Mathematics and Mechanics The Interplay 1st Edition by Luigi Morino ISBN 9783662632079 3662632071

Part I The beginning – Pre–calculus

Chapter 1 Back to high school — algebra revisited

1.1 Wanna learn math? Let’s talk bignè and babà

1.1.1 One equation with one unknown

1.1.2 Two equations with two unknowns

1.2 Where do we go from here?

1.2.1 Guidelines for reading this book

1.2.2 Axioms, theorems, and all that stuff

1.3 Natural numbers

1.3.1 Pierino. Arithmetic with natural numbers

1.3.2 Division of natural numbers

1.3.3 Fundamental theorem of arithmetic

1.3.4 Even and odd numbers

1.3.5 Greatest common divisor, least common multiple

1.4 Integers

1.4.1 Giorgetto and arithmetic with integers

1.5 Rational numbers

1.5.1 Arithmetic with rational numbers

1.5.2 The rationals are countable

1.5.3 Decimal representation of rationals

1.6 Real numbers

1.6.1 Irrational and real numbers

1.6.2 Inequalities

1.6.3 Bounded and unbounded intervals

1.6.4 Approximating irrationals with rationals

1.6.5 Arithmetic with real numbers

1.6.6 Real numbers are uncountable

1.6.7 Axiomatic formulation

1.7 Imaginary and complex numbers

1.7.1 Imaginary numbers

1.7.2 Complex numbers

1.8 Appendix. The Greek alphabet

Chapter 2 Systems of linear algebraic equations

2.1 Preliminary remarks

2.2 The 2 x 2 linear algebraic systems

2.2.1 Method by substitution

2.2.2 Gaussian elimination method

2.2.3 Summary

2.3 The 3 x 3 linear algebraic systems

2.3.1 Gaussian elimination for 3 x 3 systems

2.3.2 Illustrative examples

2.3.3 Another illustrative example; ∞^2 solutions

2.3.4 Interchanging equations and/or unknowns

2.3.5 Existence and uniqueness theorems

2.3.6 Explicit solution for a generic 3 x 3 system

2.4 The n x n linear algebraic systems

2.4.1 Gaussian elimination procedure

2.4.2 Details of the Gaussian elimination procedure

2.4.3 Existence and uniqueness theorems

Chapter 3 Matrices and vectors

3.1 Matrices

3.1.1 Column matrices and row matrices

3.1.2 Basic operations on matrices and vectors

3.2 Summation symbol. Linear combinations

3.2.1 Summation symbol. Kronecker delta

3.2.2 Linear combinations

3.3 Linear dependence of vectors and matrices

3.4 Additional operation on matrices

3.4.1 Product between two matrices

3.4.2 Products involving column and row matrices

3.4.3 Symmetric and antisymmetric matrices

3.4.4 The operation A:B

3.5 A x = b revisited. Rank of A

3.5.1 Linear independence and Gaussian elimination

3.5.2 Rank. Singular matrices

3.5.3 Linear dependence/independence of columns

3.5.4 Existence and uniqueness theorem for A x = 0

3.5.5 Existence and uniqueness theorem for A x = b

3.5.6 Positive definite matrices

3.5.7 A x = b, with A rectangular

3.6 Linearity and superposition theorems

3.6.1 Linearity of matrix–by–vector multiplication

3.6.2 Superposition in linear algebraic systems

3.7 Appendix A. Partitioned matrices

3.8 Appendix B. A tiny bit on determinants

3.8.1 Determinants of 2 x 2 matrices

3.8.2 Determinants of 3 x 3 matrices

3.9 Appendix C. Tridiagonal matrices

Chapter 4 Statics of particles in one dimension

4.1 Newton equilibrium law for single particles

4.1.1 Particles and material points

4.1.2 Forces

4.2 Equilibrium of systems of particles

4.2.1 The Newton third law of action and reaction

4.3 Longitudinal equilibrium of spring–particle chains

4.4 Spring–particle chains anchored at both ends

4.4.1 Chains of three particles

4.4.2 Chains of n particles

4.5 Particle chains anchored at one endpoint

4.6 Unanchored spring–particle chains

4.7 Appendix A. Inuence matrix

4.8 Appendix B. General spring{particle systems

4.8.1 The matrix K is positive definite/semidefinite

