Mathematics for Scientists : Fundamentals and Interactive Study Guide

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Format: Hardcover
Pub. Date: 2009-10-01
Publisher(s): Springer Verlag
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Summary

This textbook offers an accessible and highly-effective approach which is characterised by the combination of the textbook with a detailed study guide on an accompanying CD. 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 then to return to the study guide. Through interactive learning with the study guide, the 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's performance and needs and can be regarded as a total tutorial experience.More than that, the study guide aims to satisfy 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 the combination ot textbook and study guide.

Table of Contents

Prefacep. vii
Vector Algebra I: Scalars and Vectorsp. 1
Scalars and Vectorsp. 1
Addition of Vectorsp. 4
Sum of Two Vectors: Geometrical Additionp. 4
Subtraction of Vectorsp. 6
Components and Projection of a Vectorp. 7
Component Representation in Coordinate Systemsp. 9
Position Vectorp. 9
Unit Vectorsp. 10
Component Representation of a Vectorp. 11
Representation of the Sum of Two Vectors in Terms of Their Componentsp. 12
Subtraction of Vectors in Terms of their Componentsp. 13
Multiplication of a Vector by a Scalarp. 14
Magnitude of a Vectorp. 35
Vector Algebra II: Scalar and Vector Productsp. 23
Scalar Productp. 23
Application: Equation of a Line and a Planep. 26
Special Casesp. 26
Commutative and Distributive Lawsp. 27
Scalar Product in Terms of the Components of the Vectorsp. 27
Vector Productp. 30
Torquep. 30
Torque as a Vectorp. 31
Definition of the Vector Productp. 32
Special Casesp. 33
Anti-Commutative Law for Vector Productsp. 33
Components of the Vector Productp. 34
Functionsp. 39
The Mathematical Concept of Functions and its Meaning in Physics and Engineeringp. 39
Introductionp. 39
The Concept of a Functionp. 40
Graphical Representation of Functionsp. 42
Coordinate System, Position Vectorp. 42
The Linear Function: The Straight Linep. 43
Graph Plottingp. 44
Quadratic Equationsp. 47
Parametric Changes of Functions and Their Graphsp. 49
Inverse Functionsp. 50
Trigonometric or Circular Functionsp. 52
Unit Circlep. 52
Sine Functionp. 53
Cosine Functionp. 58
Relationships Between the Sine and Cosine Functionsp. 59
Tangent and Cotangentp. 61
Addition Formulaep. 62
Inverse Trigonometric Functionsp. 64
Function of a Function (Composition)p. 66
Exponential, Logarithmic and Hyperbolic Functionsp. 69
Powers, Exponential Functionp. 69
Powersp. 69
Laws of Indices or Exponentsp. 70
Binomial Theoremp. 71
Exponential Functionp. 71
Logarithm, Logarithmic Functionp. 74
Logarithmp. 74
Operations with Logarithmsp. 76
Logarithmic Functionsp. 77
Hyperbolic Functions and Inverse Hyperbolic Functionsp. 78
Hyperbolic Functionsp. 78
Inverse Hyperbolic Functionsp. 81
Differential Calculusp. 85
Sequences and Limitsp. 85
The Concept of Sequencep. 85
Limit of a Sequencep. 86
Limit of a Functionp. 89
Examples for the Practical Determination of Limitsp. 89
Continuityp. 91
Seriesp. 92
Geometric Seriesp. 93
Differentiation of a Functionp. 94
Gradient or Slope of a Linep. 94
Gradient of an Arbitrary Curvep. 95
Derivative of a Functionp. 97
Physical Application: Velocityp. 98
The Differentialp. 99
Calculating Differential Coefficientsp. 100
Derivatives of Power Functions; Constant Factorsp. 101
Rules for Differentiationp. 102
Differentiation of Fundamental Functionsp. 106
Higher Derivativesp. 112
Extreme Values and Points of Inflexion; Curve Sketchingp. 113
Maximum and Minimum Values of a Functionp. 113
Further Remarks on Points of Inflexion (Contraflexure)p. 117
Curve Sketchingp. 118
Applications of Differential Calculusp. 121
Extreme Valuesp. 121
Incrementsp. 122
Curvaturep. 123
Determination of Limits by Differentiation: L'Hôpital's Rulep. 125
Further Methods for Calculating Differential Coefficientsp. 127
Implicit Functions and their Derivativesp. 127
Logarithmic Differentiationp. 128
Parametric Functions and their Derivativesp. 129
Parametric Form of an Equationp. 129
Derivatives of Parametric Functionsp. 133
Integral Calculusp. 145
The Primitive Functionp. 145
Fundamental Problem of Integral Calculusp. 145
The Area Problem: The Definite Integralp. 147
Fundamental Theorem of the Differential and Integral Calculusp. 149
The Definite Integralp. 153
Calculation of Definite Integrals from Indefinite Integralsp. 153
Examples of Definite Integralsp. 156
Methods of Integrationp. 159
Principle of Verificationp. 159
Standard Integralsp. 159
Constant Factor and the Sum of Functionsp. 160
Integration by Parts: Product of Two Functionsp. 161
Integration by Substitutionp. 164
Substitution in Particular Casesp. 166
Integration by Partial Fractionsp. 170
Rules for Solving Definite Integralsp. 175
Mean Value Theoremp. 178
Improper Integralsp. 179
Line Integralsp. 181
Applications of Integrationp. 191
Areasp. 191
Areas for Parametric Functionsp. 194
Areas in Polar Coordinatesp. 195
Areas of Closed Curvesp. 197
