Category Archives: Math

Notes on: “Worldmath Curriculum: Fighting Eurocentrism in Mathematics”

SE Anderson, 1990: “Worldmath Curriculum: Fighting Eurocentrism in Mathematics”.  The Journal of Negro Education

Section one: “A few grim statistics”

The manuscript starts with “A few grim statistics”, looking forward to 2000 and to 2010 and providing numbers about low numbers of Black, Latino, and Native American scholars receiving PhDs in mathematics, physics, and astronomy in the 1980s.

Section two: “The Eurocentric Basis of Mathematics”

The author writes that educators have a duty to build a system “based on the assumption that any person can learn anything”, with the “true beauty of mathematics” coming from the creation of “logical systems that help explain the complexities of Nature”.  (See Joseph 1987 p 22-26).


  1. math is often specifically delinked from materialist concerns
  2. math is confined to an elite group with special gifts
  3. math discovery comes from “deductive axiomatic logic” not from “intuitive or empirical methods”
  4. math results needs to be presented in a particular style, so “new additions to mathematical knowledge” are from a small, special, Eurocentric group.

The narrative of mathematics is presented as going from the Greeks through “European men and their North American descendants”.  “African, Indian, Chinese, or Mayan contributions” receive short mentions in textbooks.

In addition, an unappealing curriculum ends up reinforcing “racist assumptions about people of color”.

Section three: “Six Pedagogical Disasters in Mathematics Education”

  1. “separate arithmetic from algebra”
  2. “teach mathematics without any historical references”
  3. “use textbooks that are elitist and cryptic”
  4. “do work and be tested as an individual” (vs in study groups)
  5. “accept the myth that mathematics is pure abstraction”
  6. “memorize”

1: the author asserts that arithmetic is taught first (for many years), followed by algebra: evidently these subjects are kept separate.  I don’t have any thoughts on this.

2: teaching math as ahistorical: a bunch of European names are attached to math facts and abstractions, without the humans themselves being introduced.

3: cryptic texts: these make math appear unaccessible and not for most people.

4: the individual: math classes are often structured competitively / individually.  this is not how people actually solve problems.

5: real math is abstract: “erudition, abstraction, and compartmentalization”, distance people from creative sources and make math seem unnatural, rather than a natural human act.

6: memorization: math problem solving is often turned into the memorization of definitions, theorems, etc.

Section four: “An alternative curriculum and pedagogy”

See Bob Moses and his “algebra project” (learning algebra in elementary school via subway rides).  See Arthur Powell and the “Writing in Math” project.  See Marilyn Frankenstein and “radical math”.  Look for alternatives “to the arithematic-algebra-precalculus-calculus `learning’ sequence that is so pervasive yet so devastating.”  College-level courses should (1) “show the interconnectedness of mathematics and real-world problems” and (2) should “show how people throughout history have created mathematical techniques to solve problems”.

Section five: “How and what I teach”

  1. Via “psychological upliftment”, “emphasizing that ordinary people create mathematical ideas and ‘do’ mathematics”.
  2. By assuming “the role of a confidence builder”.  Letting students know “they all have the intellectual capabilities to understand the material”
  3. Attributing not understanding to “my own or the textbook’s failure to communicate clearly”
  4. Choosing “the quality of mathematics knowledge” over the quantity.  “I may set out to cover six chapters… if they complete only three or four chapters and learn those well, then I am confident they can pick up the rest”

In a typical algebra class: first two classes focus on historical, cultural, sociopolitical.  Relate math to “humanity’s ongoing struggle to understand Nature”.  And to “capitalism’s attempts to control and dominate Nature”.  Emphasize that “some of the very first mathematical/scientific thinkers were African women”.  “Show how early mathematics and science led to the building of the pyramids, the Great Wall of China, and the road to Kathmandu”.  Astronomy, astrology, iron-smelting, surgery, etc.  Name that Euclid “spent 21 years studying and translating mathematical tracts in Egypt” and “Egypt is in Africa and that the people who inhabit the land were and are Africans”.  Pythagoras also studied in Egypt (and perhaps India).  The theorem attributed to him existed 1000 years before him in Babylonian documents.

The “intent… is to shatter the myth that mathematics was or is a `White man’s thing'”.  “I show how certain aspects of European mathematics could not have developed had not the Europeans traded with more advanced societies”.  Example: “The Vatican denounced Hindu-Arabic numerals”.

The research university of Bait al-Hikma is important because of Mohammed ibn-Musa al-Khwarizmi (the name “al-djabr or `algebra’ comes from a text he wrote) and the term “algorithms” is a corruption of his name (dating to his second book, “Algorithmi de Numero Indorum”).  “I also mention… the algebraist Omar Khayyam (c. 1050 – 1122 AD)”.  See Nasir Eddin al-Tusi for non-Euclidean geometry, as well.