4.9 Appendix C. International System of Units (SI)

Chapter 5 Basic Euclidean geometry

5.1 Preliminaries

5.2 Circles, arclengths, angles and π

5.3 Secants, tangents and normals

5.4 Polygons

5.5 Triangles

5.5.1 Triangle postulate

5.5.2 Similar triangles

5.5.3 Congruent triangles

5.5.4 Isosceles triangles. Pons Asinorum

5.6 Quadrilaterals

5.6.1 Rectangles and squares

5.6.2 Parallelograms

5.6.3 Rhombuses

5.7 Areas

5.8 Pythagorean theorem

5.8.1 The Pythagorean theorem in three dimensions

5.8.2 Triangle inequality

5.9 Additional results

5.9.1 Three non–coaligned points and a circle

5.9.2 Easy way to find midpoint and normal of asegment

5.9.3 Circles and right triangles

Chapter 6 Real functions of a real variable

6.1 Introducing functions

6.1.1 Powers

6.1.2 Roots

6.1.3 Range and domain

6.1.4 The one–hundred meter dash

6.1.5 Even and odd functions

6.2 Linear independence of functions

6.2.1 Linear independence of 1, x and x^2

6.3 Polynomials

6.3.1 Zeroth–degree polynomials

6.3.2 First–degree polynomials

6.3.3 Second–degree polynomials. Quadratic equations

6.3.4 Polynomials of degree n

6.3.5 Linear independence of 1, x,… , x^n

6.4 Trigonometric functions

6.4.1 Sine, cosine and tangent

6.4.2 Properties of trigonometric functions

6.4.3 Bounds for π

6.4.4 An important inequality

6.4.5 Area of a circle

6.5 Exponentiation. The functions x^α and a^x

6.5.1 Exponentiation with natural exponent

6.5.2 Exponentiation with integer exponent

6.5.3 Exponentiation with rational exponent

6.5.4 Exponentiation with real exponents

6.5.5 The functions x^α and a^x, and then more

6.6 Composite and inverse functions

6.6.1 Graphs of inverse functions

6.7 Single–valued and multi–valued functions

6.8 Appendix. More on functions

6.8.1 Functions, mappings and operators

6.8.2 Functions vs vectors

Chapter 7 Basic analytic geometry

7.1 Cartesian frames

7.1.1 Changes of axes

7.2 Types of representations

7.2.1 Two dimensions

7.2.2 Three dimensions

7.3 Straight lines on a plane

7.3.1 Reconciling two de�nitions of straight lines

7.3.2 The 2 x 2 linear algebraic system revisited

7.4 Planes and straight lines in space

7.4.1 Planes in space

7.4.2 Straight lines in space

7.5 Circles and disks

7.5.1 Polar coordinates

7.5.2 Addition formulas for sine and cosine

7.5.3 Prosthaphaeresis formulas

7.5.4 Properties of triangles and circles

7.6 Spheres and balls

7.7 Conics: the basics

7.7.1 Ellipses

7.7.2 Parabolas

7.7.3 Hyperbolas

7.8 Conics. General and canonical representations

7.9 Conics. Polar representation

7.9.1 Ellipses

7.9.2 Parabolas

7.9.3 Hyperbolas

7.9.4 Summary

7.10 Appendix. “Why conic sections?”

Part II Calculus and dynamics of a particle in one dimension

Chapter 8 Limits. Continuity of functions

8.1 Paradox of Achilles and the tortoise

8.2 Sequences and limits

8.2.1 Monotonic sequences

8.2.2 Limits of important sequences

8.2.3 Fibonacci sequence. The golden ratio

8.2.4 The coast of Tuscany.

8.2.5 Principle of mathematical induction

8.3 Limits of functions

8.3.1 Rules for limits of functions

8.3.2 Important limits of functions

8.3.3 The O and o (“big O” and “little o”) notation

8.4 Continuous and discontinuous functions

8.4.1 Discontinuous functions

8.5 Properties of continuous functions

8.5.1 Successive subdivision procedure

8.5.2 Boundedness vs continuity

8.5.3 Supremum/infimum, maximum/minimum

8.5.4 Maxima and minima of continuous functions

8.5.5 Intermediate value properties

8.5.6 Uniform continuity

8.5.7 Piecewise–continuous functions

8.6 Contours and regions

8.6.1 Two dimensions

8.6.2 Three dimensions

8.6.3 Simply– and multiply–connected regions

8.7 Functions of two variables

