Mathematical Techniques. An Introduction for the Engineering, Physical, and Mathematical Sciences

✍ Scribed by D. W. Jordan, P. Smith

• Publisher: Oxford University Press
• Year: 2008
• Tongue: English
• Leaves: 997
• Edition: 4
• Category: Library

✦ Table of Contents

Cover
Half-Title Page
Title Page
Copyright Page
Preface to the fourth edition
Brief Contents
Detailed Contents
Part 1: Elementary methods, differentiation, complex numbers
1. Standard functions and techniques
1.1 Real numbers, powers, inequalities
1.2 Coordinates in the plane
1.3 Graphs
1.4 Functions
1.5 Radian measure of angles
1.6 Trigonometric functions; properties
1.7 Inverse functions
1.8 Inverse trigonometric functions
1.9 Polar coordinates
1.10 Exponential functions; the number e
1.11 The logarithmic function
1.12 Exponential growth and decay
1.13 Hyperbolic functions
1.14 Partial fractions
1.15 Summation sign: geometric series
1.16 Infinite geometric series
1.17 Permutations and combinations
1.18 The binomial theorem
Problems
2. Differentiation
2.1 The slope of a graph
2.2 The derivative: notation and definition
2.3 Rates of change
2.4 Derivative of xn (n = 0, 1, 2, 3, … )
2.5 Derivatives of sums: multiplication by constants
2.6 Three important limits
2.7 Derivatives of ex, sin x, cos x, ln x
2.8 A basic table of derivatives
2.9 Higher-order derivatives
2.10 An interpretation of the second derivative
Problems
3. Further techniques for differentiation
3.1 The product rule
3.2 Quotients and reciprocals
3.3 The chain rule
3.4 Derivative of xn for any value of n
3.5 Functions of ax + b
3.6 An extension of the chain rule
3.7 Logarithmic differentiation
3.8 Implicit differentiation
3.9 Derivatives of inverse functions
3.10 Derivative as a function of a parameter
Problems
4. Applications of differentiation
4.1 Function notation for derivatives
4.2 Maxima and minima
4.3 Exceptional cases of maxima and minima
4.4 Sketching graphs of functions
4.5 Estimating small changes
4.6 Numerical solution of equations: Newton’s method
4.7 The binomial theorem: an alternative proof
Problems
5. Taylor series and approximations
5.1 The index notation for derivatives of any order
5.2 Taylor polynomials
5.3 A note on infinite series
5.4 Infinite Taylor expansions
5.5 Manipulation of Taylor series
5.6 Approximations for large values of x
5.7 Taylor series about other points
5.8 Indeterminate values; l’Hôpital’s rule 136
Problems
6. Complex numbers
6.1 Definitions and rules
6.2 The Argand diagram, modulus, conjugate
6.3 Complex numbers in polar coordinates
6.4 Complex numbers in exponential form
6.5 The general exponential form
6.6 Hyperbolic functions
6.7 Miscellaneous applications
Problems
Part 2: Matrix and vector algebra
7. Matrix algebra
7.1 Matrix definition and notation
7.2 Rules of matrix algebra
7.3 Special matrices
7.4 The inverse matrix
Problems
8. Determinants
8.1 The determinant of a square matrix
8.2 Properties of determinants
8.3 The adjoint and inverse matrices
Problems
9. Elementary operations with vectors
9.1 Displacement along an axis
9.2 Displacement vectors in two dimensions
9.3 Axes in three dimensions
9.4 Vectors in two and three dimensions
9.5 Relative velocity
9.6 Position vectors and vector equations
9.7 Unit vectors and basis vectors
9.8 Tangent vector, velocity, and acceleration
9.9 Motion in polar coordinates
Problems
10. The scalar product
10.1 The scalar product of two vectors
10.2 The angle between two vectors
10.3 Perpendicular vectors
10.4 Rotation of axes in two dimensions
10.5 Direction cosines
10.6 Rotation of axes in three dimensions
10.7 Direction ratios and coordinate geometry
10.8 Properties of a plane
10.9 General equation of a straight line
10.10 Forces acting at a point
10.11 Tangent vector and curvature in two dimensions
Problems
11. Vector product
11.1 Vector product
11.2 Nature of the vector p = a × b
11.3 The scalar triple product
11.4 Moment of a force
11.5 Vector triple product
Problems
12. Linear algebraic equations
12.1 Cramer’s rule
12.2 Elementary row operations
12.3 The inverse matrix by Gaussian elimination
12.4 Compatible and incompatible sets of equations
12.5 Homogeneous sets of equations
12.6 Gauss–Seidel iterative method of solution
Problems
13. Eigenvalues and eigenvectors
13.1 Eigenvalues of a matrix
13.2 Eigenvectors
13.3 Linear dependence
13.4 Diagonalization of a matrix
13.5 Powers of matrices
13.6 Quadratic forms
13.7 Positive-definite matrices
13.8 An application to a vibrating system
Problems
Part 3: Integration and differential equations
14. Antidifferentiation and area
14.1 Reversing differentiation
14.2 Constructing a table of antiderivatives
14.3 Signed area generated by a graph
14.4 Case where the antiderivative is composite
Problems
15. The definite and indefinite integral