In the 1600s, when Europeans were trying to build very large ships to carry African slaves, they used the knowledge of “Gambian, Chinese, and Indian mariners”.  “Developments in hydrodynamics (and its attendant mathematics) contributed … to the horror of .. the slave trade”.

“I further point that that calculus was created to facilitate the study of ballistics” in wars by England and Germany.  Military needs “continue to inspire many mathematicians and scientists to purse the War Machine”.

Structuring class: (1) “About two weeks into a class, I facilitate the creation of study groups” (3-4 people whom students choose).  These groups are used in and out of class.  The groups complete progress reports (no tests).  (2) “I also incorporate a weekly 15- to 20-minute class discussion” of a news article from the Science Times, to emphasize the relationships between math and the social / natural sciences.  (3) students are asked to make notecards of facts that they should bring to class.  (4) computational tools are encouraged.

Section six: “Conclusion”

“a subtle but effective form of educational genocide is taking place”.  “To offer an alternative that is genuinely egalitarian and truthful we must open our eyes to the centrality of the contributions made by the vast majority of the world’s people”.

Notes on Calculus Blue Volume 1, Chapters 3, 4, 5, 6

More of Calculus Blue by Prof Ghrist Math.

Chapter 3 is on coordinates and distance (see section 12.1 of the 6th edition of Hughes-Hallett).

01.03 (0:36) “Coordinates: intro”.  Review coordinates and see it in data.

01.03.01 (2:16) “coordinates & many dimensions”.  from curves and surfaces we’ll head on.  plane, then 3-space.  what next?  He introduces R-n.  He mentions that the coordinates may have units and that the coordinate is a point in higher dimensional space.  But why care?  (see the next examples).

01.03.02 (2:33) “Example – robot kinematics”.  robot arm with a bunch of joint angles.  There is a configuration space.  He shows a video: the tip of a robot arm is tracing a path in 3-space.

01.03.03 (1:42) “Example – wireless signals & localization”.  the phone can sense a bunch of wireless signals at once and multiple might be nonzero, so the phone has a position in signal space.  the dimension is the number of wireless routers in the building.

01.03.04 (2:04) “Example – customer preferences & profiles”.  create a preference space of how customers feel about a bunch of products.  You’ll want to cluster (group) this dataset.

01.03.05 (7:27) “Distances via coordinates”.  we need to built up algebra and geometry.  start with distance.

Examples: (1) distance from point to line (where the line is parameterized), so the shortest distance; this could be done in single variable calculus: he foreshadows that we’ll learn a better way.  (2) configuration space for four objects on a chess board, so 8 coordinates, and take a distance in that space.  (3) maximal distance between two points in a unit ball in 49-dimensional space: it is 2.  (4) how about a 49-cube?  it is 7, which is weird.

we still need more tools (not quite time for calculus).

01.03 (0:36) “the big picture”.  higher dimensional spaces exist in systems and in data.

Total time: 17:14

Chapter 4 is an intro to vectors (see sections 13.1 and 13.2 of the 6th edition of Hughes-Hallett).

01.04 (0:38) “Vectors: intro”.  This is a tool for organizing variables or data.

01.04.01 (2:38) “Vector components”.  one way to think of them is as a difference between two points.  Another is as two objects that can be added and rescaled.  We’ll work with a coordinate system and interpret vectors as movement in that space.  Stack the components vertically in vectors.  Use an underline to denote a vector.

01.04.02 (6:32) “Basic vector algebra”.  algebra: addition, rescaling, using components and acting term by term.  properties: commutative, identity, subtraction.  geometry: concatenation, so draw u, then v, then the sum.  Define the length of a vector.  Things to prove: triangle inequality, and a couple of others.  Lines and planes can be nicely parameterized using vectors. 1d line, 1 parameter, 1 vector.  2d plane, 2 parameters, 2 vectors.  Nice animation of how the two vector are used to parameterize the plane.

01.04.03 (3:52) “Standard basis vectors”.  vec i, vec j, vec k are introduced, as are vec e_k.

example: (1) take a vector and write it as a linear combination of the vec e_k vectors.  (2) do the same for a vector in 3d using vec i, vec j, vec k.  (3) Take the length of a vector sum.

01.04.04 (1:53) “Caveat & a foreshadowing of fields”.  vectors are actually independent of how you represent them.  where’s the calculus?  we need more background!  At some point, though, we’ll learn a calculus for fields of vectors (foreshadowing of vector fields).

