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).

Outcome:

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):

Boaler, J.; Staples, M. Creating mathematical futures through an equitable Teaching Approach: The Case of Railside School. Teach. Coll. Record 2008, 110, 608–645.
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
Dunleavy, T.K. Delegating Mathematical Authority as a means to Strive toward Equity. JUME 2015, 8, 62–82
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.
Horn, I.S. Strength in Numbers: Collaborative Learning in the Secondary Classroom; National Council of Teachers of Mathematics: Reston, VA, USA, 2012
Cohen, E.; Lotan, R. Designing Groupwork: Strategies for the Heterogeneous Classroom, 3rd ed.; Teachers College Press: New York, NY, USA, 2014
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.
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
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.

Course calendars: making a course schedule

I am thinking about how to build the schedule for my course. Even though I’ve taught it before, scheduling is intimidating. It has a few components:

There are topics that will be uncovered over the semester. Each topic has associated informational materials (videos and text), typical student questions, follow-up check yourself questions, an in class activity, problem set question(s), skills and procedures, and assessment questions.
The course will meet up to 37 times, but as few as 26 times (I need to decide how many days we will meet), and the topics and learning materials need to be distributed over time.
The schedule of meeting days changes each semester. This year we’ll start on a Wednesday and end on a Monday, with just one class meeting after Thanksgiving (which is a little insane – we almost always have two meetings after Thanksgiving: it looks like our semester is one MWF shorter this fall than usual).

I saw this resource from CMU, which suggests three different ways for framing out the information: https://www.cmu.edu/teaching/designteach/design/contentschedule.html

It looks like RIT also has some design tools that could be helpful: https://www.rit.edu/academicaffairs/tls/course-design/instructional-design/design-tools

Reading “Learning environment and student outcomes”

I am reading “What you do is less important than how you do it: the effects of learning environment on student outcomes”, Bonem, Fedesco, and Zissimopoulos 2019 (Learning Environment Research).

They survey a large number (14,000) students across a variety of disciplines and find students ” in highly autonomy-supportive learning environments experience significant increases in satisfaction of students’ basic psychological needs, student motivation, course evaluations and academic performance”.

They reference “self-determination theory” and distinguish between courses where the focus is on pressuring students to think like the instructor (controlling), and courses where students have more ownership of the course, with their thoughts and feelings welcome. See Ryan and Deci 2017 for information about the learning benefits of autonomy-supporting environments (environments where students “learn more conceptual knowledge, have a deeper understanding of the content, and retain information longer”).

In terms of course outcomes, autonomy, competence and relatedness are named as basic psychological needs (they say to see Deci and Ryan 1985 and 2000), and they are rated, via surveys, for each course in the study.

Autonomy: Options and choices can create a sense of autonomy.
Competence: A sense of competence is also important: this is about student perception of their progress. Perceiving progress can lead to motivation, which can lead to improved progress.
Relatedness: This is about feeling like other people in the course (instructor and students) care about them and that they are contributing to the course.

The measure of learning environment for each course in the study was set by a learning climate questionnaire (LQC, 6 questions):

I feel that my instructor provides me choices and options.
I feel understood by my instructor.
My instructor conveyed confidence in my ability to do well in the course.
My instructor encouraged me to ask questions.
My instructor listens to how I would like to do things.
My instructor tries to understand how I see things before suggesting a new way to do things.

They also use a basic psychological needs survey (7 items are about autonomy, 6 about competence and 8 about relatedness):

I feel like I can make a lot of inputs in deciding how my coursework gets done.
I really like the people in this course.
I do not feel very competent in this course.
People in this course tell me I am good at what I do.
I feel pressured in this course.
I get along with people in this course.
I pretty much keep to myself when in this course.
I am free to express my ideas and opinions in this course.
I consider the people in this course to be my friends.
I have been able to learn interesting new skills in this course.
When I am in this course, I have to do what I am told.
Most days I feel a sense of accomplishment from this course.
My feelings are taken into consideration in this course.
In this course I do not get much of a chance to show how capable I am.
People in this course care about me.
There are not many people in this course that I am close to.
I feel like I can pretty much be myself in this course.
The people in this course do not seem to like me much.
I often do not feel very capable in this course.
There is not much opportunity for me to decide for myself how to go about my coursework.
People in this course are pretty friendly towards me.

My impression is that the LQC was used to grade the learning environment of the course, and then the basic psychological needs survey was used to create dependent variables. It doesn’t seem surprising that 1, 5, 8, 11, 13, 17, 20 of the basic needs survey (the autonomy questions) would be closely related to the LQC questions, so it seems worth focusing on the relationship between the LQC score and other outcomes of the course.

The course rating, the instructor rating, and the student rating of knowledge transfer each has a positive relationship with the learning environment rating.

Looking at regression coefficients (see Table 3), competence seems perhaps more important than autonomy. Competence looks to be closely related to autonomy, though, and also to a “self determination index”. In the self determination index, students rate agreement with the following statements:

Because it allows me to develop skills that are important to me.
Because I would feel bad if I didn’t.
Because learning all I can about academic work is really essential for me.
I don’t know. I have the impression I’m wasting my time.
Because acquiring all kinds of knowledge is fundamental for me.
Because I feel I have to.
I’m not sure anymore. I think that maybe I should quit (drop the class).
Because I really enjoy it.
Because it’s a sensible way to get a meaningful experience.
Because I would feel guilty if I didn’t.
Because it’s a practical way to acquire new knowledge.
Because I really like it.
Because experiencing new things is a part of who I am.
Because that’s what I’m supposed to do.
I don’t know.I wonder if I should continue.
Because I would feel awful about myself if I didn’t.
Because it’s really fun.
Because that’s what I was told to do.

I suppose that this paper ends up setting up a list of 13 or so factors that are worth investigating further (6 learning environment questions and 7 autonomy questions), as well as highlighting the importance of what the authors are terming autonomy, competence and relatedness.

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.