Lengths of Curvesp. 198
Lengths of Curves in Polar Coordinatesp. 201
Surface Area and Volume of a Solid of Revolutionp. 202
Applications to Mechanicsp. 208
Basic Concepts of Mechanicsp. 208
Center of Mass and Centroidp. 208
The Theorems of Pappusp. 211
Moments of Inertia; Second Moment of Areap. 213
Taylor Series and Power Seriesp. 227
Introductionp. 227
Expansion of a Function in a Power Seriesp. 228
Interval of Convergence of Power Seriesp. 232
Approximate Values of Functionsp. 233
Expansion of a Function f(x) at an Arbitrary Positionp. 235
Applications of Seriesp. 237
Polynomials as Approximationsp. 237
Integration of Functions when Expressed as Power Seriesp. 240
Expansion in a Series by Integratingp. 242
Complex Numbersp. 247
Definition and Properties of Complex Numbersp. 247
Imaginary Numbersp. 247
Complex Numbersp. 248
Fields of Applicationp. 248
Operations with Complex Numbersp. 249
Graphical Representation of Complex Numbersp. 250
Gauss Complex Number Plane: Argand Diagramp. 250
Polar Form of a Complex Numberp. 251
Exponential Form of Complex Numbersp. 254
Euler's Formulap. 254
Exponential Form of the Sine and Cosine Functionsp. 255
Complex Numbers as Powersp. 255
Multiplication and Division in Exponential Formp. 258
Raising to a Power, Exponential Formp. 259
Periodicity of rej&alhpa;p. 259
Transformation of a Complex Number From One Form into Anotherp. 260
Operations with Complex Numbers Expressed in Polar Formp. 261
Multiplication and Divisionp. 261
Raising to a Powerp. 263
Roots of a Complex Numberp. 263
Differential Equationsp. 273
Concept and Classification of Differential Equationsp. 273
Preliminary Remarksp. 277
General Solution of First- and Second-Order DEs with Constant Coefficientsp. 279
Homogeneous Linear DEp. 279
Non-Homogeneous Linear DEp. 285
Boundary Value Problemsp. 291
First-Order DEsp. 291
Second-Order DEsp. 291
Some Applications of DEsp. 293
Radioactive Decayp. 293
The Harmonic Oscillatorp. 294
General Linear First-Order DEsp. 302
Solution by Variation of the Constantp. 302
A Straightforward Method Involving the Integrating Factorp. 304
Some Remarks on General First-Order DEsp. 306
Bernoulli's Equationsp. 306
Separation of Variablesp. 307
Exact Equationsp. 308
The Integrating Factor - General Casep. 311
Simultaneous DEsp. 313
Higher-Order DEs Interpreted as Systems of First-Order Simultaneous DEsp. 317
Some Advice on Intractable DEsp. 317
Laplace Transformsp. 321
Introductionp. 321
The Laplace Transform Definitionp. 321
Laplace Transform of Standard Functionsp. 322
Solution of Linear DEs with Constant Coefficientsp. 328
Solution of Simultaneous DEs with Constant Coefficientsp. 330
Functions of Several Variables; Partial Differentiation; and Total Differentiationp. 337
Introductionp. 337
Functions of Several Variablesp. 338
Representing the Surface by Establishing a Table of Z-Valuesp. 339
Representing the Surface by Establishing Intersecting Curvesp. 340
Obtaining a Functional Expression for a Given Surfacep. 343
Partial Differentiationp. 344
Higher Partial Derivativesp. 348
Total Differentialp. 350
Total Differential of Functionsp. 350
Application: Small Tolerancesp. 354
Gradientp. 356
Total Derivativep. 358
Explicit Functionsp. 358
Implicit Functionsp. 360
Maxima and Minima of Functions of Two or More Variablesp. 361
Applications: Wave Function and Wave Equationp. 367
Wave Functionp. 367
Wave Equationp. 371
Multiple Integrals; Coordinate Systemsp. 377
Multiple Integralsp. 377
Multiple Integrals with Constant Limitsp. 379
Decomposition of a Multiple Integral into a Product of Integralsp. 381
Multiple Integrals with Variable Limitsp. 382
Coordinate Systemsp. 386
Polar Coordinatesp. 387
Cylindrical Coordinatesp. 389
Spherical Coordinatesp. 391
Application: Moments of Inertia of a Solidp. 395
Transformation of Coordinates; Matricesp. 401
Introductionp. 401
Parallel Shift of Coordinates: Translationp. 404
Rotationp. 407
Rotation in a Planep. 407
Successive Rotationsp. 410
Rotations in Three-Dimensional Spacep. 411
Matrix Algebrap. 413
Addition and Subtraction of Matricesp. 415
Multiplication of a Matrix by a Scalarp. 416
Product of a Matrix and a Vectorp. 416
Multiplication of Two Matricesp. 417
Rotations Expressed in Matrix Formp. 419
Rotation in Two-Dimensional Spacep. 419
Special Rotation in Three-Dimensional Spacep. 420
Special Matricesp. 421
Inverse Matrixp. 424
Sets of Linear Equations; Determinantsp. 429
Introductionp. 429
Sets of Linear Equationsp. 429
Gaussian Elimination: Successive Elimination of Variablesp. 429
Gauss-Jordan Eliminationp. 431
Matrix Notation of Sets of Equations and Determination of the Inverse Matrixp. 432
Existence of Solutionsp. 435
Determinantsp. 438
Preliminary Remarks on Determinantsp. 438
Definition and Properties of an n-Row Determinantp. 439
Rank of a Determinant and Rank of a Matrixp. 444
Applications of Determinantsp. 445
Eigenvalues and Eigenvectors of Real Matricesp. 451
Two Case Studies: Eigenvalues of 2 x 2 Matricesp. 451
Table of Contents provided by Ingram. All Rights Reserved.

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