01.04 (0:26) “The big picture”.  Vectors carry both algebraic and geometric data.  This was our intro to them.

Total time: 16:00

Chapter 5 is on dot products (so section 13.3 of the 6th edition of Hughes-Hallett)

01.05 (0:41) “the dot product: intro”.  good data structure: geometry and algebra.

01.05.01 (1:23) “definition of the dot product”.  define it.  properties: commutative, dot with zero is zero, dot product with itself is length.

01.05.02 (3:27) “dot products & orthogonality”.  the angle between two vectors is well-defined.  memorize the geometric definition of the dot product.  use dot products to detect orthogonality.

example: (1) find an angle between two vectors with four components.  (2) can create a pair of vectors that are orthogonal.  (3) the standard basis vectors are all mutually orthogonal.

01.05.03 (3:32) “dot products as orthogonal projection”.  projected length is an important interpretation of the dot product (oriented, projected, length along an axis).  Really great animation / visualization for this!

example: (1) find the component of one vector in the direction of another.

01.05.04 (2:48) “hyperplanes & machine learning”.  use the dot product to make sense of our implicit equations for lines and planes.  Another nice animation / visualization.  hyperplanes (a “support vector machine”) separate two types of data.  with a “normal vector” to the plane you can tell which side of the plane a data point is on.

01.05.05 (1:54) “dot products and compatibility”.  love: create a preference space with a bunch of opinions.  make two vectors and then take a dot product.  a large positive dot product means two people like similar things.

01.05.06 (1:36) “foreshadowing of Fourier”.  additional math: you could learn to think of functions as infinite-dimensional vectors.  black and white foreshadowing video.

01.05 (0:27) “the big picture”.  the dot product, with algebra, geometry and applications!

Total time:  15:48

Chapter 6 is on cross products (section 13.4 of the 6th edition of Hughes-Hallett).

01.06 (0:41) “The cross product: intro”.  two more products (unique to 3d).

01.06.01 (5:35) “definition of the cross product”.  (only in 3d).  he defines it.  properties.  anti-commutativity, vec u cross vec 0 = vec 0.  cross product with itself is zero.  Look at the geometric meaning of that anti-commutativity.  The cross-product is orthogonal to both of its factors (he proves this).  Then he shows a visualization and defines the right hand rule.  The illustration involved a spinning mill.  🙂

examples: (1) three points, PQR, and find the equation of a plane by finding an orthogonal vector and put it into the formula for a plane.

01.06.02 (2:20) “computing cross products in the standard basis”.  How to remember the cross-product formula?  Use the standard basis vectors, then draw the cyclic diagram

examples: (1) find the cross product by using the standard basis vectors and expanding.

01.06.03 (2:45) “length of the cross product”.  geometric formula for the length of the cross product (a geometry problem).  the vector isn’t just orthogonal to the vec u vec u plane; its length also has a meaning.  use the cross product for a simple formula for a point to a line and the dot product for a formula from the point to a plane.

01.06.04 (3:53) “the scalar triple product”.  this product takes in 3 vectors and returns a scalar.  he gives the algebraic definition.  then he shows the cyclic repeat (5 columns) with diagonal slices for putting together the structure.  properties: there’s a cyclic permutation, anti-symmetry, and geometric meaning.  It’s the volume of a parallelepiped.

01.06.05 (1:12) “bonus: octonians”.  how about a way to multiply vectors together in another dimension?  the octonions work in the 7th dimension.  you can look them up…

01.06 (0:31) “the big picture”.  new products: cross product and scalar triple product.

Total time: 16:57

Notes on “Calculus Blue” Volume 1, Chapter 2

More notes on the Calculus Blue Multivariable Volume 1 videos on YouTube by “Prof Ghrist Math”.

Chapter 2 introduces curves in the plane and surfaces in 3-space with implicit and parametric definitions for curves in the plane and for surfaces in 3-space.  They also introduce the names and images for all of the quadratic surfaces (so we’ve left the linear world in this chapter).  This is a foreshadowing of the process of the course (start with the linear and then move to the nonlinear).

01.02.00 (0:35) “Curves and surfaces: Intro”

01.02.01 (5:10) “Implicit and parametric curves and surfaces”.  Two ways to define curves.  Implicitly or parametrically.  Implicit: “the solutions to an equation yields a curve in the plane”.  Parametric: “specifying coordinates as a function of a parameter”.  We’ll want to move between these representations.  Surfaces in 3d can also be expressed implicitly or parametrically (requires 2 parameters).

Examples: parameterization for a surface expressed implicitly (he shows the conversion where you set x = s, y = t, and then express z); go from a parameterization to an implicit equation (the example has a square root so he suggests caution).

