalar fields
most commonly encountered....
Chapter 5
Table 5.1 Legendre polynomals
for
Chapter 6
Table 6.1 Associated Legendre functions
for
,
.
Table 6.2 Spherical harmonic functions
for
,
.
Chapter 7
Table 7.1 Zeros
for which
,
,
.
Table 7.2 Spherical Bessel and Neumann functions,
and
for
.
Chapter 8
Table 8.1 Some important Laplace transforms.
List of Illustrations
Chapter 1
Figure 1.1 Vector addition and scalar multiplication.
Figure 1.2 Vectors
and
span the space of vectors confined to a plane.
Figure 1.3 Vectors
and
are a basis for the space of vectors confined to ...
Figure 1.4 Dot product of elementary vectors,
.
Figure 1.5 Coordinate systems having a common origin with axes rotated throu...
Chapter 2
Figure 2.1 Sequence of unit‐area rectangles.
Chapter 3
Figure 3.1 Displacement field
of a string under tension
.
Chapter 4
Figure 4.1 Periodic function with period
.
Figure 4.2 Periodic step function.
Figure 4.3 Partial sums of the infinite series in Eq. (4.6) in the neighborh...
Figure 4.4 Full‐wave rectification.
Figure 4.5 Periodic ramp function of period
.
Figure 4.6 Temperature distribution in a rectangular plate.
Figure 4.7 Sinc(
) versus
.
Figure 4.8 The “top hat” function.
Chapter 5
Figure 5.1 The first few Bessel functions
.
Figure 5.2 The first few Neumann functions
.
Figure 5.3 Legendre polynomials
for
.
Chapter 6
Figure 6.1 First few terms of the Legendre series for the function
.
Figure 6.2 Electrostatic potential
for a charge
displaced from the origi...
Figure 6.3 Electric dipole.
Figure 6.4 Coordinate system for the addition theorem.
Figure 6.5 Point on sphere labeled by two sets of angles.
Figure 6.6 Potential exterior to charges in
.
Figure 6.7 Conducting sphere in uniform external electric field.
Chapter 7
Figure 7.1 Diffraction from a circular aperture.
Figure 7.2 Cylindrical geometry for the Laplace equation.
Figure 7.3 Geometry of a plane wave.
Chapter 8
Figure 8.1 Complex‐valued functions map complex numbers into complex numbers...
Figure 8.2 Integration path
in the complex plane.
Figure 8.3 Region
is simply connected; regions
and
are not.
Figure 8.4 Paths of integration.
Figure 8.5 Possible branch cuts for
.
Figure 8.6 “Closing the contour” in the upper half‐plane.
Figure 8.7 Contour geometry.
Figure 8.8 Deformation of a contour to avoid poles.
Figure 8.9 Deformed paths of integration.
Figure 8.10 Contour for calculating the inverse Laplace transform.
Chapter 9
Figure 9.1 Circular drumhead geometry.
Figure 9.2 Geometry for the Graf addition theorem.
Figure 9.3 Arrow of time.
Figure 9.4 Diffusion of a localized source.
Figure 9.5 Propagation of
at speed
in one dimension.
Chapter 10
Figure 10.1 Measurement space.
Chapter 11
Figure 11.1 Force
applied to surface
with local normal
.
Figure 11.2 Points
and
infinitesimally separated, described in two coord...
Figure 11.3
coordinate system. The tangents to coordinate curves define a ...
Figure 11.4 Vector
is perpendicular to all vectors
lying in the plane.
Figure 11.5 Covariant vectors represent families of equally spaced, parallel...
Figure 11.6 Vectors of the coordinate basis
are tangent to coordinate curv...
Figure 11.7 Right‐handed and left‐handed coordinate systems. The vector
ha...
Figure 11.8 The vector cross product is a pseudovector.
Appendix A
Figure A.1 Different order of integral iteration: (a) first
and then
; (b...
Figure A.2 Green's theorem for more general regions.
