Previous Cal Poly teaching
Previous Cal Poly teaching
Embedded Files
Solutions to assignments and exams have been removed.
Please contact me if there is content behind a broken link that you would like to access.
See the links to the right for specific courses.
Math 244 (Linear Analysis I) - Spring/Fall '24, Winter/Spring '25
Math 244 (Linear Analysis I) - Spring/Fall '24, Winter/Spring '25
General Information:
Office hours: 3:10-4M, MTRF, in 25-211. Other times by appointment only.
Materials: S. W. Goode & S. A. Annin's Differential Equations & Linear Algebra, 4th Ed. (Pearson, 2016) + class notes + an internet connected device.
How to Present Work to Maximise Marks, V4, updated 09/25.
Personal Notes:
Motivation, V1, updated 3/31. (Class notes)
Solutions to ODEs (Theory), V1, updated 4/1. (Class notes)
The Integrating Factor Method, V1, updated 4/3. (Class notes)
The Integrating Factor Method II, V1, updated 4/6. (Class notes) --> extra Youtube video with another example!
Mixing Problems, V1, updated 4/7. (Class notes)
Introduction to Matrices, V1, updated 4/9. (Class notes)
Matrix Algebra, V1, updated 4/10. (Class notes)
EROs, REF, and RREF, V1, updated 4/13. (Class notes) --> Read this lesson from Khan academy for more examples of performing elementary row operations.
RREF, and Gaussian Elimination, V1, updated 4/14. (Class notes)
Gaussian Elimination II, V1, updated 4/16. (Class notes) --> Solving a Linear System (example 1 and example 2 and example 3) from Khan Academy.
Invertible Matrices, and Determinants, V1, updated 4/18. (Class notes)
Determinants II, then Intro to Vector Spaces, V1, updated 4/21. (Class notes)
Vector Spaces and Subspaces, V1, updated 4/23. (Class notes)
Subspaces and Span, V1, updated 4/24. (Class notes)
Span, and Spanning Sets, V1, updated 4/25. (No class notes -- on blackboard)
Linear Independence, V1, updated 4/28. (Class notes)
Bases, V1, updated 4/30. (Class notes). Blackboard Examples: Chapter 4, V1, updated 4/30. (No class notes)
Eigenvalues and Eigenvectors, V1, updated 5/3. (Class notes)
Eigenvalues and Eigenvectors II, V1, updated 5/5. (Class notes)
Diagonalisation, V1, updated 5/8. (Class notes)
Linear ODEs of Order n, V1, updated 5/11. (Class notes)
Homogeneous Linear ODEs of Order n, V1, updated 5/12. (Class notes)
Real Solutions to Homogeneous Linear ODEs of Order n, V1, updated 5/13. (Class notes)
Nonhomogeneous Equations I, V1, updated 5/15. (Class notes)
Nonhomogeneous Equations II, V1, updated 5/17. (Class notes)
Mass-Spring Systems, V1, updated 5/19. (Class notes)
Forced Mass-Spring Systems, then Intro to Systems of ODEs, V1, updated 5/20. (Class notes)
Theory of First-Order Linear Systems, V1, updated 5/22. (No class notes -- on blackboard)
Real Constant Coefficient Vector ODEs with a Nondefective Coefficient Matrix, V1, updated 5/23. (Class notes)
Nondefective Coefficient Matrix: Complex Solutions, V1, updated 5/27. (Class notes -- calculations presented on blackboard)
Defective Coefficient Matrices I, V1, updated 5/30. (Class notes)
Defective Coefficient Matrices II, V1, updated 6/2. (Class notes)
Defective Coefficient Matrices III, and Additional Examples, V1, updated 6/3. (Class notes)
The course is now finished -- good luck in your final!
Assignments:
Take-Home Assignment 1/THA1 Solutions, V1, updated 4/10.
Take-Home Assignment 2/THA2 Solutions, V1, updated 4/17.
Class Assignment 1 Solutions, V1, updated 4/22.
Take-Home Assignment 3/THA3 Solutions, V2, updated 5/9.
Class Assignment 2 Solutions, V1, updated 5/12.