15.1 Signed area as the sum of strips
15.2 Numerical illustration of the sum formula
15.3 The definite integral and area
15.4 The indefinite-integral notation
15.5 Integrals unrelated to area
15.6 Improper integrals
15.7 Integration of complex functions: a new type of integral
15.8 The area analogy for a definite integral
15.9 Symmetric integrals
15.10 Definite integrals having variable limits
Problems
16. Applications involving the integral as a sum
16.1 Examples of integrals arising from a sum
16.2 Geometrical area in polar coordinates
16.3 The trapezium rule
16.4 Centre of mass, moment of inertia
Problems
17. Systematic techniques for integration
17.1 Substitution method for ∫ f(ax + b) dx
17.2 Substitution method for ∫ f(ax2 + b)x dx
17.3 Substitution method for ∫ cosmax sinnax dx (m or n odd)
17.4 Definite integrals and change of variable
17.5 Occasional substitutions
17.6 Partial fractions for integration
17.7 Integration by parts
17.8 Integration by parts: definite integrals
17.9 Differentiating with respect to a parameter
Problems
18. Unforced linear differential equations with constant coefficients
18.1 Differential equations and their solutions
18.2 Solving first-order linear unforced equations
18.3 Solving second-order linear unforced equations
18.4 Complex solutions of the characteristic equation
18.5 Initial conditions for second-order equations
Problems
19. Forced linear differential equations
19.1 Particular solutions for standard forcing terms
19.2 Harmonic forcing term, by using complex solutions
19.3 Particular solutions: exceptional cases
19.4 The general solutions of forced equations
19.5 First-order linear equations with a variable coefficient
Problems
20. Harmonic functions and the harmonic oscillator
20.1 Harmonic oscillations
20.2 Phase difference: lead and lag
20.3 Physical models of a differential equation
20.4 Free oscillations of a linear oscillator
20.5 Forced oscillations and transients
20.6 Resonance
20.7 Nearly linear systems
20.8 Stationary and travelling waves
20.9 Compound oscillations; beats
20.10 Travelling waves; beats
20.11 Dispersion; group velocity
20.12 The Doppler effect
Problems
21. Steady forced oscillations: phasors, impedance, transfer functions
21.1 Phasors
21.2 Algebra of phasors
21.3 Phasor diagrams
21.4 Phasors and complex impedance
21.5 Transfer functions in the frequency domain
21.6 Phasors and waves; complex amplitude
Problems
22. Graphical, numerical, and other aspects of first-order equations
22.1 Graphical features of first-order equations
22.2 The Euler method for numerical solution
22.3 Nonlinear equations of separable type
22.4 Differentials and the solution of first-order equations
22.5 Change of variable in a differential equation
Problems
23. Nonlinear differential equations and the phase plane
23.1 Autonomous second-order equations
23.2 Constructing a phase diagram for (x, ẋ )
23.3 (x, ẋ ) phase diagrams for other linearequations; stability
23.4 The pendulum equation
23.5 The general phase plane
23.6 Approximate linearization
23.7 Classification of linear equilibrium points
23.8 Limit cycles
23.9 A numerical method for phase paths
Problems
Part 4: Transforms and Fourier Series
24. The Laplace transform
24.1 The Laplace transform
24.2 Laplace transforms of tn, e±t, sin t, cos t
24.3 Scale rule; shift rule; factors tn and ekt
24.4 Inverting a Laplace transform
24.5 Laplace transforms of derivatives
24.6 Application to differential equations
24.7 The unit function and the delay rule
24.8 The division rule for f(t)/t
Problems
25. Laplace and z transforms: applications
25.1 Division by s and integration
25.2 The impulse function
25.3 Impedance in the s domain
25.4 Transfer functions in the s domain
25.5 The convolution theorem
25.6 General response of a system from its impulsive response
25.7 Convolution integral in terms of memory
25.8 Discrete systems
25.9 The z transform
25.10 Behaviour of z transforms in the complex plane
25.11 z transforms and difference equations
Problems
26. Fourier series
26.1 Fourier series for a periodic function
26.2 Integrals of periodic functions
26.3 Calculating the Fourier coefficients
26.4 Examples of Fourier series
26.5 Use of symmetry: sine and cosine series
26.6 Functions defined on a finite range: half-range series
26.7 Spectrum of a periodic function
26.8 Obtaining one Fourier series from another
26.9 The two-sided Fourier series
Problems
27. Fourier transforms
27.1 Sine and cosine transforms
27.2 The exponential Fourier transform
27.3 Short notations: alternative expressions
27.4 Fourier transforms of some basic functions
27.5 Rules for manipulating transforms
27.6 The delta function and periodic functions
27.7 Convolution theorem for Fourier transforms
27.8 The shah function