8.7.1 Non–interchangeability of limits

8.7.2 Continuity for functions of two variables

8.7.3 Piecewise continuity for two variables

8.7.4 Boundedness for functions of two variables

8.7.5 Uniform continuity in two variables

8.8 Appendix. Bolzano–Weierstrass theorem

8.8.1 Accumulation points

8.8.2 Bolzano–Weierstrass theorem

8.8.3 Cauchy sequence

8.8.4 A rigorous proof of Theorem 73, p. 308

Chapter 9 Differential calculus

9.1 Derivative of a function

9.1.1 Differentiability implies continuity

9.1.2 Differentials. Leibniz notation

9.1.3 A historical note on infinitesimals

9.1.4 Basic rules for derivatives

9.1.5 Higher–order derivatives

9.1.6 Notations for time derivatives

9.2 Derivative of x^α

9.2.1 The case α = 0

9.2.2 The case α = n = 1, 2,…

9.2.3 The case α = –1

9.2.4 The case α = –2, –3,…

9.2.5 The case α = 1/q

9.2.6 The case α rational

9.2.7 The case α real

9.3 Additional algebraic derivatives

9.4 Derivatives of trigonometric and inverse functions

9.5 More topics on differentiable functions

9.5.1 Local maxima/minima of differentiable functions

9.5.2 Rolle theorem and mean value theorem

9.5.3 The l’Hôpital rule

9.5.4 Existence of discontinuous derivative

9.6 Partial derivatives

9.6.1 Multiple chain rule, total derivative

9.6.2 Second– and higher–order derivatives

9.6.3 Schwarz theorem on second mixed derivative

9.6.4 Partial derivatives for n variables

9.7 Appendix A. Curvature of the graph of f(x)

9.7.1 Convex and concave functions

9.7.2 Osculating circle. Osculating parabola

9.7.3 Curvature vs f”(x)

9.8 Appendix B. Linear operators. Linear mappings

9.8.1 Operators

9.8.2 Linear operators

9.8.3 Linear–operator equations

9.8.4 Uniqueness and superposition theorems

9.8.5 Linear mappings

9.9 Appendix C. Finite di�erence approximations

Chapter 10 Integral calculus

10.1 Integrals

10.1.1 Properties of integrals

10.1.2 Integrals of piecewise–continuous functions

10.2 Primitives

10.2.1 Primitives of elementary functions

10.3 Use of primitives to evaluate integrals

10.3.1 First fundamental theorem of calculus

10.3.2 Second fundamental theorem of calculus

10.4 Rules for evaluating integrals

10.4.1 Integration by substitution

10.4.2 Integration by parts

10.5 Applications of integrals and primitives

10.5.1 Separation of variables

10.5.2 Linear independence of sin x and cos x

10.5.3 Length of a line. Arclength

10.6 Derivative of an integral

10.6.1 The cases a = a(x) and b = b(x)

10.6.2 The case f = f(u; x); the Leibniz rule

10.6.3 General case: a = a(x), b = b(x), f = f(u; x)

10.7 Appendix A. Improper integrals

10.7.1 Improper integrals of the first kind

10.7.2 Improper integrals of the second kind

10.7.3 Riemann integrable functions

10.8 Appendix B. Darboux integral

Chapter 11 Rectilinear single–particle dynamics

11.1 Kinematics

11.1.1 Rectilinear motion, velocity and acceleration

11.1.2 The one–hundred meter dash again

11.1.3 What is a frame of reference?

11.2 The Newton second law in one dimension

11.2.1 What is an inertial frame of reference?

11.3 Momentum

11.4 A particle subject to no forces

11.4.1 The Newton first law in one dimension

11.5 A particle subject to a constant force

11.5.1 A particle subject only to gravity

11.6 Undamped harmonic oscillators. Free oscillations

11.6.1 Harmonic motions and sinusoidal functions

11.7 Undamped harmonic oscillators. Periodic forcing

11.7.1 Harmonic forcing with Ω ≠ ω

11.7.2 Harmonic forcing with Ω = ω. Resonance

11.7.3 Near resonance and beats (Ω ≅ ω)

11.7.4 Periodic, but not harmonic, forcing

11.7.5 An anecdote: “Irrational mechanics”

11.8 Undamped harmonic oscillators. Arbitrary forcing

11.9 Mechanical energy

11.9.1 Kinetic energy and work