Figure A.3 Green's theorem for surfaces.
Figure A.4 Generalized curvilinear coordinates.
Figure A.5 Cylindrical coordinates. (a) The three mutually perpendicular sur...
Figure A.6 Spherical coordinates. (a) The three mutually perpendicular surfa...
Appendix C
Figure C.1
versus
.
Appendix D
Figure D.1 Boundary value problem.
Figure D.2 Boundary curve
.
Figure D.3 Characteristic curves for the one‐dimensional wave equation.
Figure D.4 Domain of dependence for
.
Figure D.5 Boundary curve tangent to a characteristic at point
.
Guide
Cover
Table of Contents
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Mathematical Methods in Physics, Engineering, and Chemistry
BRETT BORDEN AND JAMES LUSCOMBE
Naval Postgraduate SchoolMonterey, CA, USA
This edition first published 2020© 2020 John Wiley & Sons, Inc.
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The right of Brett Borden and James Luscombe to be identified as the authors of this work has been asserted in accordance with law.
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Library of Congress Cataloging-in-Publication data applied for
ISBN: 9781119579656
Cover Design: WileyCover Image: © FrankRamspott/Getty Images
Preface
Mathematics is deeply ingrained in physics, in how it's taught and how it's practiced. Courses in mathematical methods of physics are core components of physics curricula, at the advanced undergraduate and beginning graduate levels. In our experience, textbooks that seek to provide a comprehensive coverage of mathematical methods tend not to mesh well with the needs of today's students, who face curricula continually under pressure to squeeze more content into the allotted time. The amount of mathematics one could be called upon to know in a scientific career is daunting – hence, the temptation to try to cover it all in textbooks. We have developed, over years of teaching, a set of notes outlining the essentials of the subject, which has turned into this book.1 Our goal has been to emphasize topics that the majority of students will require in the course of their studies. Not every student will go on to a career in theoretical physics, and thus not every student need be exposed to specialized topics at this point in their education.
Following is a sketch of the contents of this book.
Linear algebra
: What's more important in the education of scientists and engineers, calculus or linear algebra? We opt for the latter. Students are assumed to have had vector calculus (a standard component of Calculus III), a summary of which is provided in
Appendix A
. We start with vector spaces – a robust, unifying concept that pertains to much of the mathematics students will encounter in their studies. We introduce Dirac notation – never too early to see in a scientific education – but only when it makes sense. It's not a law of nature that we use Dirac notation, yet students must be well versed in it. An overarching theme of the book is
operators
. We develop linear transformations and matrices in
Chapter 1
, even though it's assumed that students already have a certain dexterity with matrices. A key topic is that of function spaces, which underlies much of what's done throughout the book in terms of representing functions with various expansions – Fourier series, for example.
Chapter 1
is a review of a full course in linear algebra; students sufficiently prepared could begin with
Chapter 2
.
Partial differential equations
: A pervasive concept in physics is that of
fields
, the behavior of which in space and time is described by partial differential equations (PDEs). To study physics at the advanced undergraduate level, one must be proficient in solving PDEs and that provides another overarching theme: Boundary value problems.
Chapters 2
–
7
form the backbone of a fairly standard one‐quarter, upper‐division course in math methods.
Chapters 8
–
11
cover separate topics that could form the basis of a follow‐on course – or a graduate‐level course appropriate for some instructional programs. Ambitious programs could cover the entire book in one semester.
Chapter 2 starts with second‐order differential equations with variable coefficients. We find that an early introduction to Sturm–Liouville theory provides a unifying framework for discussing second‐order differential equations. The key notion of the completeness of solutions to the Sturm–Liouville eigenvalue problem is of vital importance to solving boundary value problems. We restrict ourselves to homogeneous boundary conditions at this point (inhomogeneous problems are considered in Chapter 9). In Chapter 3 we review the three types of PDEs most commonly encountered – wave equation, diffusion equation, and Laplace equation. We introduce the method of separation of variables in the three main coordinate systems used in science and engineering (Cartesian, spherical, cylindrical) and how PDEs separate into systems of ordinary differential equations (ODEs). In Chapter 5, we develop the Frobenius method for solving ODEs with variable coefficients. Along the way, Fourier analysis is introduced in Chapter 4.