01.02.02 (0:29) “Some examples please…?”.  it’s important to learn the quadratic surfaces.

01.02.03 (5:33) “Examples of quadratic surfaces”.  start with the sphere, then modify to get the ellipsoids, change a sign to get a hyperboloid, with two negatives signs get a 2-sheeted hyperboloid, then an elliptic paraboloid, hyperbolic paraboloids, cones (degenerate hyperbola), cylinders.

Then the narrator provides a reassurance: this isn’t about memorizing these or drawing pictures of these; it’s just worth recognizing them.  Then the narrator mentions these won’t come back for a while and names surface integrals as an application.  These are in there as a motivation to build up geometry and algebra skills.

01.02 (0:26) “The big picture”.  Previous chapter was lines and planes.  This one was curves and surfaces.  Progression from linear to nonlinear is what will happen in the course.

Notes on “Calculus Blue” Volume 1, Chapter 1

These notes are on the Calculus Blue videos by Ghrist on YouTube.  He emphasizes that the math will involve substantial (and worthwhile) work, which I really appreciate.

01 (0:51) “Vectors & matrices: Intro”  “Your journey is not a short one”.  To learn “calculus, the mathematics of the nonlinear”, prepare with “the mathematics of the linear”.

01 (3:25) “Prologue” definition of multivariate (multiple inputs and multiple outputs).  Asks why do we care?  (graphs of surfaces, arbitrary dimensions).  linear algebra will “help us do calculus”.  “Calculus involves approximating nonlinear functions with linear functions”, so start with “the mathematics of linear multivariable functions”.

Why learn about vectors and matrices?  machine learning, statistics, information from data, geometry (distance, area, volume), determinants will help calculate areas and volumes.

algebra + work + fun.

01.01.00 (0:35) “Lines & planes: intro”.

01.01.01 (3:50) “Formulae for lines & planes”.  Lines in the plane: y = mx + b, (y-y0) = m(x-x0) (point slope form), x/a + y/b = 1 (intercept form).

Example: a line passing through a point with a particular slope; a line passing through two points.

Orthogonal: (the orthogonal slope is the negative reciprocal).

01.01.02 (3:33)  “Implicit planes in 3d”.  These are analogous to lines in the plane.  n1(x-x0) + n2(y-y0)+n3(z-z0) = 0 (point slope form).  x/a + y/b + z/c = 1 (intercept form) –> he says intercept form shows up in economics.

example: equation of a plane passing through a point and parallel to another plane.

01.01.03 (5:21) “parameterized lines in 3d”.  add an “auxiliary variable” (a parameter).  x(t) = 3r-5, y(r) = r+3, z(r) = -4r+1.  The name of the parameter doesn’t matter, and shifts or changes to the parameter that happens in all three equations doesn’t matter.

Examples: find a line through two points in 3-space; find a line orthogonal to a plane and through a specific point.

What will happen in higher dimensions?  “hyperplanes”, “subspaces”.

01.01.04 (1:52) “Bonus! Machine learning”.  hyperplanes come up in analyzing data.  A space of images.  A “support vector machine is a hyperplane that optimally separates two types of data points”.  The video illustrates how flat planes usually won’t cut it to separate two datasets, so we’ll need nonlinear ideas (i.e. calculus).

01.01 (0:25) “The big picture”: lines and planes are the start of the story!

Notes on “How a Detracked Mathematics Approach Promoted Respect, Responsibility, and High Achievement”

Boaler, 2006.  “How a Detracked Mathematics Approach Promoted Respect, Responsibility, and High Achievement”.   Theory Into Practice, 45:1, 40-46

This article is about a high school math program with high and equitable math achievement, where mixed-ability approaches led to “higher overall attainment and more equitable outcomes”.  The students in this study developed “extremely positive intellectual relations” with peers across culture, social class, gender, and attainment “through a collaborative problem-solving approach”.

The article describes a problem solving approach that was used ad the school and that enables these outcomes.  The problem solving approach (“complex instruction”) involved “additional strategies to make group work successful”.  The author identifies seven factors: “The first four (multidimensional classrooms, student roles, assigning competence, and student responsibility) are recommended in the complex instruction approach; the last three (high expectations, effort over ability, and learning practices) were consonant with the approach and they were important to the high and equitable results that were achieved.”


Ingredients in the approach:

(1) Multidimensionality:  In some classrooms success is about “executing procedures correctly and quickly”.  Here, success requires a range of abilities where “no one student ‘will be good on all these abilities’ and … each student will be ‘good on at least one'”.  Giving students “group-worthy problems”: “open-ended problems that illustrated important mathematical concepts, allowed for multiple representations, and had several possible solution paths (Horn, 2005).”  Students were able to identify: “asking good questions, rephrasing problems, explaining well, being logical, justifying work, considering answers, and using manipulatives” as contributing to success in mathematics.

