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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
The book assumes next to no prior knowledge of the topic. The first part introduces the core mathematics, always in conjunction with the physical context. In the second part of the book, a series of examples showcases some of the more conceptually advanced areas of physics, the presentation of which draws on the developments in the first part. A large number of problems helps students to hone their skills in using the presented mathematical methods. Solutions to the problems are available to instructors on an associated password-protected website for lecturers.
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Veröffentlichungsjahr: 2016
Ähnliche
Table of Contents
Cover
Related Titles
Title Page
Copyright
Dedication
Preface
Part I: Mathematics
Chapter 1: Functions of One Variable
1.1 Limits
1.2 Elementary Calculus
1.3 Integration
1.4 The Binomial Expansion
1.5 Taylor's Series
1.6 Extrema
1.7 Power Series
1.8 Basic Functions
1.9 First-Order Ordinary Differential Equations
1.10 Trigonometric Functions
Problems
Chapter 2: Complex Numbers
2.1 Exponential Function of a Complex Variable
2.2 Argand Diagrams and the Complex Plane
2.3 Complex Logarithm
2.4 Hyperbolic Functions
2.5 The Simple Harmonic Oscillator
Problems
Chapter 3: Vectors in
3.1 Basic Operation
3.2 Kinematics in Three Dimensions
3.3 Coordinate Systems
3.4 Central Forces
3.5 Rotating Frames
Problems
Chapter 4: Vector Spaces
4.1 Formal Definition of a Vector Space
4.2 Fourier Series
4.3 Linear Operators
4.4 Change of Basis
Problems
Chapter 5: Functions of Several Variables
5.1 Partial Derivatives
5.2 Extrema under Constraint
5.3 Multiple Integrals
Problems
Chapter 6: Vector Fields and Operators
6.1 The Gradient Operator
6.2 Work and Energy in Vectorial Mechanics
6.3 A Little Fluid Dynamics
6.4 Surface Integrals
6.5 The Divergence Theorem
6.6 Stokes' Theorem
Problems
Chapter 7: Generalized Functions
7.1 The Dirac Delta Function
7.2 Green's Functions
7.3 Delta Function in Three Dimensions
Problems
Chapter 8: Functions of a Complex Variable
8.1 Limits
8.2 Power Series
8.3 Fluids Again
8.4 Complex Integration
Problems
Part II: Physics
Chapter 9: Maxwell's Equations: A Very Short Introduction
9.1 Electrostatics: Gauss's Law
9.2 The No Magnetic Monopole Rule
9.3 Current
9.4 Faraday's Law
9.5 Ampère's Law
9.6 The Wave Equation
9.7 Gauge Conditions
Problems
Chapter 10: Special Relativity: Four-Vector Formalism
10.1 Lorentz Transformation
10.2 Minkowski Space
10.3 Four-Velocity
10.4 Electrodynamics
10.5 Transformation of the Electromagnetic Fields
Problems
Chapter 11: Quantum Theory
11.1 Bohr Atom
11.2 The de Broglie Hypothesis
11.3 The Schrödinger Wave Equation [13]
11.4 Interpretation of the Wave function
11.5 Atom
11.6 Formalism
11.7 Probabilistic Interpretation
11.8 Time Evolution
11.9 The Stern–Gerlach Experiment
Problems
Chapter 12: An Informal Treatment of Variational Principles and their History
12.1 Sin and Death
12.2 The Calculus of Variations
12.3 Constrained Variations
12.4 Hamilton's Equations
12.5 Phase Space
12.6 Fixed Points
Problems
Appendix A: Conic Sections
A.1 Polar Coordinates
A.2 Intersection of a Cone and a Plane
Appendix B: Vector Relations
B.1 Products
B.2 Differential Operator Relations
B.3 Coordinates
Bibliography
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Part I: Mathematics
Begin Reading
List of Illustrations
Chapter 1: Functions of One Variable
Figure 1.1 If is a continuous function on if we pick any value, , that is between the value of and the value of and draw a line straight out from this point, the line will hit the graph in at least one point with an value between and .
