This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Calculus Illustrated

This is a page of *Calculus Illustrated* by Peter Saveliev, a textbook for undergraduates. Its major feature is an introduction to discrete calculus, which is something everybody should be familiar with.

Below is an early but complete draft. It is about 1000 pages long with about 1300 illustrations.

If you have any questions, please email me or use the facebook page (until it moves elsewhere). Related lectures are posted on this YouTube channel. The channel also shows some physics simulations for Part IV Differential equations.

## Contents

## Part 0: Precalculus

- Chapter 1. Calculus of sequences

1 What is calculus about? 2 The real number line 3 Sequences 4 Repeated addition and repeated multiplication 5 How to find formulas for sequences 6 The algebra of exponents 7 The Binomial Formula 8 The sequence of differences: velocity 9 The sequence of the sums: displacement 10 Sums of differences and differences of sums: motion 11 The algebra of sums and differences

- Chapter 2. Introduction to sets and functions

1 Sets and relations 2 Functions 3 Sequences as functions 4 How numerical functions emerge: optimization 5 How numerical functions emerge: motion 6 Set building 7 The xy-plane: where graphs live... 8 Linear relations 9 Relations vs. functions 10 A function as a black box 11 Give the function a domain... 12 The graph of a function 13 Algebra creates functions 14 Algebra creates functions, continued 15 The arithmetic operations on functions 16 The image: the range of values of a function

- Chapter 3. Compositions of functions

1 Operations on sets 2 Piece-wise defined functions 3 Numerical functions are transformations of the real number line 4 Functions with regularities: one-to-one and onto 5 Compositions of functions 6 The inverse of a function 7 Transforming the axes transforms the plane 8 Change of variables 9 Changing variables transforms the graphs of functions 10 The graph of a quadratic polynomial is a parabola 11 The algebra of compositions 12 Solving equations

- Chapter 4. Classes of functions

1 The simplest functions 2 Monotonicity and extreme values of functions 3 Functions with symmetries 4 Quadratic polynomials 5 Polynomial functions 6 Rational functions 7 The root functions 8 The exponential function 9 The logarithmic function 10 Algebra of logarithms 11 The Euclidean plane: distances 12 From geometry to calculus 13 Trigonometric functions 14 The Euclidean plane: angles 15 Solving inequalities

## Part I: Differential calculus

- Chapter 5. Limits of sequences

1 Limits of sequences: long-term trends 2 The definition of limit 3 Algebra of sequences and limits 4 Can we add infinities? Subtract? Divide? Multiply? 5 More properties of limits of sequences 6 Theorems of Introductory Analysis 7 Compositions 8 Famous limits

- Chapter 6. Limits and continuity of functions

1 Limits of functions: small scale trends 2 Limits under algebraic operations 3 Discontinuity: what to avoid 4 Continuity of transformations 5 Continuity under algebraic operations 6 The transcendental functions 7 Limits and continuity under compositions 8 Continuity of the inverse 9 More on limits and continuity 10 Global properties of continuous functions 11 Large-scale behavior and asymptotes 12 Limits and infinity 13 Continuity and accuracy 14 The ε-δ definition of limit 15 Flowchart for limit computation

- Chapter 7. The derivative

1 The Tangent Problem 2 Location - velocity - acceleration 3 The rate of change: the difference quotient of a function 4 The limit of the difference quotient: the derivative 5 The derivative is the instantaneous rate of change 6 The existence of the derivative: differentiability 7 The derivative as a function 8 Basic differentiation 9 Basic differentiation, continued 10 Shooting a cannon...