Special functions
: We cover the most commonly encountered special functions. The gamma function is treated in
Appendix C
, and the properties of Bessel functions, Legendre polynomials, and spherical harmonics are covered in
Chapters 6
and
7
. We omit special functions seen only in other courses, e.g. the Laguerre and Hermite polynomials.
Complex analysis
: It's never quite obvious where a chapter on complex analysis should go. We place the theory of analytic functions (
Chapter 8
) before the chapter on Green functions (
Chapter 9
). We cover the standard topics of contour integration and Cauchy's theorem. We develop the
nonstandard
topics of the approximation of integrals (steepest descent and stationary phase) and the analytic signal (Hilbert transform, the Paley–Weiner and Titchmarsh theorems). The latter is important in applications involving signal processing, which many students end up doing in their thesis work, and is natural to include in a discussion on analytic functions.
Green functions
: Inhomogeneous differential equations are naturally solved using the method of Green functions. We illustrate (in
Chapter 9
) the Green function method for the inhomogeneous Helmholtz, diffusion, and wave equations.
Integral equations
: Many key equations occur as integral equations – in scattering theory, for example, either quantum or electromagnetic. We cover integral equations in
Chapter 10
, a topic not always included in books at this level. Yet, it's important for students to understand that the separation‐of‐variables method (introduced earlier in the book) is often not realistic – it relies on the boundaries of systems having rather simple shapes (spherical, cylindrical, etc.). Numerical solutions of PDEs are often based on integral‐equation approaches.
Tensor analysis
: The book ends with an introduction to tensors,
Chapter 11
. Tensors are motivated as a mathematical tool for treating systems featuring anisotropy, and for their fundamental role in establishing covariant equations. We present tensors with sufficient depth so as to provide a foundation for their use in the special theory of relativity. The covariant derivative, developed at the end of the chapter, would be the starting point for more advanced applications.
Brett BordenJames LuscombeMonterey, California
Note
1
A condensed and partial version of these notes was published as [
1
].
1Vectors and linear operators
1.1 The linearity of physical phenomena
Much of the mathematics employed in science and engineering is predicated on the linearity of physical phenomena. As just one among many possible examples, linearity in quantum mechanics means that if two quantum wavefunctions, and , separately satisfy the time‐dependent Schrödinger equation, , then so does any linear combination, where and are constants. Underlying linearity is superposition: The net physical effect, or system response, due to two or more sources (stimuli) is the sum (superposition) of the responses that would have been obtained from each acting individually.
Example.
The electrostatic potential at location , produced by a fixed continuous charge distribution described by charge density function , is obtained as a superposition of the potentials produced by infinitesimal charges ,
where is a constant that depends on the system of units employed.
Not all physical phenomena are linear. Human senses, for example, are not responsive unless the strength of various stimuli exceeds a threshold value. Finding the self‐consistent charge distribution in semiconductor devices is a nonlinear problem; in contrast to the fixed charge distribution in the aforementioned example, charges are able to move and respond to the potentials they generate. The general theory of relativity (the fundamental theory of gravitation) is nonlinear for the same reason – masses respond to the gravitational potential they generate. Nonlinear theories are among the most difficult in physics. Let's learn to walk before we run. There are sufficiently many physical effects having linearity as a common feature that it's not surprising there is a common body of mathematical techniques to treat them – and so, the subject of this chapter. We note that nonlinear effects can be described in terms of linear theories when the strength of their sources is sufficiently small.1
Figure 1.1 Vector addition and scalar multiplication.