Figure 1.2 Over the range shown, the function is invertible, is not.
Figure 1.3 (a) Approximating the value of the integral as , (b) picking a point where and approximating integral as , and (c) picking another point where and approximating integral as .
Figure 1.4 Plot of . , solid line; , dashed line.
Figure 1.5 Plot of .
Chapter 2: Complex Numbers
Figure 2.1 Argand diagram for .
Figure 2.2 Plots of the hyperbolic functions and .
Figure 2.3 Example of a potential showing points of stable and unstable equilibrium.
Figure 2.4 The simple pendulum consists of a heavy weight attached to a fixed point by a massless string. The string is of length . It is displaced from equilibrium through some small angle and allowed to oscillate.
Figure 2.5 The magnitude of and plotted for pendulum released from rest with for a pendulum of length m resulting in an of 1/s.
Figure 2.6 Example of underdamped motion. The long dashed curve shows the oscillation, with angular frequency the dotted curves envelopes , the solid curve show the resultant motion (Eq. (2.68)).
Chapter 3: Vectors in ℝ3
Figure 3.1 (a) To add vector to vector , place the tail of at the head of . The sum, , is a vector from the tail of to the head of . (b) To subtract from vector either add and or place the head of to the head of . extends from the tail of to the tail of .
Figure 3.2 Vectors define a plane the component of along the direction of is and perpendicular to it is
Figure 3.3 The vector product , it is a vector of magnitude perpendicular to the plane containing and with sense as shown. It is the direction with which a right hand screw would turn as we rotate from
to
. Alternatively the sense of the vector product can be visualized with the right-hand rule. If you curl the fingers of your right hand so that they follow a rotation from vector to vector , then the thumb will point in the direction of the vector product.
Figure 3.4 Left- and right-handed Cartesian coordinates: If we fix our and axes as shown, then in a right-handed system the axis will point out of the paper and in a left-handed one it will point in.
Figure 3.5 is the volume of the parallelepiped spanned by , and .
Figure 3.6 (a) The line through , parallel to , is given by (b). The plane defined by .
Figure 3.7 Solutions to Eq. (3.42) for and different values of .
Figure 3.8 A rigid body rotating about a stationary axis through 0. This axis is perpendicular to the plane of the diagram. The reference point moves through a circle. (Reproduced by kind permission of Edward Whelan)
Chapter 4: Vector Spaces
Figure 4.1 The function of Example 4.1 together with the Fourier series of Eq. (4.9) truncate to (a) , (b) , and (c) terms.
Figure 4.2 The operator acts on the vector and rotates it through to turn it into the vector . Note .
Chapter 5: Functions of Several Variables
Figure 5.1 Comparison of the analytic solution to Eq. (1.63), solid line, compared with various fourth-order Runge–Kutta calculations defined by the step size as given in (5.42): —crosses; —half-filled squares.
Figure 5.2 Comparison of the analytic solution to Eq. (5.49), represented as solid line, compared with the fourth-order Runge–Kutta calculation, , represented as crosses.
Figure 5.3 We note that in case (a) of Example 5.7 we have an ellipse while in case (b) we have a hyperbola. In the second case, we remain on the positive branch of the hyperbola since it is assumed that we start on this branch.
Figure 5.4 The region is the projection of on the – plane. Reproduced with kind permission from Edward Whelan.
Figure 5.5 The region is divided into subrectangles. Assume that there are intervals in the direction and intervals in the direction. Reproduced with kind permission from Edward Whelan.
Figure 5.6 The region of integration for Example 5.9.
Figure 5.7 The area of integration is divided up into grids using the new coordinates, where and are constants.
Figure 5.8 is an infinitesimal area bounded by the curves .
Figure 5.9 When evaluating double integrals in cartesian coordinates the element of area is , in polar it is .
