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E-raamat: Mathematics for Physicists and Engineers: Fundamentals and Interactive Study Guide

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 27-Jun-2014
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783642541247
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Mathematics for Physicists and Engineers: Fundamentals and Interactive Study GuideSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 27-Jun-2014
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783642541247
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This textbook offers an accessible and highly approved approach which is characterized by the combination of the textbook with a detailed study guide available online at our repository extras.springer.com. This study guide divides the whole learning task into small units which the student is very likely to master successfully. Thus he or she is asked to read and study a limited section of the textbook and to return to the study guide afterwards. Working with the study guide his or her learning results are controlled, monitored and deepened by graded questions, exercises, repetitions and finally by problems and applications of the content studied. Since the degree of difficulties is slowly rising the students gain confidence and experience their own progress in mathematical competence thus fostering motivation. Furthermore in case of learning difficulties he or she is given supplementary explanations and in case of individual needs supplementary exercises and applications. So the sequence of the studies is individualized according to the individual performance and needs and can be regarded as full tutorial course. The study guide satisfies two objectives simultaneously: firstly it enables students to make effective use of the textbook and secondly it offers advice on the improvement of study skills. Empirical studies have shown that the student's competence for using written information has improved significantly by using this study guide.

The new edition includes a new chapter on Fourier integrals and Fourier transforms, numerous sections had been updated, 30 new problems with solutions had been added. The interactive study guide has seen a substantial update.



In its new edition, this book offers an all-new chapter on Fourier integrals and Fourier transforms, plus numerous updates to existing material, 30 new problems with solutions and a substantial update to the contents of the online interactive study guide.

Arvustused

From the book reviews:

In this mathematics textbook, youll find possibly all the mathematics youd need for first years of an undergraduate physics course - algebra, geometry, calculus and statistics. With the book there is also an online study guide designed to help the student advance quite quickly. If taken alone, the textbook is as any other mathematics textbook for physicists, together with the study guide it is an excellent combination, especially if you prefer to study alone . (Kadri Tinn, AstroMadness.com, February, 2015)