This breadth was key: that there are multiple paths to an answer, with interaction and explanation central to the work.

(2) Roles: “facilitator, team captain, recorder or reporter, or resource manager”.  If each student has something important to do in the group, they are needed for the group to work.  The teachers reinforced the centrality of each role by pausing to ask facilitators to help with answer checking, etc.  This helps with the reliance of students on each other.

(3) Assigning competence: “public, intellectual, specific” feedback that is also relevant can lift students up.  This can reinforce the breadth of contributions that are valued.  I suppose I can imagine naming that a student has done a great job questioning how the problem worked or digging deeper into the underlying concept.  Specificity is important so that students know what is being praised.

(4) Student responsibility: creating responsibility for each other’s learning, and taking that seriously by “rating the quality of conversations groups had”, or giving “group tests” (this comes in multiple flavors).  In one version of a group test, the students work through the test together, but write it up individually, and the instructor grades only one of the individual write-ups (at random).  That will then be the grade on the test for all of the students in the group.  Another way to create inter-student responsibility is to ask a follow up question to one student in the group, and if they can’t answer it, give the group more time to talk together before returning to that same student with the question.

Justification and reasoning were also centered.  They emphasized to students the responsibility “to help someone who asked for help, but also to ask if they needed help”.

(5) High expectations: complex problems with high-level follow-up.  “Teachers would leave groups to work through their understanding rather than providing them with small structured questions that led them to the correct answer”.

(6) Effort over ability: math success is about hard work and continuing to try.  This message needs to come through.

(7) Learning practices: point out the process of what students are doing (things like fully formulating a question that they want to ask, or thinking about whether their answer is reasonable).



Relational equity: this was a learning outcome of being in the classroom, where students developed respectful relationships.


For more on Complex Instruction, see:

Cohen, E. (1994). Designing groupwork. New York: Teachers College Press.

Cohen, E., & Lotan, R. (Eds.). (1997). Working for equity in heterogeneous classrooms: Sociological the- ory in practice. New York: Teachers College Press.

Reading “High School Algebra Students Busting the Myth about Mathematical Smartness: Counterstories to the Dominant Narrative “Get It Quick and Get It Right””

Dunleavy 2018, “High School Algebra Students Busting the Myth about Mathematical Smartness: Counterstories to the Dominant Narrative “Get It Quick and Get It Right””.  Education Sciences.

I’m reading a paper about a high school Algebra I course that uses the principles of “complex instruction” (which I still need to look up).  The article highlights how students in the class, through their group process, made time for multiple solution strategies (so were not racing to complete problems), valued explanations and justification, felt “assigned competence” by their teacher (validating their contributions), and saw it as their role to help each other.

For more on complex instruction (references cited in the paper):

  1. Boaler, J.; Staples, M. Creating mathematical futures through an equitable Teaching Approach: The Case of Railside School. Teach. Coll. Record 2008, 110, 608–645.
  2. Jilk, L.M.; Erickson, S. Shifting students’ beliefs about competence by integrating mathematics strengths into tasks and participation norms. In Access and Equity: Promoting High-Quality Mathematics in Grades 6–8; Fernandes, A., Crespo, S., Civil, M., Eds.; NCTM: Reston, VA, USA, 2017; pp. 11–26
  3. Dunleavy, T.K. Delegating Mathematical Authority as a means to Strive toward Equity. JUME 2015, 8, 62–82
  4. Featherstone, H.; Crespo, S.; Jilk, L.M.; Oslund, J.A.; Parks, A.N.; Wood, M.B. Smarter Together! Collaboration nd Equity in the Elementary Math Classroom; National Council of Teachers of Mathematics: Reston, VA, USA, 2011.
  5. Horn, I.S. Strength in Numbers: Collaborative Learning in the Secondary Classroom; National Council of Teachers of Mathematics: Reston, VA, USA, 2012
  6. Cohen, E.; Lotan, R. Designing Groupwork: Strategies for the Heterogeneous Classroom, 3rd ed.; Teachers College Press: New York, NY, USA, 2014
  7. Cohen, E. Equity in heterogeneous classrooms: A challenge for teachers and sociologists. In Working for Equity in Heterogeneous Classrooms: Sociological Theory in Practice; Cohen, E.G., Lotan, R.A., Eds.; Teachers College Press: New York, NY, USA, 1997; pp. 3–14.
  8. Dunleavy, T.K. “Ms. Martin is secretly teaching us!” High school Mathematics Practices of a Teacher Striving toward Equity. Ph.D. Thesis, University of Washington, Seattle, WA, USA, 2013
  9. Horn, I. Fast kids, slow kids, lazy kids: Framing the mismatch problem in mathematics teachers’ conversations. J. Learn. Sci. 2007, 16, 37–79.