11.9.2 Conservative forces and potential energy

11.9.3 A deeper look: U = U(x; t)

11.9.4 Conservation of mechanical energy

11.9.5 Illustrative examples

11.9.6 Minimum–energy theorem for statics

11.9.7 Dynamics solution via energy considerations

Chapter 12 Logarithms and exponentials

12.1 Natural logarithm

12.1.1 Properties of ln x

12.2 Exponential

12.2.1 Properties of exponential

12.3 Hyperbolic functions

12.4 Inverse hyperbolic functions

12.4.1 Derivatives of inverse hyperbolic functions

12.4.2 Explicit expressions

12.5 Primitives involving logarithm and exponentials

12.6 Appendix. Why “hyperbolic” functions?

12.6.1 What is u?

Chapter 13 Taylor polynomials

13.1 Polynomials of degree one. Tangent line

13.2 Polynomials of degree two; osculating parabola

13.3 Polynomials of degree n

13.4 Taylor polynomials of elementary functions

13.4.1 Algebraic functions

13.4.2 Binomial theorem

13.4.3 Trigonometric functions and their inverses

13.4.4 Logarithms, exponentials, and related functions

13.5 Lagrange form of Taylor polynomial remainder

13.6 Appendix A. On local maxima and minima again

13.7 Appendix B. Finite differences revisited

13.8 Appendix C. Linearization

Chapter 14 Damping and aerodynamic drag

14.1 Dampers. Damped harmonic oscillators

14.1.1 Dampers and dashpots

14.1.2 Modeling damped harmonic oscillators

14.2 Damped harmonic oscillator. Free oscillations

14.2.1 Small{damping solution

14.2.2 Large–damping solution

14.2.3 Critical–damping solution

14.2.4 Continuity of solution in terms of c

14.3 Damped harmonic oscillator. Harmonic forcing

14.4 Damped harmonic oscillator. Arbitrary forcing

14.5 Illustrative examples

14.5.1 Model of a car suspension system

14.5.2 Time to change the shocks of your car!

14.5.3 Other problems governed by same equations

14.6 Aerodynamic drag

14.6.1 Free fall of a body in the air

14.6.2 A deeper analysis. Upward initial velocity

14.7 Energy theorems in the presence of damping

14.8 Appendix A. Stability of x + px + qx = 0

14.9 Appendix B. Negligible–mass systems

14.9.1 The problem with no forcing

14.9.2 The problem with harmonic forcing

14.9.3 The problem with arbitrary forcing

Part III Multivariate calculus and mechanics in three dimensions

Chapter 15 Physicists’ vectors

15.1 Physicists’ vectors

15.1.1 Multiplication by scalar and addition

15.1.2 Linear independence of physicists’ vectors

15.1.3 Projections

15.1.4 Base vectors and components

15.2 Dot (or scalar) product

15.2.1 Projections vs components in R^3

15.2.2 Parallel{ and normal{vector decomposition

15.2.3 Applications

15.2.4 Dot product in terms of components

15.2.5 Lines and planes revisited

15.2.6 Tangent and normal to a curve on a plane

15.3 Cross (or vector) product

15.3.1 Cross product in terms of components

15.4 Composite products

15.4.1 Scalar triple product

15.4.2 Vector triple product

15.4.3 Another important vector product

15.5 Change of basis

15.6 Gram–Schmidt procedure for physicists’ vectors

15.7 Appendix. A property of parabolas

Chapter 16 Inverse matrices and bases

16.1 Zero, identity and diagonal matrices

16.1.1 The zero matrix

16.1.2 The identity matrix

16.1.3 Diagonal matrices

16.2 The inverse matrix

16.2.1 Definition and properties

16.2.2 The problem A X = B. Inverse matrix again

16.3 Bases and similar matrices

16.3.1 Linear vector spaces R^n and C^n and their bases

16.3.2 Similar matrices

16.3.3 Diagonalizable matrices

16.3.4 Eigenvalue problem in linear algebra

16.4 Orthogonality in R^n

16.4.1 Projections vs components in R^n

16.4.2 Orthogonal matrices in R^n

16.4.3 Orthogonal transformations

16.4.4 Gram{Schmidt procedure in R^n