Preface and Introduction vii
1 Vector Algebra I: Scalars and Vectors
1(22)
1.1 Scalars and Vectors
1(3)
1.2 Addition of Vectors
4(2)
1.2.1 Sum of Two Vectors: Geometrical Addition
4(2)
1.3 Subtraction of Vectors
6(1)
1.4 Components and Projection of a Vector
7(2)
1.5 Component Representation in Coordinate Systems
9(5)
1.5.1 Position Vector
9(1)
1.5.2 Unit Vectors
10(1)
1.5.3 Component Representation of a Vector
11(1)
1.5.4 Representation of the Sum of Two Vectors in Terms of Their Components
12(1)
1.5.5 Subtraction of Vectors in Terms of their Components
13(1)
1.6 Multiplication of a Vector by a Scalar
14(1)
1.7 Magnitude of a Vector
15(8)
2 Vector Algebra II: Scalar and Vector Products
23(16)
2.1 Scalar Product
23(7)
2.1.1 Application: Equation of a Line and a Plane
26(1)
2.1.2 Special Cases
26(1)
2.1.3' Commutative and Distributive Laws
27(1)
2.1.4 Scalar Product in Terms of the Components of the Vectors
27(3)
2.2 Vector Product
30(9)
2.2.1 Torque
30(1)
2.2.2 Torque as a Vector
31(1)
2.2.3 Definition of the Vector Product
32(1)
2.2.4 Special Cases
33(1)
2.2.5 Anti-Commutative Law for Vector Products
33(1)
2.2.6 Components of the Vector Product
34(5)
3 Functions
39(32)
3.1 The Mathematical Concept of Functions and its Meaning in Physics and Engineering
39(3)
3.1.1 Introduction
39(1)
3.1.2 The Concept of a Function
40(2)
3.2 Graphical Representation of Functions
42(5)
3.2.1 Coordinate System, Position Vector
42(1)
3.2.2 The Linear Function: The Straight Line
43(1)
3.2.3 Graph Plotting
44(3)
3.3 Quadratic Equations
47(2)
3.4 Parametric Changes of Functions and Their Graphs
49(1)
3.5 Inverse Functions
50(2)
3.6 Trigonometric or Circular Functions
52(12)
3.6.1 Unit Circle
52(1)
3.6.2 Sine Function
53(5)
3.6.3 Cosine Function
58(1)
3.6.4 Relationships Between the Sine and Cosine Functions
59(2)
3.6.5 Tangent and Cotangent
61(1)
3.6.6 Addition Formulae
62(2)
3.7 Inverse Trigonometric Functions
64(2)
3.8 Function of a Function (Composition)
66(5)
4 Exponential, Logarithmic and Hyperbolic Functions
71(16)
4.1 Powers, Exponential Function
71(5)
4.1.1 Powers
71(1)
4.1.2 Laws of Indices or Exponents
72(1)
4.1.3 Binomial Theorem
73(1)
4.1.4 Exponential Function
73(3)
4.2 Logarithm, Logarithmic Function
76(4)
4.2.1 Logarithm
76(2)
4.2.2 Operations with Logarithms
78(1)
4.2.3 Logarithmic Functions
79(1)
4.3 Hyperbolic Functions and Inverse Hyperbolic Functions
80(7)
4.3.1 Hyperbolic Functions
80(3)
4.3.2 Inverse Hyperbolic Functions
83(4)
5 Differential Calculus
87(60)
5.1 Sequences and Limits
87(6)
5.1.1 The Concept of Sequence
87(1)
5.1.2 Limit of a Sequence
88(3)
5.1.3 Limit of a Function
91(1)
5.1.4 Examples for the Practical Determination of Limits
91(2)
5.2 Continuity
93(1)
5.3 Series
94(2)
5.3.1 Geometric Series
95(1)
5.4 Differentiation of a Function
96(6)
5.4.1 Gradient or Slope of a Line
96(1)
5.4.2 Gradient of an Arbitrary Curve
97(2)
5.4.3 Derivative of a Function
99(1)
5.4.4 Physical Application: Velocity
100(1)
5.4.5 The Differential
101(1)
5.5 Calculating Differential Coefficients
102(12)
5.5.1 Derivatives of Power Functions; Constant Factors
103(1)
5.5.2 Rules for Differentiation
104(4)
5.5.3 Differentiation of Fundamental Functions
108(6)
5.6 Higher Derivatives
114(1)
5.7 Extreme Values and Points of Inflexion; Curve Sketching
115(8)
5.7.1 Maximum and Minimum Values of a Function
115(4)
5.7.2 Further Remarks on Points of Inflexion (Contraflexure)
119(1)
5.7.3 Curve Sketching
120(3)
5.8 Applications of Differential Calculus
123(6)
5.8.1 Extreme Values
123(1)
5.8.2 Increments
124(1)
5.8.3 Curvature
125(2)
5.8.4 Determination of Limits by Differentiation: L'Hopital's Rule