1.2 Vector spaces
A basic idea in elementary mechanics is that of vector, quantities having two attributes: direction and magnitude. The position vector (physical displacement relative to an origin) is the prototype vector. Anything called “vector” must share the properties of the prototype. From a mathematical perspective, vectors have two main properties: Vectors can be added to produce new vectors, and vectors can be multiplied by numbers to produce new vectors. These operations are depicted in Figure 1.1: the parallelogram rule of vector addition2; and scalar multiplication, , where is a constant. Many physical quantities have a vector nature (which is why we study vectors): velocity , acceleration , electric field , etc. We'll refer to the vectors used in three‐dimensional position space as elementary vectors.
In more advanced physics (such as quantum mechanics), we encounter quantities that generalize elementary vectors in two ways – as existing in spaces of dimension greater than 3, and/or for which their components are not real valued (complex valued, for example). Such quantities may not have direction and magnitude in the traditional sense, yet they can be combined mathematically in the same way as elementary vectors.
A vector space is a set of mathematical objects (called vectors) having the two main properties of vectors: The sum of vectors is a vector, , and each vector when multiplied by a scalar is a vector, . The scalar is chosen from another set, the field3. If is a real (complex) number, then is referred to as a real (complex) vector space.4 Vector spaces, as collections of objects that combine linearly, provide a common mathematical “platform” to describe diverse physical phenomena.
Definition.
A vector space over a fieldis a set of elementsthat satisfy the following requirements.
To every pair of vectors
and
in
, there corresponds a vector
, called the sum of
and
, such that
addition is commutative,
,
addition is associative,
,
there exists a unique vector
(the additive identity or
null vector
) obeying
for every
, and
to every
, there exists a unique vector
(the
negative vector
) such that
.
To every pair
and
, there corresponds a vector
, called the product of
and
, such that
scalar multiplication is associative,
and
there exists a multiplicative identity 1 such that
for every
.
For scalars
and
,
scalar multiplication is distributive with respect to vector addition
, and
multiplication by vectors is distributive with respect to scalar addition
.
It may be shown as a result of these axioms [2, p. 32] that holds for every , where 0 is the scalar zero.5 The axioms in group (A) prescribe how vectors add, those in (B) prescribe the product of scalars and vectors, and those in (C) prescribe the connection between the additive and multiplicative structures of a vector space.
Examples
The concept of vector space is quite robust. As the following examples show, diverse collections of mathematical objects meet the requirements of a vector space. To check that a given set is indeed a vector space, one should, to be meticulously correct, verify that each of the requirements are met. Often it's enough to check that the sum of two elements in the set remains in the set.
The set of all elementary vectors
,
, etc., comprises a vector space under the parallelogram law of addition and multiplication by numbers.
The set of all
‐tuples of numbers (which could be complex)
is called
‐dimensional Euclidean space, denoted
. To show that
‐tuples form a vector space, the operations of vector addition and scalar multiplication must be specified. These operations are defined
componentwise
, with
and
. For real‐valued scalars,
is denoted
; for complex‐valued scalars,
is denoted
.
The set of all
infinite sequences
of numbers
having the property that
is finite, with addition and scalar multiplication defined componentwise, is a
sequence space
,
. Convergent infinite series do not naturally come to mind as “vectors,” but they satisfy the requirements of a vector space.
6
The set of all continuous functions of a real variable
, with addition and scalar multiplication defined pointwise,
and
, is called a
function space
.
The set of all
square‐integrable functions
of a real variable
x
for which
is finite (for specified limits of integration), with addition and scalar multiplication defined pointwise, is a vector space known as
. (This is the “arena” of quantum mechanics.
7
,
8
)
The set of all polynomials
, in the variable
, of degree at most
, and with real coefficients, is called a
polynomial vector space
.