27.9 Energy in a signal: Rayleigh’s theorem
27.10 Diffraction from a uniformly radiating strip
27.11 General source distribution and the inverse transform
27.12 Transforms in radiation problems
Problems
Part 5: Multivariable calculus
28. Differentiation of functions of two variables
28.1 Depiction of functions of two variables
28.2 Partial derivatives
28.3 Higher derivatives
28.4 Tangent plane and normal to a surface
28.5 Maxima, minima, and other stationary points
28.6 The method of least squares
28.7 Differentiating an integral with respect to a parameter
Problems
29. Functions of two variables: geometry and formulae
29.1 The incremental approximation
29.2 Small changes and errors
29.3 The derivative in any direction
29.4 Implicit differentiation
29.5 Normal to a curve
29.6 Gradient vector in two dimensions
Problems
30. Chain rules, restricted maxima, coordinate systems
30.1 Chain rule for a single parameter
30.2 Restricted maxima and minima: the Lagrange multiplier
30.3 Curvilinear coordinates in two dimensions
30.4 Orthogonal coordinates
30.5 The chain rule for two parameters
30.6 The use of differentials
Problems
31. Functions of any number of variables
31.1 The incremental approximation; errors
31.2 Implicit differentiation
31.3 Chain rules
31.4 The gradient vector in three dimensions
31.5 Normal to a surface
31.6 Equation of the tangent plane
31.7 Directional derivative in terms of gradient
31.8 Stationary points
31.9 The envelope of a family of curves
Problems
32. Double integration
32.1 Repeated integrals with constant limits
32.2 Examples leading to repeated integrals with constant limits
32.3 Repeated integrals over non-rectangular regions
32.4 Changing the order of integration for non-rectangular regions
32.5 Double integrals
32.6 Polar coordinates
32.7 Separable integrals
32.8 General change of variable; the Jacobian determinant
Problems
33. Line integrals
33.1 Evaluation of line integrals
33.2 General line integrals in two and three dimensions
33.3 Paths parallel to the axes
33.4 Path independence and perfect differentials
33.5 Closed paths
33.6 Green’s theorem
33.7 Line integrals and work
33.8 Conservative fields
33.9 Potential for a conservative field
33.10 Single-valuedness of potentials
Problems
34. Vector fields: divergence and curl
34.1 Vector fields and field lines
34.2 Divergence of a vector field
34.3 Surface and volume integrals
34.4 The divergence theorem; flux of a vector field
34.5 Curl of a vector field
34.6 Cylindrical polar coordinates
34.7 General curvilinear coordinates
34.8 Stokes’s theorem
Problems
Part 6: Discrete mathematics
35. Sets
35.1 Notation
35.2 Equality, union, and intersection
35.3 Venn diagrams
Problems
36. Boolean algebra: logic gates and switching functions
36.1 Laws of Boolean algebra
36.2 Logic gates and truth tables
36.3 Logic networks
36.4 The inverse truth-table problem
36.5 Switching circuits
Problems
37. Graph theory and its applications
37.1 Examples of graphs
37.2 Definitions and properties of graphs
37.3 How many simple graphs are there?
37.4 Paths and cycles
37.5 Trees
37.6 Electrical circuits: the cutset method
37.7 Signal-flow graphs
37.8 Planar graphs
37.9 Further applications
Problems
38. Difference equations
38.1 Discrete variables
38.2 Difference equations: general properties
38.3 First-order difference equations and the cobweb
38.4 Constant-coefficient linear difference equations
38.5 The logistic difference equation
Problems
Part 7: Probability and statistics
39. Probability
39.1 Sample spaces, events, and probability
39.2 Sets and probability
39.3 Frequencies and combinations
39.4 Conditional probability
39.5 Independent events
39.6 Total probability
39.7 Bayes’ theorem
Problems
40. Random variables and probability distributions
40.1 Probability distributions
40.2 The binomial distribution
40.3 Expected value and variance
40.4 Geometric distribution
40.5 Poisson distribution
40.6 Other discrete distributions
40.7 Continuous random variables and distributions
40.8 Mean and variance of continuous random variables
40.9 The normal distribution
Problems
41. Descriptive statistics
41.1 Representing data
41.2 Random samples and sampling distributions
41.3 Sample mean and variance, and their estimation
41.4 Central limit theorem
41.5 Regression
Problems
Part 8: Projects
42. Applications projects using symbolic computing
42.1 Symbolic computation
42.2 Projects
Self-tests: Selected answers
Answers to selected problems
Appendices
A Some algebraical rules
B Trigonometric formulae
C Areas and volumes
D A table of derivatives
E Tables of indefinite and definite integrals
F Laplace transforms, inverses, and rules
G Exponential Fourier transforms and rules
H Probability distributions and tables
I Dimensions and units
Further reading
Index