127(2)
5.9 Further Methods for Calculating Differential Coefficients
129(2)
5.9.1 Implicit Functions and their Derivatives
129(1)
5.9.2 Logarithmic Differentiation
130(1)
5.10 Parametric Functions and their Derivatives
131(16)
5.10.1 Parametric Form of an Equation
131(5)
5.10.2 Derivatives of Parametric Functions
136(11)
6 Integral Calculus
147(46)
6.1 The Primitive Function
147(2)
6.1.1 Fundamental Problem of Integral Calculus
147(2)
6.2 The Area Problem: The Definite Integral
149(2)
6.3 Fundamental Theorem of the Differential and Integral Calculus
151(4)
6.4 The Definite Integral
155(6)
6.4.1 Calculation of Definite Integrals from Indefinite Integrals
155(3)
6.4.2 Examples of Definite Integrals
158(3)
6.5 Methods of Integration
161(16)
6.5.1 Principle of Verification
161(1)
6.5.2 Standard Integrals
161(1)
6.5.3 Constant Factor and the Sum of Functions
162(1)
6.5.4 Integration by Parts: Product of Two Functions
163(3)
6.5.5 Integration by Substitution
166(2)
6.5.6 Substitution in Particular Cases
168(4)
6.5.7 Integration by Partial Fractions
172(5)
6.6 Rules for Solving Definite Integrals
177(3)
6.7 Mean Value Theorem
180(1)
6.8 Improper Integrals
181(2)
6.9 Line Integrals
183(10)
7 Applications of Integration
193(36)
7.1 Areas
193(7)
7.1.1 Areas for Parametric Functions
196(1)
7.1.2 Areas in Polar Coordinates
197(2)
7.1.3 Areas of Closed Curves
199(1)
7.2 Lengths of Curves
200(4)
7.2.1 Lengths of Curves in Polar Coordinates
203(1)
7.3 Surface Area and Volume of a Solid of Revolution
204(6)
7.4 Applications to Mechanics
210(19)
7.4.1 Basic Concepts of Mechanics
210(1)
7.4.2 Center of Mass and Centroid
210(3)
7.4.3 The Theorems of Pappus
213(2)
7.4.4 Moments of Inertia; Second Moment of Area
215(14)
8 Taylor Series and Power Series
229(20)
8.1 Introduction
229(1)
8.2 Expansion of a Function in a Power Series
230(4)
8.3 Interval of Convergence of Power Series
234(1)
8.4 Approximate Values of Functions
235(2)
8.5 Expansion of a Function ƒ (x) at an Arbitrary Position
237(2)
8.6 Applications of Series
239(10)
8.6.1 Polynomials as Approximations
239(3)
8.6.2 Integration of Functions when Expressed as Power Series
242(2)
8.6.3 Expansion in a Series by Integrating
244(5)
9 Complex Numbers
249(26)
9.1 Definition and Properties of Complex Numbers
249(3)
9.1.1 Imaginary Numbers
249(1)
9.1.2 Complex Numbers
250(1)
9.1.3 Fields of Application
250(1)
9.1.4 Operations with Complex Numbers
251(1)
9.2 Graphical Representation of Complex Numbers
252(4)
9.2.1 Gauss Complex Number Plane: Argand Diagram
252(1)
9.2.2 Polar Form of a Complex Number
253(3)
9.3 Exponential Form of Complex Numbers
256(7)
9.3.1 Euler's Formula
256(1)
9.3.2 Exponential Form of the Sine and Cosine Functions
257(1)
9.3.3 Complex Numbers as Powers
257(3)
9.3.4 Multiplication and Division in Exponential Form
260(1)
9.3.5 Raising to a Power, Exponential Form
261(1)
9.3.6 Periodicity of rejα
261(1)
9.3.7 Transformation of a Complex Number From One Form into Another
262(1)
9.4 Operations with Complex Numbers Expressed in Polar Form
263(12)
9.4.1 Multiplication and Division
263(2)
9.4.2 Raising to a Power
265(1)
9.4.3 Roots of a Complex Number
265(10)
10 Differential Equations
275(48)
10.1 Concept and Classification of Differential Equations
275(4)
10.2 Preliminary Remarks
279(2)
10.3 General Solution of First- and Second-Order DEs with Constant Coefficients
281(12)
10.3.1 Homogeneous Linear DE
281(6)
10.3.2 Non-Homogeneous Linear DE
287(6)
10.4 Boundary Value Problems
293(2)
10.4.1 First-Order DEs
293(1)
10.4.2 Second-Order DEs
293(2)
10.5 Some Applications of DEs
295(9)