16.5 Complex matrices

16.5.1 Orthonormality in C^n

16.5.2 Projections vs components in C^n

16.5.3 Hermitian matrices

16.5.4 Positive de�nite matrices in C^n

16.5.6 Gram{Schmidt procedure in C^n

16.6 Appendix A. Gauss–Jordan elimination

16.7 Appendix B. The L-U decomposition, and more

16.7.1 Triangular matrices. Nilpotent matrices

16.7.2 The L-U decomposition

16.7.3 Tridiagonal matrices

Chapter 17 Statics in three dimensions

17.1 Applied vectors and forces

17.2 Newton equilibrium law of a particle

17.2.1 Illustrative examples

17.3 Statics of n particles

17.3.1 General formulation

17.3.2 Statics of spring{connected particles

17.4 Chains of spring{connected particles

17.4.1 Exact formulation for spring–particle chains

17.4.2 Equilibrium in the stretched configuration

17.4.3 The loaded con�guration (planar case)

17.4.4 Linearized formulation (planar case)

17.5 Moment of a force

17.5.1 Properties of moments

17.6 Particle systems

17.6.1 The Newton third law

17.6.2 Resultant force

17.6.3 Resultant moment

17.6.4 Equipollent systems. Torques and wrenches

17.6.5 Considerations on constrained particle systems

17.7 Rigid systems. Rigid bodies

17.7.1 Levers and balance scales

17.7.2 Rigid system subject to weight. Barycenter

17.8 Appendix A. Friction

17.8.1 Static friction

17.8.2 Dynamic friction

17.9 Appendix B. Illustrative examples of friction

17.9.1 Pushing a tall stool

17.9.2 Ruler on two fingers

17.9.3 Particle on a slope with friction, no drag

17.9.4 Particle on a slope with friction and drag

Chapter 18 Multivariate differential calculus

18.1 Differential forms and exact differential forms

18.2 Lines in two and three dimensions revisited

18.2.1 Lines in two dimensions

18.2.2 Lines in three dimensions

18.3 Gradient, divergence, and curl

18.3.1 Gradient and directional derivative

18.3.2 Divergence

18.3.3 The Laplacian

18.3.4 Curl

18.3.5 Combinations of grad, div, and curl

18.3.6 The del operator, and more

18.4 Taylor polynomials in several variables

18.4.1 Taylor polynomials in two variables

18.4.2 Taylor polynomials for n > 2 variables

18.5 Maxima and minima for functions of n variables

18.5.1 Constrained problems. Lagrange multipliers

Chapter 19 Multivariate integral calculus

19.1 Line integrals in two dimensions

19.2 Path{independent line integrals

19.2.1 Simply connected regions

19.2.2 Multiply connected regions

19.3 Double integrals

19.3.1 Double integrals over rectangular regions

19.3.2 Darboux double integrals

19.3.3 Double integrals over arbitrary regions

19.4 Evaluation of double integrals by iterated integrals

19.5 Green theorem

19.5.1 Consequences of the Green theorem

19.5.2 Green theorem vs Gauss and Stokes theorems

19.5.3 Irrotational, lamellar, conservative, potential

19.5.4 Exact di�erential forms revisited

19.5.5 Integrating factor

19.6 Integrals in space

19.6.1 Line integrals in space

19.6.2 Surface integrals

19.6.3 Triple (or volume) integrals

19.6.4 Three{dimensional Gauss and Stokes theorems

19.7 Time derivative of volume integrals

19.8 Appendix. Integrals in curvilinear coordinates

19.8.1 Polar coordinates. Double integrals

19.8.2 Cylindrical coordinates. Volume integrals

19.8.3 Spherical coordinates. Volume integrals

19.8.4 Integration using general coordinates

19.8.5 Orthogonal curvilinear coordinates

19.8.6 Back to cylindrical and spherical coordinates

Chapter 20 Single particle dynamics in space

20.1 Velocity, acceleration and the Newton second law

20.2 Illustrative examples

20.2.1 Particle subject to no forces. Newton first law

20.2.2 Particle subject to a constant force

20.2.3 Sled on snowy slope with friction (no drag)