Take-Home Assignment 4/THA4 Solutions, V1, updated 5/21.
Take-Home Assignment 5/THA5 Solutions, V1, updated 5/28.
Class Assignment 3 Solutions, V1, updated 6/2.
Exam material
Practice Midterm/Practice Midterm Solutions/Midterm Guide, V1, updated 4/28.
Q1 SOLUTION for Midterm (Spring 2025), V1, updated 5/13.
Practice Final/Practice Final Solutions/Final Guide, V1, updated 6/4.
Final Bingo, V1, updated 6/6.
Slope Fields (not covered in Spring 2025)
Slope Fields (not covered in Spring 2025)
A slope field (or "direction field") for a first-order ODE is a graphical representation of the "field of slopes" of the solution curves to the equation. At a point (x, y), we draw a small vertical line meant to be tangent to the solution curve passing through that point: the slope of this line is given by the first-order ODE exactly! Try plotting some slope fields using the Geogebra link above.
We are able to create such plots -- draw a single line as tangent to the unique solution curve at any* point (a, b) -- because of Picard's Theorem. This was one of the many mathematical discoveries of French Mathematician Émile Picard (1856-1941). As well as his research career, he was deeply invested in education: at the École Centrale des Arts et Manufactures between 1894 and 1937, he trained over 10,000 engineers in mathematics. There is a great quote attributed to him on MacTutor: "True rigour is productive, being distinguished in this from another rigour which is purely formal and tiresome, casting a shadow over the problems it touches."
*For the ODE y' = f(x, y), f and its partial derivative with respect to y need to be continuous on an open rectangle around (a,b).
"Gaussian Elimination" is an algorithm for solving systems of linear equations, by applying "elementary row operations" to a matrix to reduce it to a simpler form (row-echelon form -- incidentally, the word echelon comes from the French word échelon, which means the "rung" or step of a ladder. Can you guess why the name applies?).
According to Wikipedia, "The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject" -- Carl Friedrich Gauss in 1810 developed a notation for row reduction that was subsequently widely adopted, but he was not the inventor of the method! This method appears (without proof) as a calculation tool in a Chinese mathematical text approximately 2000 years ago (see the above article from the American Mathematical Society for more information). In Europe, the method was introduced by Isaac Newton in the early 1700s, as he found there was not a systematic or complete account of solving simultaneous linear equations available in the algebra textbooks of the time. According to the AMS article, the method is taught today in universities via matrices in a presentation described by the English mathematician Alan Turing, the "father" of computer science.
Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
A linear transformation can be thought of as a transformation of vectors that does some combination of rotating, stretching, and shearing. To the left is a shearing transformation applied to the Mona Lisa: if every point in the image is represented by a vector pointing from the centre of the image to that point, multiplying (on the left) by a certain 2 x 2 matrix A brings us from the left to the right image.
Vectors off of the centre horizontal line such as v are sheared by A. However vectors such as v remain pointing in the same direction. v is an eigenvector of this transformation: it is a nonzero vector that satisfies the equation Av = λv, where λ is a scalar (in this example, it appears λ = 1).
An eigenvector v is a nonzero vector whose direction remains unchanged by the transformation v --> Av. It may be stretched/contracted, hence the equation Av = λv.
One particularly cool feature of second order linear ODEs (with constant coefficients) is that those of the form y'' + by' + cy = F(t), where c > 0 and b is greater than or equal to 0, can be physically interpreted as describing the motion of a mass on a spring. "b" is the damping constant (measuring resistive forces in the system, such as drag or friction), "c" is the spring constant (which measures how rigid the spring is), and "F(t)" is the forcing term (for external forces acting on the system).
To the right is an example of simple harmonic motion: no external forces or damping is present (mathematically, b = 0 and F(t) = 0) and (with c = 1) the solutions to y'' + y = 0 are y(t) = c_1 sin(t) + c_2 cos(t).
Below we have a damped mass-spring system (with no external forces, i.e. F(t) = 0): the left is underdamped and the right is overdamped. You can see the effect damping has on the (short and long term) behaviour of the system!