10.5.1 Radioactive Decay
295(1)
10.5.2 The Harmonic Oscillator
296(8)
10.6 General Linear First-Order DEs
304(4)
10.6.1 Solution by Variation of the Constant
304(2)
10.6.2 A Straightforward Method Involving the Integrating Factor
306(2)
10.7 Some Remarks on General First-Order DEs
308(7)
10.7.1 Bernoulli's Equations
308(1)
10.7.2 Separation of Variables
309(1)
10.7.3 Exact Equations
310(3)
10.7.4 The Integrating Factor -- General Case
313(2)
10.8 Simultaneous DEs
315(4)
10.9 Higher-Order DEs Interpreted as Systems of First-Order Simultaneous DEs
319(1)
10.10 Some Advice on Intractable DEs
319(4)
11 Laplace Transforms
323(16)
11.1 Introduction
323(1)
11.2 The Laplace Transform Definition
323(1)
11.3 Laplace Transform of Standard Functions
324(6)
11.4 Solution of Linear DEs with Constant Coefficients
330(2)
11.5 Solution of Simultaneous DEs with Constant Coefficients
332(7)
12 Functions of Several Variables; Partial Differentiation; and Total Differentiation
339(40)
12.1 Introduction
339(1)
12.2 Functions of Several Variables
340(6)
12.2.1 Representing the Surface by Establishing a Table of Z-Values
341(1)
12.2.2 Representing the Surface by Establishing Intersecting Curves
342(3)
12.2.3 Obtaining a Functional Expression for a Given Surface
345(1)
12.3 Partial Differentiation
346(6)
12.3.1 Higher Partial Derivatives
350(2)
12.4 Total Differential
352(8)
12.4.1 Total Differential of Functions
352(4)
12.4.2 Application: Small Tolerances
356(2)
12.4.3 Gradient
358(2)
12.5 Total Derivative
360(3)
12.5.1 Explicit Functions
360(2)
12.5.2 Implicit Functions
362(1)
12.6 Maxima and Minima of Functions of Two or More Variables
363(6)
12.7 Applications: Wave Function and Wave Equation
369(10)
12.7.1 Wave Function
369(4)
12.7.2 Wave Equation
373(6)
13 Multiple Integrals; Coordinate Systems
379(24)
13.1 Multiple Integrals
379(2)
13.2 Multiple Integrals with Constant Limits
381(3)
13.2.1 Decomposition of a Multiple Integral into a Product of Integrals
383(1)
13.3 Multiple Integrals with Variable Limits
384(4)
13.4 Coordinate Systems
388(9)
13.4.1 Polar Coordinates
389(2)
13.4.2 Cylindrical Coordinates
391(2)
13.4.3 Spherical Coordinates
393(4)
13.5 Application: Moments of Inertia of a Solid
397(6)
14 Transformation of Coordinates; Matrices
403(28)
14.1 Introduction
403(3)
14.2 Parallel Shift of Coordinates: Translation
406(3)
14.3 Rotation
409(6)
14.3.1 Rotation in a Plane
409(3)
14.3.2 Successive Rotations
412(1)
14.3.3 Rotations in Three-Dimensional Space
413(2)
14.4 Matrix Algebra
415(6)
14.4.1 Addition and Subtraction of Matrices
417(1)
14.4.2 Multiplication of a Matrix by a Scalar
418(1)
14.4.3 Product of a Matrix and a Vector
418(1)
14.4.4 Multiplication of Two Matrices
419(2)
14.5 Rotations Expressed in Matrix Form
421(2)
14.5.1 Rotation in Two-Dimensional Space
421(1)
14.5.2 Special Rotation in Three-Dimensional Space
422(1)
14.6 Special Matrices
423(3)
14.7 Inverse Matrix
426(5)
15 Sets of Linear Equations; Determinants
431(22)
15.1 Introduction
431(1)
15.2 Sets of Linear Equations
431(9)
15.2.1 Gaussian Elimination: Successive Elimination of Variables
431(2)
15.2.2 Gauss--Jordan Elimination
433(1)
15.2.3 Matrix Notation of Sets of Equations and Determination of the Inverse Matrix
434(3)
15.2.4 Existence of Solutions
437(3)
15.3 Determinants
440(13)
15.3.1 Preliminary Remarks on Determinants
440(1)
15.3.2 Definition and Properties of an n-Row Determinant
441(5)
15.3.3 Rank of a Determinant and Rank of a Matrix
446(1)
15.3.4 Applications of Determinants
447(6)
16 Eigenvalues and Eigenvectors of Real Matrices
453(10)
16.1 Two Case Studies: Eigenvalues of 2 X 2 Matrices
453(3)