20.2.4 Sled on snowy slope with friction and drag

20.2.5 Newton gravitation and Kepler laws

20.3 Intrinsic velocity and acceleration components

20.3.1 A heavy particle on a frictionless wire

20.3.2 The circular pendulum

20.3.3 The Huygens isochronous pendulum

20.4 The d’Alembert principle of inertial forces

20.4.1 Illustrative example: a car on a turn

20.5 Angular momentum

20.5.1 Spherical pendulum

20.6 Energy

20.6.1 Energy equation

20.6.2 Potential vs conservative force fields

20.6.3 Potential energy for forces of interest

20.6.4 Minimum–energy theorem for a single particle

20.7 Dynamics solutions, via energy equation

20.7.1 A bead moving along a wire, again

20.7.2 Exact circular pendulum equation

20.8 Appendix A. Newton and the Kepler laws

20.8.1 Newton law of gravitation

20.8.2 Planets on circular orbits

20.8.3 The Kepler laws

20.8.4 Gravity vs gravitation

20.9 Appendix B. Cycloids, tautochrones, isochrones

20.9.1 Cycloids

20.9.2 Tautochrones

20.9.3 From cycloids to tautochrones

Chapter 21 Dynamics of n-particle systems

21.1 Dynamics of n-particle systems

21.2 Dynamics of mass{spring chains

21.2.1 Two–particle chain. Beats. Energy transfer

21.2.2 A more general problem

21.3 What comes next?

21.4 Momentum

21.4.1 Center of mass

21.5 Angular momentum

21.5.1 Useful expressions for angular momentum

21.6 Energy

21.6.1 Kinetic energy. The König theorem

21.6.2 Decomposition of the energy equation

21.6.3 Potential energy for two particles

21.6.4 Potential energy for fields of interest

21.7 Applications

21.7.1 How come cats always land on their feet?

21.7.2 Two masses connected by a spring

21.7.3 The Kepler laws revisited

21.8 Appendix A. Sun–Earth–Moon dynamics

21.9 Appendix B. Potential energy for n particles

21.9.1 Conservative external forces

21.9.2 Conservative internal forces

21.9.3 Energy conservation for conservative fields

Chapter 22 Dynamics in non–inertial frames

22.1 Frames in relative motion with a common point

22.2 Poisson formulas

22.2.1 Transport theorem

22.3 Frames of reference in arbitrary motion

22.3.1 Point of the moving frame

22.3.2 Point in the moving frame

22.4 Apparent inertial forces

22.4.1 Some qualitative examples

22.4.2 Gravity vs gravitation revisited

22.4.3 Mass on a spinning wire

22.4.4 A spinning mass–spring system

22.4.5 Spherical pendulum, again

22.4.6 Foucault pendulum

22.5 Inertial frames revisited

22.6 Appendix A. The tides

22.6.1 A relatively simple model

22.6.2 So! Is it the Sun, or is it the Moon?

22.6.3 Wrapping it up

22.7 Appendix B. Time derivative of rotation matrix

Chapter 23 Rigid bodies

23.1 Kinematics of rigid bodies

23.1.1 Instantaneous axis of rotation

23.2 Momentum and angular momentum equations

23.2.1 Momentum equation

23.2.2 Angular momentum equation

23.2.3 Euler equations for rigid–body dynamics

23.2.4 Principal axes of inertia

23.2.5 On the stability of frisbees and oval balls

23.3 Energy equation

23.3.1 Comments

23.4 Planar motion

23.4.1 Momentum and angular momentum equations

23.4.2 Energy equation

23.5 A disk rolling down a slope

23.5.1 Solution from P_0 to P_1

23.5.2 Solution from P_1 to P_2

23.5.3 Solution after P2

23.6 Pivoted bodies

23.6.1 Illustrative example: my toy gyroscope

23.7 Hinged bodies

23.7.1 Illustrative example: realistic circular pendulum

23.8 Appendix A. Moments of inertia

23.9 Appendix B. Tensors

23.9.1 Matrices vs tensors

23.9.2 Dyadic product. Dyadic tensor

23.9.3 Second–order tensors

23.9.4 Tensor–by–vector product

23.9.5 Tensors components and similar matrices

23.9.6 Transpose of a tensor

23.9.7 Symmetric and antisymmetric tensors

23.9.8 An interesting formula

References

Astronomical data

Index