1.2.1 A word on notation
We've used the abstract symbols , and in defining the axioms of vector spaces. This notation – Dirac notation – is widely used in quantum physics and quantum mechanics and is a prominent reason physical scientists study vector spaces. Dirac notation was invented by (you guessed it) P.A.M. Dirac, one of the founders of quantum mechanics.9 In quantum mechanics, physical states of systems are represented as elements of vector spaces (often denoted ) because it's been found through much experimentation that states of quantum systems combine like elements of vector spaces. As seen in the aforementioned examples, membership in vector spaces applies to a variety of mathematical objects, and hence the use of abstract notation is apt. Dirac could have used to indicate vectors (elements of vector spaces) as applied to quantum states; arrows, however, imply a spatial nature that's not apropos of quantum states. Vectors symbolized as are known as kets. Kets are another word for vectors (physics vocabulary lesson). The word ket is derived from “bracket”; Dirac also introduced another notation, (referred to as bras), that we'll discuss in Section 1.3. A benefit of Dirac notation is that kets indicate vectors without specifying a representation.10 With all this said, Dirac notation is not universally used outside of quantum mechanics.11 In many ways Dirac notation is clumsy, such as in its representation of the norm of a vector. Other notations for vectors, such as bold symbols are also used. To work with vector‐space concepts in solving problems, one must be able to handle a variety of notations for elements of vector spaces. In what follows, we'll interchangeably use different notations for vectors, such as , , or .
1.2.2 Linear independence, bases, and dimensionality
The definition of vector space does not specify the dimension of the space. The key notion for that purpose is linear independence.
Definition.
A set of vectors (elements of a vector space) is linearly independent if for scalars, the linear combination holds only for the trivial case. Otherwise the set is linearly dependent.
Linear independence means that every nontrivial linear combination of vectors is different from zero. Thus, no member of a linearly independent set can be expressed as a linear combination of the other vectors in the set.
Definition.
A vector space is ‐dimensional if it contains linearly independent vectors, but not .
The notation is used to indicate the dimension of . Vector spaces containing an infinite number of linearly independent vectors are said to be infinite dimensional; and are examples. We work for the most part in this chapter with finite‐dimensional vector spaces; we touch on infinite‐dimensional spaces in Section 1.6.
A set of vectors , each an element of the vector space , spans the space if any vector can be expressed as a linear combination, .
Definition.
A set of vectors is a basis if (i) it spans the space and (ii) is linearly independent.
Thus, a vector space is ‐dimensional if (and only if) it has a basis of vectors.12
Definition.
The numbers in the linear combination are the components (or coordinates) ofwith respect to the basis.
Vector components are unique with respect to a given basis. Suppose has a representation in the basis , with . Assume we can find another representation in the same basis, with . Subtract the two formulas for : . Because the basis set is linearly independent, it must be that . If we change the basis, however (a generic task, frequently done), the vector components will change as a result; see Section 1.4.7.
Examples
Elementary vectors confined to a plane are elements of a two‐dimensional vector space. We can use as a basis the unit vectors
and
shown in
Figure 1.2
Any vector
in the plane can be constructed from a linear combination,
;
and
span the space. They're also linearly independent:
can't be expressed in terms of
. Basis vectors, however, don't have to be orthogonal (such as they are in
Figure 1.2
).
Any
two non colinear vectors
and
, such as shown in
Figure 1.3
, can serve as a basis. No linear combination of
and
can sum to zero:
. (Try it!) The only way we can force
with
and
is if
, i.e. if
is not independent of
. We can easily add
three
vectors confined to a plane to produce zero,
, in which case
is linearly dependent on
and
. The maximum number of linearly independent vectors restricted to a plane is two, and hence, the set of all vectors confined to a plane is a two‐dimensional vector space. Note that while there are an unlimited number of possible vectors confined to a plane, a plane is a two‐dimensional space.
Figure 1.2 Vectors and span the space of vectors confined to a plane.
Figure 1.3 Vectors and are a basis for the space of vectors confined to a plane.