Meiss: Differential Dynamical Systems (chaos)

I am reading James Meiss’ text Differential Dynamical Systems (SIAM).  I am specifically interested in how he tells the story of chaos.

In the Preface, he mentions the following: That  Chapter 5 focuses on invariant manifolds:

  • stable and unstable sets
  • heteroclinic orbits
  • stable manifolds
  • local stable manifold theorem
  • global stable manifolds
  • center manifolds

That the “stable and unstable manifolds, proved to exist for a hyperbolic saddle, give rise to one prominent mechanism for chaos — heteroclinic intersection”.

That Chapter 7 is background for understanding chaos (“Lyapunov exponents, transitivity, fractals, etc”):

  • chaos
  • Lyapunov exponents / definition / properties
  • strange attractors / Hausdorff dimension / strange, nonchaotic attractors

And that in Chapter 8 he’ll discuss Melnikov’s method (onset of chaos): sections 8.12 and 8.13.

He notes that he doesn’t discuss discrete dynamics (maps).


After the preface, doing a word search for “chaos” or “chaotic”:

Chaos next comes up in the examples in section 1.4: Meiss introduces an example called the “ABC flow” from Arnold 1965.  He mentions this is a “prototype chaotic system” and introduces the idea that “nearby trajectories will often diverge exponentially quickly in time”.  Then he defines the Lyapunov exponent.

Section 1.7 is about Quadratic ODEs: the simplest chaotic systems, after the Lorenz model is introduced in section 1.6.  The introduction of the Lorenz model includes an image of the setup, a mention of convective rolls, the idea of the Galerkin truncation, etc.  So he introduces this system by deriving the ODEs for it.

In section 1.7 he says “informally, chaos corresponds to aperiodic motion that exhibits ‘sensitive dependence on initial conditions'”.  He’ll provide a formal definition in chapter 7.  He mentions that 3-dimensional systems “are the lowest dimensional autonomous ODEs that can exhibit chaos”.  There is a chart of Sprott’s quadratic chaotic differential equations (the simplest quadratic systems with chaos).

In section 4.1 Definitions, “orbits can be quasiperiodic, aperiodic, or chaotic”.  When he introduces orbits, he introduces the idea of a periodic orbit as well as other options.

Meiss returns to chaos in section 5.2 Heteroclinic orbits.  (See Diacu and Holmes 1996 for the story of Poincare, his mistake, and its correction).  He defines a heteroclinic orbit as an orbit that is backward asymptotic to one invariant set and forward asymptotic to a different one.  The homoclinic orbit (doubly asymptotic) is then a special case that is forward and backward asymptotic to the same invariant set.

In a 2d system, if a branch of W^u intersects a branch of W^s then the branches coincide.  Orbits that separate phase space are called separatrices: “they separate phase space into regions that cannot communicate”.  In section 8.13, we’ll see that higher-dimensional systems are different from 2d ones, and that this doesn’t have to happen!  Meiss also defines “saddle connection” and mentions that Hamiltonian systems in the plane often have separatrices.

Chaos comes up again in section 5.5 Global stable manifolds.  The global set comes from flowing the local set backward in time.  For finite time, it will be smooth.  To think about its structure in general, Meiss introduces the idea of an “embedding”.  He also defines an “immersion” and notes that “an immersion is locally a smooth surface”.  Note that immersions can cross themselves.  The topologist’s sine curve is an example that “has infinitely many oscillations and accumulates upon the interval [-1,1] on the y-axis”.  “We will see later that the global stable manifold can have this accumulation problem: indeed, this is one of the indications of chaos”.

Next, in section 6.6, when the Poincaré-Bendixson theorem is introduced, it is introduced as a statement that “There is no chaos in two dimensions”.  So Meiss is building intuition for the idea of chaos from the very first day in the course, and is distinguishing between it and what happens in 2d, even as he introduces 2d.

Chapter 7 focuses on chaotic dynamics, so is where the story will be more fully built out.  The chapter begins with quotes from Poincaré and Lorenz and an informal definition.  To formalize it will require defining “unpredictable” and “sensitive dependence”.  This chapter occurs before the chapter on bifurcations.