Figure 5.10 The -axis coincides with the axis of the cone. (Reproduced with kind permission of Edward Whelan)
Chapter 6: Vector Fields and Operators
Figure 6.1 (a) Cylindrical polar coordinates, . (b) Spherical polar coordinates . In both cases, the associated unit vectors in the direction of increase of the corresponding coordinates are also shown.
Figure 6.2 An infinitesimal element of the curve has a length .
Figure 6.3 An infinitesimal parallelepiped through which a fluid flows. (Reproduced with kind permission of Edward Whelan)
Figure 6.4 Circulation around an infinitesimal loop.
Figure 6.5 For a rectangle, we can always choose the orientation of our axis so that the long side of the rectangle lies along the -axis and the short side along the so that the entire rectangle lies in the – plane.
Figure 6.6 The direction of the outward drawn normal and hence, is fixed by the way the boundary is traversed by a right-hand rule.
Figure 6.7 The integral over the volume is divided into a large number of rectangular parallelepipeds. The net flux through interior faces will exactly cancel. (Reproduced with kind permission of Edward Whelan)
Figure 6.8 (a) An arbitrary-shaped loop in two dimensions can be reduced to a series of tiny rectangular loops. (b) The circulation from adjacent rectangles will cancel along their common boundary.
Figure 6.9 Examples of possible surfaces that have the same boundary curve . The upper and lower hemispheres and the disk . (Reproduced by kind permission of Edward Whelan)
Chapter 8: Functions of a Complex Variable
Figure 8.1 The exists and is equal to if when given an arbitrarily small real positive number , we can find another real positive number such that every complex number in the open circle in the – plane centered on with radius is mapped into the open circle in the – plane, centered on with radius .
Figure 8.2 is analytic on and in . A cut is made between and along the line connecting and , two cuts are shown to make the picture clear but actually they coincide.
Figure 8.3 is the semicircular arc radius , centered on the origin, and in the upper half plane ().The closed contour consists of the real line and the arc.
Figure 8.4 Contour for the evaluation of the integral .
Figure 8.5 The contour for Example 8.6.
Chapter 9: Maxwell's Equations: A Very Short Introduction
Figure 9.1 (a) A positive charge acts like a source and the net flux through the surface is positive. (b) A negative charge acts like a sink and the net flux is negative. (c) If there is no net charge within the surface, then there is no net flux. (Figure reproduced by kind permission of Edward Whelan)
Figure 9.2 Field lines for a negative and a positive charges, the electric force at any point is target to the field.
Figure 9.3 In Example 9.2, we consider a uniformly charged spherical particle and we consider as Gaussian surfaces two concentric spherical surfaces: one smaller than the shell and thus containing no charge and one bigger containing a charge of .
Figure 9.4 If a piece of perfect conducting material is placed in an electric field, the charges will distribute themselves until there is no net field within the conductor. (Figure reproduced with kind permission of Edward Whelan)
Figure 9.5 Magnetic field lines for a bar magnet. Note the similarity to the electric field in Figure 9.2.
Figure 9.6 A simple electric circuit in which the emf, (a battery) maintains a steady current, , through a resistor.
Figure 9.7 A rectangular loop is moved with a velocity through a region of thickness , where a constant magnetic field is maintained. The position of the loop is measured by the distance between the left edge of the magnetic field and the right hand of the loop.
Figure 9.8 We have a cavity of such unspecified shape the net charged carried by the conductor is and in addition there is a charge of within the cavity. (Figure reproduced with kind permission of Edward Whelan)
Chapter 10: Special Relativity: Four-Vector Formalism
Figure 10.1 Two inertial frames and in standard configuration. The frame is moving with speed along the -axis of .
Figure 10.2 The surface of the light cone corresponds to those events that can only be connected to the present by signals traveling at the speed of light. Events within the light cone can be connected by a signal with a speed less than “c.” Points outside the light cone can have no causal connection with the present.
