
Mathematics for Scientists : Fundamentals and Interactive Study Guide
by Weltner, Klaus; Weber, Wolfgang J.; Schuster, Peter-
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Summary
Table of Contents
Preface | p. vii |
Vector Algebra I: Scalars and Vectors | p. 1 |
Scalars and Vectors | p. 1 |
Addition of Vectors | p. 4 |
Sum of Two Vectors: Geometrical Addition | p. 4 |
Subtraction of Vectors | p. 6 |
Components and Projection of a Vector | p. 7 |
Component Representation in Coordinate Systems | p. 9 |
Position Vector | p. 9 |
Unit Vectors | p. 10 |
Component Representation of a Vector | p. 11 |
Representation of the Sum of Two Vectors in Terms of Their Components | p. 12 |
Subtraction of Vectors in Terms of their Components | p. 13 |
Multiplication of a Vector by a Scalar | p. 14 |
Magnitude of a Vector | p. 35 |
Vector Algebra II: Scalar and Vector Products | p. 23 |
Scalar Product | p. 23 |
Application: Equation of a Line and a Plane | p. 26 |
Special Cases | p. 26 |
Commutative and Distributive Laws | p. 27 |
Scalar Product in Terms of the Components of the Vectors | p. 27 |
Vector Product | p. 30 |
Torque | p. 30 |
Torque as a Vector | p. 31 |
Definition of the Vector Product | p. 32 |
Special Cases | p. 33 |
Anti-Commutative Law for Vector Products | p. 33 |
Components of the Vector Product | p. 34 |
Functions | p. 39 |
The Mathematical Concept of Functions and its Meaning in Physics and Engineering | p. 39 |
Introduction | p. 39 |
The Concept of a Function | p. 40 |
Graphical Representation of Functions | p. 42 |
Coordinate System, Position Vector | p. 42 |
The Linear Function: The Straight Line | p. 43 |
Graph Plotting | p. 44 |
Quadratic Equations | p. 47 |
Parametric Changes of Functions and Their Graphs | p. 49 |
Inverse Functions | p. 50 |
Trigonometric or Circular Functions | p. 52 |
Unit Circle | p. 52 |
Sine Function | p. 53 |
Cosine Function | p. 58 |
Relationships Between the Sine and Cosine Functions | p. 59 |
Tangent and Cotangent | p. 61 |
Addition Formulae | p. 62 |
Inverse Trigonometric Functions | p. 64 |
Function of a Function (Composition) | p. 66 |
Exponential, Logarithmic and Hyperbolic Functions | p. 69 |
Powers, Exponential Function | p. 69 |
Powers | p. 69 |
Laws of Indices or Exponents | p. 70 |
Binomial Theorem | p. 71 |
Exponential Function | p. 71 |
Logarithm, Logarithmic Function | p. 74 |
Logarithm | p. 74 |
Operations with Logarithms | p. 76 |
Logarithmic Functions | p. 77 |
Hyperbolic Functions and Inverse Hyperbolic Functions | p. 78 |
Hyperbolic Functions | p. 78 |
Inverse Hyperbolic Functions | p. 81 |
Differential Calculus | p. 85 |
Sequences and Limits | p. 85 |
The Concept of Sequence | p. 85 |
Limit of a Sequence | p. 86 |
Limit of a Function | p. 89 |
Examples for the Practical Determination of Limits | p. 89 |
Continuity | p. 91 |
Series | p. 92 |
Geometric Series | p. 93 |
Differentiation of a Function | p. 94 |
Gradient or Slope of a Line | p. 94 |
Gradient of an Arbitrary Curve | p. 95 |
Derivative of a Function | p. 97 |
Physical Application: Velocity | p. 98 |
The Differential | p. 99 |
Calculating Differential Coefficients | p. 100 |
Derivatives of Power Functions; Constant Factors | p. 101 |
Rules for Differentiation | p. 102 |
Differentiation of Fundamental Functions | p. 106 |
Higher Derivatives | p. 112 |
Extreme Values and Points of Inflexion; Curve Sketching | p. 113 |
Maximum and Minimum Values of a Function | p. 113 |
Further Remarks on Points of Inflexion (Contraflexure) | p. 117 |
Curve Sketching | p. 118 |
Applications of Differential Calculus | p. 121 |