For the space
of infinite sequences
such that
is finite, a basis (called the
standard basis
) is
,
,
with 1 in the
th slot. Each
belongs to the space (the sum of the squares of its elements is finite). The vectors
span the space,
, and they are linearly independent. Thus,
is infinite dimensional: It has an infinite number of basis vectors.
For the space of
th‐degree polynomials in
,
, a basis is
,
,
,
. The vectors
span the space,
, and they're linearly independent.
13
The space is therefore
‐dimensional.
The set of all functions
that satisfy the differential equation
is a two‐dimensional function space: Any solution
can be expressed as a linear combination of basis functions (vectors)
and
so that
.
1.2.3 Subspaces
Definition.
A subset of a vector space is called a subspace of if is a vector space under the same rules of addition and scalar multiplication introduced in .14
Examples
The zero vector
of
is a subspace of
.
The whole space
can be considered a subspace of
if a set can be considered a subset of itself. It's easier to allow the whole space to be a subspace (especially when we come to infinite‐dimensional spaces).
For
the three‐dimensional position space, any plane in
passing through the origin is a subspace of
.
For the space of
‐tuples, for any vector
, all vectors obtained by setting
form a subspace.
1.2.4 Isomorphism of ‐dimensional spaces
Two vector spaces over the same field and having the same dimension are instances of the “same” vector space. Consider elementary vectors in three‐dimensional space. Once a basis , , and has been chosen, so that , there is a correspondence between and the three‐tuple , and this correspondence exists between every element of each set. This “sameness” of vector spaces is called an isomorphism.
Definition.
Two vector spaces and are isomorphic if it's possible to establish a one‐to‐one correspondence between and such that if and ,
the vector which this correspondence associates with
is
, and
the vector which this correspondence associates with
is
.
All vector spaces of dimension are isomorphic. Consider that for isomorphic spaces and , if are in and are their counterparts in , the equation corresponds to . Thus the counterparts in of linearly independent vectors in are also linearly independent. The maximum number of linearly independent vectors in is therefore the same as that in , i.e. the dimensions of and are the same.
1.2.5 Dual spaces
Definition.
Linear scalar‐valued functions of vectors are called linear functionals.15
Linear functionals assign a scalar to each vector such that for any vectors and and scalars and , the property of linearity holds:
Let be a basis for . Any vector can be expressed as a linear combination, . The action of a linear functional is (by definition)
where . The value of acting on is determined by the values of acting on the basis vectors for . Because bases are not unique, one would seemingly have reason to worry that the value of is basis dependent. Such is not the case, however, but to prove that here would be getting ahead of ourselves.16
As a consequence of linearity, consider that
Thus, maps the zero vector onto the scalar zero. This property rules out adding a constant to Eq. (1.1), as in ; implies .
Linear functionals form a vector space in their own right, called the dual space, often denoted17, which is closely connected with . The set of all linear functionals forms a vector space because they have the properties of a linear space:
(
,
.
Definition.
For an ‐dimensional vector space, the dual space is the vector space whose elements are linear functionals on . Addition and scalar multiplication in follow from the rules of addition and scalar multiplication in.
Relative to a given basis in , every linear functional is uniquely determined by the ‐tuple , where . This correspondence is preserved under vector addition and multiplication of vectors by scalars; it follows that is isomorphic to . The dual space associated with is ‐dimensional.
1.3 Inner products and orthogonality
Thus, vectors – elements of vector spaces – generalize elementary vectors. Vectors in ‐dimensional spaces generalize the notion of direction. Direction is a concept associated with our experience in three‐dimensional space, and is specified by the components of a vector relative to a given set of basis vectors. We can't intuitively speak of direction in spaces of arbitrary dimension, but we can continue to speak of the components of vectors relative to bases in spaces of any dimension. What about the other half of the maxim that vectors have direction and magnitude? We've so far ignored the notion of the length of a vector. We address that point now.
1.3.1 Inner products
Can vectors be multiplied by vectors? Nothing in the definition of vector space tells us how to do that. One way is with the inner product,18 a rule that associates a scalar to a pair of vectors. Vector spaces equipped with the additional property of an inner product are known as inner product spaces.19
Definition.
