Tag Archives: book

Differential equations and SageMath

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The files below were on my teaching page when I was a college teacher. Since I retired, they disappeared. Samuel Lelièvre found an archived copy on the web, so I’m posting them here.

The files are licensed under the Attribution-ShareAlike Creative Commons license.

  1. Partial fractions handout, pdf
  2. Introduction to matrix determinants handout, pdf
  3. Impulse-response handout, pdf
  4. Introduction to ODEs, pdf
  5. Initial value problems, pdf
  6. Existence and uniqueness, pdf
  7. Euler’s method for numerically approximating solutions to DEs, pdf.
    Includes both 1st order DE case (with Euler and improved Euler) and higher order DE and systems of DEs cases, without improved Euler.
  8. Direction fields and isoclines, pdf
  9. 1st order ODEs, separable and linear cases, pdf
  10. A falling body problem in Newtonian mechanics, pdf
  11. A mixing problem, pdf
  12. Linear ODEs, I, pdf
  13. Linear ODEs, II, pdf
  14. Undetermined coefficients for non-homogeneous 2nd order constant coefficient ODEs, pdf
  15. Variation of parameters for non-homogeneous 2nd order constant coefficient ODEs, pdf.
  16. Annihilator method for non-homogeneous 2nd order constant coefficient ODEs, pdf.
    I found students preferred (the more-or-less equivalent) undetermined coefficient method, so didn’t put much effort into these notes.
  17. Springs, I, pdf
  18. Springs, II, pdf
  19. Springs, III, pdf
  20. LRC circuits, pdf
  21. Power series methods, I, pdf
  22. Power series methods, II, pdf
  23. Introduction to Laplace transform methods, I, pdf
  24. Introduction to Laplace transform methods, II, pdf
  25. Lanchester’s equations modeling the battle between two armies, pdf
  26. Row reduction/Gauss elimination method for systems of linear equations, pdf.
  27. Eigenvalue method for homogeneous constant coefficient 2×2 systems of 1st order ODEs, pdf.
  28. Variation of parameters for first order non-homogeneous linear constant coefficient systems of ODEs, pdf.
  29. Electrical networks using Laplace transforms, pdf
  30. Separation of variables and the Transport PDE, pdf
  31. Fourier series, pdf.
  32. one-dimensional heat equation using Fourier series, pdf.
  33. one-dimensional wave equation using Fourier series, pdf.
  34. one-dimensional Schroedinger’s wave equation for a “free particle in a box” using Fourier series, pdf.
  35. All these lectures collected as one pdf (216 pages).
    While licensed Attribution-ShareAlike CC, in the US this book is in the public domain, as it was written while I was a US federal government employee as part of my official duties. A warning – it has lots of typos. The latest version, written with Marshall Hampton, is a JHUP book, much more polished, available on amazon and the JHUP website. Google “Introduction to Differential Equations Using Sage”.

Course review: pdf

Love, War, and Zombies, pdf
This set of slides is of a lecture I would give if there was enough time towards the end of the semester

Integral Calculus and SageMath

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Long ago, using LaTeX I assembled a book on Calculus II (integral calculus), based on notes of mine, Dale Hoffman (which was written in word), and William Stein. I ran out of energy to finish it and the source files mostly disappeared from my HD. Recently, Samuel Lelièvre found a copy of the pdf of this book on the internet (you can download it here). It’s licensed under a Creative Commons Attribution 3.0 United States License. You are free to print, use, mix or modify these materials as long as you credit the original authors.

Table of Contents

0 Preface

1 The Integral
1.1 Area
1.2 Some applications of area
1.2.1 Total accumulation as “area”
1.2.2 Problems
1.3 Sigma notation and Riemann sums
1.3.1 Sums of areas of rectangles
1.3.2 Area under a curve: Riemann sums
1.3.3 Two special Riemann sums: lower and upper sums
1.3.4 Problems
1.3.5 The trapezoidal rule
1.3.6 Simpson’s rule and Sage
1.3.7 Trapezoidal vs. Simpson: Which Method Is Best?
1.4 The definite integral
1.4.1 The Fundamental Theorem of Calculus
1.4.2 Problems
1.4.3 Properties of the definite integral
1.4.4 Problems
1.5 Areas, integrals, and antiderivatives
1.5.1 Integrals, Antiderivatives, and Applications
1.5.2 Indefinite Integrals and net change
1.5.3 Physical Intuition
1.5.4 Problems
1.6 Substitution and Symmetry
1.6.1 The Substitution Rule
1.6.2 Substitution and definite integrals
1.6.3 Symmetry
1.6.4 Problems

2 Applications
2.1 Applications of the integral to area
2.1.1 Using integration to determine areas
2.2 Computing Volumes of Surfaces of Revolution
2.2.1 Disc method
2.2.2 Shell method
2.2.3 Problems
2.3 Average Values
2.3.1 Problems
2.4 Moments and centers of mass
2.4.1 Point Masses
2.4.2 Center of mass of a region in the plane
2.4.3 x-bar For A Region
2.4.4 y-bar For a Region
2.4.5 Theorems of Pappus
2.5 Arc lengths
2.5.1 2-D Arc length
2.5.2 3-D Arc length

3 Polar coordinates and trigonometric integrals
3.1 Polar Coordinates
3.2 Areas in Polar Coordinates
3.3 Complex Numbers
3.3.1 Polar Form
3.4 Complex Exponentials and Trigonometric Identities
3.4.1 Trigonometry and Complex Exponentials
3.5 Integrals of Trigonometric Functions
3.5.1 Some Remarks on Using Complex-Valued Functions

4 Integration techniques
4.1 Trigonometric Substitutions
4.2 Integration by Parts
4.2.1 More General Uses of Integration By Parts
4.3 Factoring Polynomials
4.4 Partial Fractions
4.5 Integration of Rational Functions Using Partial Fractions
4.6 Improper Integrals
4.6.1 Convergence, Divergence, and Comparison

5 Sequences and Series
5.1 Sequences
5.2 Series
5.3 The Integral and Comparison Tests
5.3.1 Estimating the Sum of a Series
5.4 Tests for Convergence
5.4.1 The Comparison Test
5.4.2 Absolute and Conditional Convergence
5.4.3 The Ratio Test
5.4.4 The Root Test
5.5 Power Series
5.5.1 Shift the Origin
5.5.2 Convergence of Power Series
5.6 Taylor Series
5.7 Applications of Taylor Series
5.7.1 Estimation of Taylor Series

6 Some Differential Equations
6.1 Separable Equations
6.2 Logistic Equation

7 Appendix: Integral tables

If you were a math textbook …

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If I were a Springer-Verlag Graduate Text in Mathematics, I would be David Eisenbud’s Commutative Algebra with a view towards Algebraic Geometry.

I am an attempt to write on commutative algebra in a way that includes the geometric ideas that played a great role in its formation; with a view, in short, towards Algebraic Geometry. I cover the material that graduate students studying Algebraic Geometry – and in particular those studying the book Algebraic Geometry by Robin Hartshorne – should know. The reader should have had one year of basic graduate algebra.

Which Springer GTM would you be? The Springer GTM Test