Extreme Values | p. 121 |
Increments | p. 122 |
Curvature | p. 123 |
Determination of Limits by Differentiation: L'Hôpital's Rule | p. 125 |
Further Methods for Calculating Differential Coefficients | p. 127 |
Implicit Functions and their Derivatives | p. 127 |
Logarithmic Differentiation | p. 128 |
Parametric Functions and their Derivatives | p. 129 |
Parametric Form of an Equation | p. 129 |
Derivatives of Parametric Functions | p. 133 |
Integral Calculus | p. 145 |
The Primitive Function | p. 145 |
Fundamental Problem of Integral Calculus | p. 145 |
The Area Problem: The Definite Integral | p. 147 |
Fundamental Theorem of the Differential and Integral Calculus | p. 149 |
The Definite Integral | p. 153 |
Calculation of Definite Integrals from Indefinite Integrals | p. 153 |
Examples of Definite Integrals | p. 156 |
Methods of Integration | p. 159 |
Principle of Verification | p. 159 |
Standard Integrals | p. 159 |
Constant Factor and the Sum of Functions | p. 160 |
Integration by Parts: Product of Two Functions | p. 161 |
Integration by Substitution | p. 164 |
Substitution in Particular Cases | p. 166 |
Integration by Partial Fractions | p. 170 |
Rules for Solving Definite Integrals | p. 175 |
Mean Value Theorem | p. 178 |
Improper Integrals | p. 179 |
Line Integrals | p. 181 |
Applications of Integration | p. 191 |
Areas | p. 191 |
Areas for Parametric Functions | p. 194 |
Areas in Polar Coordinates | p. 195 |
Areas of Closed Curves | p. 197 |
Lengths of Curves | p. 198 |
Lengths of Curves in Polar Coordinates | p. 201 |
Surface Area and Volume of a Solid of Revolution | p. 202 |
Applications to Mechanics | p. 208 |
Basic Concepts of Mechanics | p. 208 |
Center of Mass and Centroid | p. 208 |
The Theorems of Pappus | p. 211 |
Moments of Inertia; Second Moment of Area | p. 213 |
Taylor Series and Power Series | p. 227 |
Introduction | p. 227 |
Expansion of a Function in a Power Series | p. 228 |
Interval of Convergence of Power Series | p. 232 |
Approximate Values of Functions | p. 233 |
Expansion of a Function f(x) at an Arbitrary Position | p. 235 |
Applications of Series | p. 237 |
Polynomials as Approximations | p. 237 |
Integration of Functions when Expressed as Power Series | p. 240 |
Expansion in a Series by Integrating | p. 242 |
Complex Numbers | p. 247 |
Definition and Properties of Complex Numbers | p. 247 |
Imaginary Numbers | p. 247 |
Complex Numbers | p. 248 |
Fields of Application | p. 248 |
Operations with Complex Numbers | p. 249 |
Graphical Representation of Complex Numbers | p. 250 |
Gauss Complex Number Plane: Argand Diagram | p. 250 |
Polar Form of a Complex Number | p. 251 |
Exponential Form of Complex Numbers | p. 254 |
Euler's Formula | p. 254 |
Exponential Form of the Sine and Cosine Functions | p. 255 |
Complex Numbers as Powers | p. 255 |
Multiplication and Division in Exponential Form | p. 258 |
Raising to a Power, Exponential Form | p. 259 |
Periodicity of rej&alhpa; | p. 259 |
Transformation of a Complex Number From One Form into Another | p. 260 |
Operations with Complex Numbers Expressed in Polar Form | p. 261 |
Multiplication and Division | p. 261 |
Raising to a Power | p. 263 |
Roots of a Complex Number | p. 263 |
Differential Equations | p. 273 |
Concept and Classification of Differential Equations | p. 273 |
Preliminary Remarks | p. 277 |
General Solution of First- and Second-Order DEs with Constant Coefficients | p. 279 |
Homogeneous Linear DE | p. 279 |
Non-Homogeneous Linear DE | p. 285 |
Boundary Value Problems | p. 291 |
First-Order DEs | p. 291 |