- Chapter 8. Differentiation

1 Differentiation over addition and constant multiple: the linearity 2 Differentiation over compositions: the Chain Rule 3 Differentiation over multiplication and division 4 The rate of change of the rate of change 5 Repeated differentiation 6 Change of variables and the derivative 7 Implicit differentiation and related rates 8 Radar gun: the math 9 The derivative of the inverse function 10 Reversing differentiation

- Chapter 9. The main theorems of differential calculus

1 Monotonicity, extreme points, and the derivative 2 Optimization of functions 3 What the derivative says about the difference quotient: The Mean Value Theorem 4 Monotonicity and the sign of the derivative 5 Concavity and the sign of the second derivative 6 Derivatives and extrema 7 Anti-differentiation: the derivative of what function? 8 Antiderivatives 9 The limit of the difference quotient is the derivative

- Chapter 10. Applications of differential calculus

1 Solving equations numerically: bisection and Newton's method 2 How to compare functions, L'Hopital's Rule 3 Linearization 4 The accuracy of the best linear approximation 5 Flows: a discrete model 6 Motion under forces 7 Exponential models 8 Functions of several variables 9 Optimization examples

## Part II: Integral calculus

- Chapter 11. The Riemann integral

1 The Area Problem 2 The total value of a function: the Riemann sum 3 The Fundamental Theorem of Calculus 4 How to approximate areas and displacements 5 The limit of the Riemann sums: the Riemann integral 6 Properties of the Riemann sums and the Riemann integrals 7 The Fundamental Theorem of Calculus, continued

- Chapter 12. Integration

1 Linear change of variables in integral 2 Integration by substitution: compositions 3 Change of variables in integrals 4 Change of variables in definite integrals 5 Trigonometric and other inverse substitutions 6 Integration by parts: products 7 Approaches to integration 8 The areas of infinite regions: improper integrals 9 Properties of proper and improper definite integrals

- Chapter 13. Applications of integral calculus

1 The area between two graphs 2 The linear density and the mass 3 The center of mass 4 The radial density and the mass 5 Flow rate 6 Work 7 The average value of a function 8 Numerical integration 9 Lengths of curves 10 The coordinate system for dimension 3 11 Volumes via cross-sections 12 Volumes of solids of revolution

- Chapter 14. Several variables

1 A ball is thrown... 2 Introduction to parametric curves 3 Introduction to functions of several variables 4 Introduction to calculus of several variables 5 Differential equations 6 The centroid of a flat object 7 Alternative coordinate systems 8 Discrete forms 9 Differential forms

- Chapter 15. Series

1 From linear to quadratic approximations 2 The Taylor polynomials 3 Sequences of functions 4 Infinite series 5 Examples of series 6 From finite sums via limits to series 7 Divergence 8 Series with non-negative terms 9 Comparison of series 10 Absolute convergence 11 The Ratio Test and the Root Test 12 Power series 13 Calculus of power series

## Part III: Calculus in higher dimensions

- Chapter 16. Functions in multidimensional spaces

1 Multiple variables, multiple dimensions 2 Euclidean spaces and Cartesian systems of dimensions 1, 2, 3,... 3 Geometry of distances 4 Sequences and topology in ${\bf R}^n$ 5 The coordinate-wise treatment of sequences 6 Vectors 7 Algebra of vectors 8 Components of vectors 9 Lengths of vectors 10 Parametric curves 11 Partitions of the Euclidean space 12 Discrete forms 13 Angles between vectors and the dot product 14 Projections and decompositions of vectors 15 Sequences of vectors and their limits

- Chapter 17. Parametric curves

1 Parametric curves 2 Limits 3 Continuity 4 Location - velocity - acceleration 5 The change and the rate of change: the difference and the difference quotient 6 The instantaneous rate of change: derivative 7 Computing derivatives 8 Properties of difference quotients and derivatives 9 Compositions and the Chain Rule 10 What the derivative says about the difference quotient: the Mean Value Theorem 11 Sums and integrals 12 The Fundamental Theorem of Calculus 13 Algebraic properties of sums and integrals 14 The rate of change of the rate of change: the second difference quotient and the second derivative 15 Reversing differentiation: antiderivatives 16 The speed 17 Curves vs. parametric curves 18 The curvature 19 The arc-length parametrization 20 Re-parametrization 21 Lengths of curves 22 Arc-length integrals: weight 23 The helix