An inner product for vector space over field associates a scalar, denoted , with every pair of vectors (for ), such that it satisfies:
,
,
,
with equality holding for
,(positive definite)
where () denotes complex conjugation.20,21 Requirements (i) and (ii) imply that .
Definition.
The norm (or length) of a vector , denoted , is the square root of the inner product of a vector with itself, .
Examples
The
dot product
of elementary vector analysis satisfies the definition of inner product,
, where
is the angle between the vectors when their “tails” are joined together (see
Figure 1.4
) and
denotes the magnitude of
.
For
, the space of
‐tuples, let
and
. The operation
satisfies the requirements of an inner product.
For the space
of infinite sequences, if
and
, then
satisfies the requirements for an inner product. The condition for
, that
is finite, is the requirement that the norm
is finite.
For the space
of square‐integrable functions
and
, the following rule meets the requirements of an inner product:
Figure 1.4 Dot product of elementary vectors, .
The condition for , that is finite, is again the requirement that the norm is finite. That is finite (for all ) implies that the inner product of two vectors is finite.22 Note that the complex conjugate in Eq. (1.3) ensures a positive result for .
Dirac notation has been used in Eq. (1.3) to indicate the inner product, instead of . The notation arises from the fact that a linear functional can be defined for each23 by the requirement that for every vector . Linear functionals are written in Dirac notation simply as (termed bras), so that ; operates on to produce . For the inner product of (Eq. (1.3)), the functional is the operator , where is a placeholder for the vector , so that .
1.3.2 The Schwarz inequality
From the dot product, the angle between elementary vectors can be inferred from the relation . For to be calculated in this way, it must be the case that for all and . This inequality finds its generalization for any inner product in the Schwarz inequality:
where equality holds when and are linearly dependent.
To prove the Schwarz inequality, introduce , where is any complex number. Because the inner product is positive definite, we must have , or
The inequality (1.5) is valid for all , and thus it remains valid for any particular . The Schwarz inequality follows under the substitution in (1.5).24
1.3.3 Vector norms
The norm has the following properties, for any vectors and , and for any scalar ,
, with equality holding only for
,
, and
. (triangle inequality)
Properties (i) and (ii) follow by definition. Property (iii) can be derived using the Schwarz inequality (see Exercise 1.7): If , , and are envisioned as forming the sides of a triangle, the length of the side is less than the sum of the lengths of the other two sides (hence, the “triangle inequality”).
1.3.4 Orthonormal bases and the Gram–Schmidt process
With vector spaces endowed with an inner product, we can define orthogonality.
Definition.
Vectors , in an inner‐product space are orthogonal if.
Elementary vectors are orthogonal when the angle between them is 90; when . The definition of orthogonality generalizes to any inner product, .
Any set of linearly independent vectors that spans a space is a basis for that space. Nothing requires basis vectors to be orthogonal. Bases comprised of orthogonal vectors, however, have many convenient properties.25 In this section, we show that an orthonormal basis (orthogonal vectors having unit norm) can always be constructed from an arbitrary basis. No loss of generality is implied therefore by working with orthonormal bases.
Definition.
A set of vectors is called an orthonormal set if
where is the Kronecker delta symbol.
Example.
The basis for , is an orthonormal set.
Let be an orthonormal basis for . We can express any vector as the linear combination . The components are easily found when the basis is orthonormal because
In a general basis, a given set of scalars determines a vector, in an orthonormal basis, the coefficients are determined as in Eq. (1.6), . In this case, is simply the projection of along the “direction” . From Eq. (1.6), any vector can be expressed as a linear combination of orthonormal basis vectors in the form
Moreover, in terms of this orthonormal basis, with and ,
This result is sometimes used to define the inner product (but it requires that and have a known expansion in terms of an orthonormal basis). Equation (1.8)