Second-Order DEs | p. 291 |
Some Applications of DEs | p. 293 |
Radioactive Decay | p. 293 |
The Harmonic Oscillator | p. 294 |
General Linear First-Order DEs | p. 302 |
Solution by Variation of the Constant | p. 302 |
A Straightforward Method Involving the Integrating Factor | p. 304 |
Some Remarks on General First-Order DEs | p. 306 |
Bernoulli's Equations | p. 306 |
Separation of Variables | p. 307 |
Exact Equations | p. 308 |
The Integrating Factor - General Case | p. 311 |
Simultaneous DEs | p. 313 |
Higher-Order DEs Interpreted as Systems of First-Order Simultaneous DEs | p. 317 |
Some Advice on Intractable DEs | p. 317 |
Laplace Transforms | p. 321 |
Introduction | p. 321 |
The Laplace Transform Definition | p. 321 |
Laplace Transform of Standard Functions | p. 322 |
Solution of Linear DEs with Constant Coefficients | p. 328 |
Solution of Simultaneous DEs with Constant Coefficients | p. 330 |
Functions of Several Variables; Partial Differentiation; and Total Differentiation | p. 337 |
Introduction | p. 337 |
Functions of Several Variables | p. 338 |
Representing the Surface by Establishing a Table of Z-Values | p. 339 |
Representing the Surface by Establishing Intersecting Curves | p. 340 |
Obtaining a Functional Expression for a Given Surface | p. 343 |
Partial Differentiation | p. 344 |
Higher Partial Derivatives | p. 348 |
Total Differential | p. 350 |
Total Differential of Functions | p. 350 |
Application: Small Tolerances | p. 354 |
Gradient | p. 356 |
Total Derivative | p. 358 |
Explicit Functions | p. 358 |
Implicit Functions | p. 360 |
Maxima and Minima of Functions of Two or More Variables | p. 361 |
Applications: Wave Function and Wave Equation | p. 367 |
Wave Function | p. 367 |
Wave Equation | p. 371 |
Multiple Integrals; Coordinate Systems | p. 377 |
Multiple Integrals | p. 377 |
Multiple Integrals with Constant Limits | p. 379 |
Decomposition of a Multiple Integral into a Product of Integrals | p. 381 |
Multiple Integrals with Variable Limits | p. 382 |
Coordinate Systems | p. 386 |
Polar Coordinates | p. 387 |
Cylindrical Coordinates | p. 389 |
Spherical Coordinates | p. 391 |
Application: Moments of Inertia of a Solid | p. 395 |
Transformation of Coordinates; Matrices | p. 401 |
Introduction | p. 401 |
Parallel Shift of Coordinates: Translation | p. 404 |
Rotation | p. 407 |
Rotation in a Plane | p. 407 |
Successive Rotations | p. 410 |
Rotations in Three-Dimensional Space | p. 411 |
Matrix Algebra | p. 413 |
Addition and Subtraction of Matrices | p. 415 |
Multiplication of a Matrix by a Scalar | p. 416 |
Product of a Matrix and a Vector | p. 416 |
Multiplication of Two Matrices | p. 417 |
Rotations Expressed in Matrix Form | p. 419 |
Rotation in Two-Dimensional Space | p. 419 |
Special Rotation in Three-Dimensional Space | p. 420 |
Special Matrices | p. 421 |
Inverse Matrix | p. 424 |
Sets of Linear Equations; Determinants | p. 429 |
Introduction | p. 429 |
Sets of Linear Equations | p. 429 |
Gaussian Elimination: Successive Elimination of Variables | p. 429 |
Gauss-Jordan Elimination | p. 431 |
Matrix Notation of Sets of Equations and Determination of the Inverse Matrix | p. 432 |
Existence of Solutions | p. 435 |
Determinants | p. 438 |
Preliminary Remarks on Determinants | p. 438 |
Definition and Properties of an n-Row Determinant | p. 439 |
Rank of a Determinant and Rank of a Matrix | p. 444 |
Applications of Determinants | p. 445 |
Eigenvalues and Eigenvectors of Real Matrices | p. 451 |
Two Case Studies: Eigenvalues of 2 x 2 Matrices | p. 451 |
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