Two very simple linear examples with sensitive dependence are given to build some intuition for the stretching between nearby trajectories that happens for some initial conditions in a system with sensitive dependence, and to show that sensitive dependence alone is not “chaotic”.

Aperiodicity or “wanders everywhere” on an invariant set is the next idea introduced, leading to a definition of “transitive”.  Then “a flow is chaotic on a compact invariant set X if the flow is transitive and exhibits sensitive dependence on X”.  This gives us an idea of mixing and unpredictability.

For a lot of systems, their trajectories look chaotic “when solved numerically”.  Chaos was verified in the Lorenz system for r = 28 in Tucker 2002.

Note: I also should read the text for references to the Lorenz system, because that system is used as an example.


Blanchard, Devaney, and Hall 3rd edition (2006): Differential Equations. Sections 1.1-1.4, 1.8

Chapter 1: First order differential equations.  They present a goal: predicting a future value of a quantity modeled by a differential equation.

  • Section 1.1a.  Modeling via differential equations.  a: Introduce the idea of a model.  Distinguish between the independent variable (time), dependent variables (dependent on time) and parameters (don’t depend on time but can be adjusted).
  • Section 1.1b.  Modeling via differential equations.  b: Unlimited population growth.  P’ = k P is the equation (exponential growth).   Define first-order, ordinary differential equation, equilibrium solution, initial condition, qualitative analysis.  Introduce initial-value-problem, and solution.  Guess and check method of finding a solution.  Particular solution vs general solution.  Example comparing to United States population (annual census since 1790).
  • Section 1.1c.  Modeling via differential equations.  c: Logistic population growth.  Add a second assumption (at some level of population growth will become negative).  Logistic population model, nonlinear, equilibria.  They do a qualitative analysis and create approximate solutions.
  • Section 1.1d.  Modeling via differential equations.  d: Predator prey systems.  Add assumptions about fox and rabbit interactions.  They generate a first order system and define the solution to a system.
  • Section 1.1e.  Modeling via differential equations.  e: Analytic, qualitative, and numerical approaches.  Here they name that there are three approaches.
  • Section 1.2a.  Analytic technique: separation of variables.  a.  What is a differential equation and what is a solution?
  • Section 1.2b.  Analytic technique: separation of variables.  b.  Initial-value problems and the general solution.
  • Section 1.2c.  Analytic technique: separation of variables.  c.  Initial-value problems and the general solution.
  • Section 1.2d.  Analytic technique: separation of variables.  d.  Separable equations
  • Section 1.2e.  Analytic technique: separation of variables.  e.  Missing solutions
  • Section 1.2f.  Analytic technique: separation of variables.  f. Getting stuck
  • Section 1.2g.  Analytic technique: separation of variables.  g. A savings model
  • Section 1.2h.  Analytic technique: separation of variables.  h. A mixing problem
  • Section 1.3a.  Qualitative technique: slope fields.  a. The geometry of dy/dt = f(t,y)
  • Section 1.3b.  Qualitative technique: slope fields.  b. Slope fields
  • Section 1.3c.  Qualitative technique: slope fields.  c. Important special cases
  • Section 1.3d.  Qualitative technique: slope fields.  d. Analytic versus qualitative analysis
  • Section 1.3e.  Qualitative technique: slope fields.  e. The mixing problem revisited
  • Section 1.3f.  Qualitative technique: slope fields.  f. An RC circuit
  • Section 1.3g.  Qualitative technique: slope fields.  g. Combining qualitative with quantitative results
  • Section 1.4a.  Numerical technique: Euler’s method.  a. Stepping along the slope field
  • Section 1.4b.  Numerical technique: Euler’s method.  b. Euler’s method
  • Section 1.4c.  Numerical technique: Euler’s method.  c. Approximating an autonomous equation
  • Section 1.4d.  Numerical technique: Euler’s method.  d. A non-autonomous example
  • Section 1.4e.  Numerical technique: Euler’s method.  e. An RC circuit with periodic input
  • Section 1.4f.  Numerical technique: Euler’s method.  f. Errors in numerical methods
  • Leave existence and uniqueness (1.5), equilibria and the phase line (1.6), bifurcations (1.7), integrating factors (1.9) to a later course.
  • Section 1.8a.  Linear equations.  a.  Linear differential equations
  • Section 1.8b.  Linear equations.  b.  Linearity principles
  • Section 1.8c.  Linear equations.  c.  Solving linear equations
  • Section 1.8d.  Linear equations.  d.  Qualitative analysis
  • Section 1.8e.  Linear equations.  e.  Second guessing


Varburg and Purcell 7th edition. Differential equations (mainly chapter 18)