16.2 General Method for Finding Eigenvalues
456(2)
16.3 Worked Example: Eigenvalues of a 3 X 3 Matrix
458(2)
16.4 Important Facts on Eigenvalues and Eigenvectors
460(3)
17 Vector Analysis: Surface Integrals, Divergence, Curl and Potential
463(30)
17.1 Flow of a Vector Field Through a Surface Element
463(3)
17.2 Surface Integral
466(2)
17.3 Special Cases of Surface Integrals
468(4)
17.3.1 Flow of a Homogeneous Vector Field Through a Cuboid
468(2)
17.3.2 Flow of a Spherically Symmetrical Field Through a Sphere
470(2)
17.3.3 Application: The Electrical Field of a Point Charge
472(1)
17.4 General Case of Computing Surface Integrals
472(5)
17.5 Divergence of a Vector Field
477(3)
17.6 Gauss's Theorem
480(2)
17.7 Curl of a Vector Field
482(4)
17.8 Stokes' Theorem
486(1)
17.9 Potential of a Vector Field
487(3)
17.10 Short Reference on Vector Derivatives
490(3)
18 Fourier Series; Harmonic Analysis
493(16)
18.1 Expansion of a Periodic Function into a Fourier Series
493(5)
18.1.1 Evaluation of the Coefficients
494(3)
18.1.2 Odd and Even Functions
497(1)
18.2 Examples of Fourier Series
498(5)
18.3 Expansion of Functions of Period 2L
503(1)
18.4 Fourier Spectrum
504(5)
19 Fourier Integrals and Fourier Transforms
509(10)
19.1 Transition from Fourier Series to Fourier Integral
509(2)
19.2 Fourier Transforms
511(4)
19.2.1 Fourier Cosine Transform
511(1)
19.2.2 Fourier Sine Transform, General Fourier Transform
512(2)
19.2.3 Complex Representation of the Fourier Transform
514(1)
19.3 Shift Theorem
515(1)
19.4 Discrete Fourier Transform, Sampling Theorems
516(1)
19.5 Fourier Transform of the Gaussian Function
516(3)
20 Probability Calculus
519(12)
20.1 Introduction
519(1)
20.2 Concept of Probability
520(7)
20.2.1 Random Experiment, Outcome Space and Events
520(1)
20.2.2 The Classical Definition of Probability
521(1)
20.2.3 The Statistical Definition of Probability
521(2)
20.2.4 General Properties of Probabilities
523(2)
20.2.5 Probability of Statistically Independent Events. Compound Probability
525(2)
20.3 Permutations and Combinations
527(4)
20.3.1 Permutations
527(1)
20.3.2 Combinations
528(3)
21 Probability Distributions
531(18)
21.1 Discrete and Continuous Probability Distributions
531(6)
21.1.1 Discrete Probability Distributions
531(3)
21.1.2 Continuous Probability Distributions
534(3)
21.2 Mean Values of Discrete and Continuous Variables
537(2)
21.3 The Normal Distribution as the Limiting Value of the Binomial Distribution
539(10)
21.3.1 Properties of the Normal Distribution
542(2)
21.3.2 Derivation of the Binomial Distribution
544(5)
22 Theory of Errors
549(20)
22.1 Purpose of the Theory of Errors
549(1)
22.2 Mean Value and Variance
550(4)
22.2.1 Mean Value
550(1)
22.2.2 Variance and Standard Deviation
551(1)
22.2.3 Mean Value and Variance in a Random Sample and Parent Population
552(2)
22.3 Mean Value and Variance of Continuous Distributions
554(2)
22.4 Error in Mean Value
556(1)
22.5 Normal Distribution: Distribution of Random Errors
557(1)
22.6 Law of Error Propagation
558(2)
22.7 Weighted Average
560(1)
22.8 Curve Fitting: Method of Least Squares, Regression Line
561(3)
22.9 Correlation and Correlation Coefficient
564(5)
Answers 569(26)
Index 595
Prof. Dr. Klaus Weltner is based at the University of Frankfurt, Germany.



Sebastian John teaches at Cambridge University, UK.



Dr. Wolfgang J. Webern is based in the University of Frankfurt, Germany.



Dr. Peter Schuster teaches at the University of Leeds, UK. 



Professor Jean Grosjean is based in the School of Engineering at the University of Bath, UK.
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