Chapter 11: Quantum Theory
Figure 11.1 , solid line; , short dashed line; for , dashed dotted; , dotted; , long dashed.
Figure 11.2 (a) The wavefunction in Eq. (11.48) and its square for and : , solidline; , dashed line. (b) The wavefunction in Eq. (11.52) with and : , solid line; , dashed line.
Figure 11.3 Graph of for showing allowed bands (shaded areas) separated by gaps.
Figure 11.4 Graph of for , dashed line; , solid line.
Figure 11.5 Schematic diagram of Stern–Gerlach apparatus, reproduced with kind permission of Edward Whelan.
Appendix A: Conic Sections
Figure A.1 Plotted in the – plane. (a) The ellipse ; (b) both branches of the hyperbola , also shown are the lines ; (c) the parabola .
Figure A.2 Intersection curves for a plane and a right circular cone. Reproduced with kind permission of Edward Whelan.
List of Tables
Chapter 11: Quantum Theory
Table 11.1 Table (spin components)
Table 11.2 Table (spin components)
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Colm T. Whelan
A first Course in Mathematical Physics
Author
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Old Dominion University
Department of Physics
4600 Elkhorn Avenue
VA
United States
Cover
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For Agnieszka, of course
Preface
This book is an elementary introduction to the mathematics needed for students taking undergraduate classes in the physical sciences. My ambition is to present in a simple and easily intelligible way the core material they will need for their courses and help them to uncover the character of the physical laws, which can sometimes, unfortunately, be obscured by a lack of understanding of and sympathy with the precise mathematical language in which, perforce, they have to be expressed. I have emphasized the direct connection between the conceptual basis of the physics the students are about to learn and the mathematics they are studying. The first part of the book introduces the core mathematics and while I have included numerous applications in this section I felt that a fuller understanding could only be achieved if the physical context was given a more complete introduction. In the second part of the book, I have given a series of brief overviews of some of the more beautiful and conceptually stimulating areas of physics.
The material in this book is designed for a one- or two-semester course to be ideally taken at the start of the sophomore year. I assume the readers have no previous knowledge. The first two chapters give a brief overview of some of the more elementary mathematics one might hope that students bring with them from high school and their first year. The choice of material included in the course involved some painful pruning. I am all too aware of the numerous shortcuts I have taken, the many important interesting and beautiful diversions I have ignored, and the rigor I have forsaken all with one principle object in view to equip the students with a basic mathematical toolkit that will allow them to enjoy and profit from their higher level physics courses. I rather hope that the student will be sufficiently intrigued by the exposure to the interesting topics covered in this text to seek out some other more formal and rigorous specialist mathematics courses. A large number of problems have been included, for none of which is the use of a calculator needed. These form an integral part of the book and students are strongly encouraged to attempt all of them.
C T WNorfolk, May, 28, 2015
Part I
Mathematics
Chapter 1Functions of One Variable
1.1 Limits
It is often said that most mathematical errors, which get published, follow the word “clearly” and involve the improper interchange of two limits. In simple terms, a “limit” is the number that a function or sequence “approaches” as the input or index approaches some value. For example, we will say that the sequence approaches the limit 0 as moves to infinity Or, in other words, we can make arbitrarily small by choosing big enough. We often write this as
We can also take the limit of a function, for example, if then
A sequence of numbers is said to converge to a limit if we can make the difference arbitrarily small by making big enough. If such a limit point does not exist, then we state that the sequence diverges. For example, the sequence of integers
is unbounded as , while the sequence
oscillates and never settles down to a limit. More formally, we state
Definition 1.1
Let f be a function defined on a real interval then the limit as exists if there exists a number such that given a number no matter how small, we can find a number , where for all satisfying
we have
Notice that we do not necessarily let ever reach but only get infinitesimally close to it. If in fact , then we state that the function is continuous. Intuitively, a function that is continuous on some interval will take on all values between
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