  • Section 5.2: What is a diff eq?  Provides an example and two solution methods before defining diff eq (and doesn’t define a solution…).  Then presents separation of variables via an example.  Then a falling body example and an escape velocity example.
  • Section 7.5: exponential growth and decay.  They motivate y’ = ky via population growth, then solve by separation.  Example 1: doubling time.  Example 2: growth time.  Example 3: radioactive decay.  Example 4 and 5: compound interest.
  • Extra note: in section 3.10 they present little-o notation.
  • Chapter 18: differential equations.  Section 18.1: linear first order equations.  They define “differential equations”, “ordinary differential equation of order n”, “solution”, “general solution”, “particular solution”, “linear”.  They introduce the method of integrating factors, and apply to a mixture problem, to a circuit, and to a battery.
  • Section 18.2: second order homogeneous equations.  They define “independent” for solutions, the auxiliary equation, and use it to provide a solution to a diff eq with constant coefficients.  They don’t intro Euler’s formula but assume it in the complex roots example.  Then on to higher order equations.
  • Section 18.3: the nonhomogeneous equation.  they provide general / particular solution info, do the method of undetermined coefficients, and variation of parameters.
  • Section 18.4: Applications of second order equations.  A vibrating spring, simple harmonic motion, damping, overdamped, critically damped.   Electric circuits.
  • This text goes through a laundry-list of methods.  Perhaps not so much motivation.

Courant (and John) 1965, Differential Equations: Chapter 9.

In the intro to Chapter 9 they note that we’ve already seen differential equations in Chapter 3, p. 223, and on p.312, and in Chapter 4 (see p 405).  So I’ll start there.

  • Section 3.4: First encounter: in “Some Applications of the Exponential Function”, y’ = ay is introduced.  “Since Eq. (8) expresses a relation between the function and its derivative, it is called the differential equation of the exponential function”.  They show that the exponential function is the unique solution (this argument is worthwhile, actually, because it is a small proof).
  • Section 3.4: more y’ = ay.  Examples associated with the exponential function: compound interest, radioactive decay, Newton’s law of cooling, atmospheric pressure with height above the surface of the Earth,  the law of mass action (chemical reactions), switching on and off an electric circuit.  Newton’s law of cooling,  the law of mass action, and the electric circuit involve differential equations.
  • Section 3.16a: Differential equations of trigonometric functions.  In 3.16a they intro diff eqs.  Diff eqs move beyond equations y’ = f(x) to “more general relationships between y and derivatives of y”.
  • Section 3.16b: Define sine and cosine via a differential equation (u” + u = 0) and an initial condition.  “Any function u = F(x) satisfying the equation, …, is a solution.”  They then show that shifts of solutions are solutions and linear combos are solutions, and scalar multiples are solutions, so the properties of linearity.  Initial conditions single out a specific solution.  They also derive cos(x+y) = cos x cos y – sin x sin y using the differential equation.  They also note that pi/2 can then be defined via “the smallest positive value of x for which cos x = 0.”
  • Section 4.4a: Newton’s law of motion, a relationship “from which we hope to determine the motion”.  They define diff eq and solution again.
  • Section 4.4b/c: Motion of falling bodies and motion constrained to a curve.
  • Section 4.5: free fall of a body in the air (find terminal velocity under this model)
  • Section 4.6: simplest elastic vibration: motion of a spring.
  • Section 4.7abcde: motion on a given curve.  The differential equation and its solution.  Particle sliding down a curve.  Discussion of the motion.  The ordinary pendulum.  The cycloidal pendulum.
  • Section 4.8abc: Motion in a gravitational field.  Newton’s universal law of gravitation.  Circular motion about the center of attraction.  Radial motion – escape velocity.
  • Chapter 9:  They summarize the differential differential equations that have been encountered above.  This chapter is differential equations for the simplest types of vibration.
  • 9.1ab: Vibration problems of mechanics and physics.  The simplest mechanical vibrations (forced second order constant coefficient equation).  Electrical oscillations (similar).
  • 9.2abc: Solution of the homogeneous equation.  Free oscillations. a:The formal solution.  b:Interpretation of the solution.  c: Fulfilment of given initial conditions. Uniqueness of the solution. In the formal solution they construct the characteristic equation and distinguish the three cases of roots.  For the solutions in complex form they introduce Euler’s formula.  In interpretation they introduce “damping”, “damped harmonic oscillations”, “attenuation constant”, “natural frequency”.
  • 9.3abcde: The nonhomogeneous equation.  Forced oscillations.  a: general remards.  Superposition.  b: Solution of the nonhomogeneous equation.  c: The resonance curve.  d: Further discussion of the oscillation.  e: Remarks on the construction of recording instruments.