- Chapter 18. Functions of several variables

1 Overview of functions 2 Linear functions and planes in ${\bf R}^3$ 3 An example of a non-linear function 4 Graphs 5 Limits 6 Continuity 7 The partial differences and difference quotients 8 The average and the instantaneous rates of change 9 Linear approximations and differentiability 10 Partial differentiation and optimization 11 The second difference quotient with respect to a repeated variable 12 The second difference and the difference quotient with respect to mixed variables 13 The second partial derivatives

- Chapter 19. The gradient

1 Overview of differentiation 2 Gradients vs. vector fields 3 The change of a function of several variables: the difference 4 The rate of change of a function of several variables: the gradient 5 Algebraic properties of the difference quotients and the gradients 6 Compositions and the Chain Rule 7 The gradient is perpendicular to the level curves 8 Monotonicity of functions of several variables 9 Differentiation and anti-differentiation 10 When is anti-differentiation possible? 11 When is a vector field a gradient?

- Chapter 20. The integral

1 Volumes and the Riemann sums 2 Properties of the Riemann sums 3 The Riemann integral over rectangles 4 The weight as the 3d Riemann sum 5 The weight as the 3d Riemann integral 6 Lengths, areas, volumes, and beyond 7 Outside the sandbox 8 Triple integrals 9 The n-dimensional case 10 The center of mass 11 The expected value 12 Gravity

- Chapter 21. Vector fields

1 What are vector fields? 2 Motion under forces: a discrete model 3 The algebra and geometry of vector fields 4 Summation along a curve: flow and work 5 Line integrals: work 6 Sums along closed curves reveal exactness 7 Path-independence of integrals 8 How a ball is spun by the stream 9 The Fundamental Theorem of Discrete Calculus of degree 2 10 Green's Theorem: the Fundamental Theorem of Calculus for vector fields in dimension

## Part IV: Differential equations

- Chapter 22. Ordinary differential equations

1 Discrete models: how to set up differential equations 2 Solution sets of ODEs 3 Change of variables in ODEs 4 Separation of variables in ODEs 5 The method of integrating factors 6 Euler's method: back to discrete 7 Qualitative analysis of ODEs 8 How large is the difference between discrete and continuous? 9 Linearization of ODEs 10 Motion under forces: ODEs of second order

- Chapter 23. Vector and complex variables

1 Where do matrices come from? 2 Transformations of the plane 3 Linear operators 4 Examples of linear operators 5 The determinant of a matrix 6 It's a stretch: eigenvalues and eigenvectors 7 Linear operators with real eigenvalues 8 How complex numbers emerge 9 Classification of quadratic polynomials 10 The complex plane ${\bf C}$ is the Euclidean space ${\bf R}^2$ 11 Multiplication of complex numbers: ${\bf C}$ isn't just ${\bf R}^2$ 12 Complex functions 13 Complex linear operators 14 Linear operators with complex eigenvalues 15 Complex calculus 16 Series and power series 17 Solving ODEs with power series

- Chapter 24. Systems of ODEs

1 The predator-prey model 2 Qualitative analysis of the predator-prey model 3 Solving the Lotka–Volterra equations 4 Vector fields and systems of ODEs 5 Discrete systems of ODEs 6 Qualitative analysis of systems of ODEs 7 The vector notation and linear systems 8 Classification of linear systems 9 Classification of linear systems, continued

- Chapter 25. Applications of ODEs

1 Pursuit curves 2 ODEs of second order as systems 3 Vector ODEs of second order: a double spring 4 A pendulum 5 Planetary motion 6 The two- and three-body problems 7 A cannon is fired... 8 Boundary value problems

- Chapter 26. Partial differential equations

1 Heat transfer in dimension $1$: a rod 2 The heat equation with respect to difference quotients 3 The heat equation with respect to derivatives 4 Heat transfer in dimension $2$: a plate 5 Wave propagation in dimension $1$: springs and strings 6 The wave equation with respect to derivatives 7 Wave propagation in dimension $2$: a membrane

- Appendix: Notation
- Appendix: Excel simulations
- Appendix: Modelling projects
- Appendix: What shape of sword is best for cutting? (a prototype project, Part III)
- Appendix: Calculus exercises: advanced
- Appendix: Transformations