A double mass-spring system can be modelled as a first-order linear system with 4 equations, i.e. a system x' = Ax where A is a 4 x 4 matrix. In general, A may be defective or nondefective. If you can solve a system with a 4 x 4 coefficient matrix, you can determine the equations of motion of a double mass-spring system! The link (right) is a customisable simulation of such a system (see the "Time Graph" tab for a plot of the position of the masses with respect to time). The mathematical setup for a double mass-spring system without damping as a first-order linear system with 4 equations is explained on the first half of this webpage.
Math 241 (Calculus IV) - Winter/Spring/Fall '24
Math 241 (Calculus IV) - Winter/Spring/Fall '24
General Information:
Office hours: 3:10-4PM, MTWR, in 25-211. Often outside building 25. Other times by appointment only.
Materials: J. Stewart’s Calculus, 8th Ed. (Cengage Publishing, 2016) + class notes + an internet connected device.
How to Present Work to Maximise Marks, V4, updated 09/25.
First day review material:
Lines and Planes Recap, given out in class.
Warning! Differentiability, V1, updated 10/14.
Extra Example: Lagrange Multipliers
Assignments:
Take Home Assignment 1/THA1 Solutions
Take-Home Assignment 2/THA2 Solutions
Class Assignment 1/CA1 Solutions
Take-Home Assignment 3/THA3 Solutions (V2) <-- updated to fix a sign error & typo!
Class Assignment 2/CA2 Solutions
Take-Home Assignment 4/THA4 Solutions
Take-Home Assignment 5/THA5 Solutions
Take-Home Assignment 6/THA6 Solutions
Class Assignment 3/CA3 Solutions
Exam Material:
Practice Midterm/Practice Midterm Solutions/Midterm Bingo
Practice Final/Practice Final Solutions/Final Bingo
The course is now finished. Good luck, and see you in Finals week!
Personal Notes:
Introduction, V1, updated 9/23. (Class notes)
Surfaces and Traces, V1, updated 9/23. (Class notes)
Functions of Two Variables, V1, updated 9/24. (Class notes)
Plotting, V1, updated 9/25. (Class notes) --> Youtube examples (1 and 2) of calculating the domain of a function.
Limits and Continuity, V1, updated 9/28. (Class notes).
Want to see it all again? Watch Professor Retsek's "Limits" lecture.Partial Derivatives, V1, updated 9/30. (Class notes) --> see the "Partial Derivatives are everywhere" links below!
Tangent Planes and Approximation, V1, updated 10/1. (Class notes -- S12's class cancelled)
Differentiability, and Differentials, V1, updated 10/2. (Class notes)
The Chain Rule, V1, updated 10/5. (Class notes -- V2)
Directional Derivatives and the Gradient, V1, updated 10/7. (Class notes)
More on the Gradient, V1, updated 10/10. (Class notes)
Maxs and Mins, V1, updated 10/13. (Class notes)
Maxs and Mins II, V1, updated 10/14. (Class notes)
Lagrange Multipliers, V1, updated 10/15. (Class notes)
Double Integrals, V1, updated 10/16. (Class notes)
Double Integrals II, V1, updated 10/20. (Class notes)
Double Integrals III, V1, updated 10/21. (Class notes -- did not cover p. 62 or 63)
Double Integrals: Polar Coordinates, V1, updated 10/22. (Class notes)
Triple Integrals, V1, updated 10/29. (Class notes)
Triple Integrals II, V1, updated 10/30. (Class notes)
Cylindrical Coordinates, V1, updated 11/4. (Class notes)
Spherical Coordinates, V1, updated 11/4. (Class notes)
Multiple Integrals: Applications, V1, updated 11/5. (Class notes)
Vector Fields, V1, updated 11/12. (Class notes)
Line Integrals, V1, updated 11/13. (Class notes)
Line Integrals II, V1, updated 11/13. (Class notes)
The Fundamental Theorem of Calculus (Again), V1, updated 11/15. (Class notes)
The Fundamental Theorem of Calculus (Again) II, V1, updated 11/18. (Class notes, V2 -- includes Examples 77 & 78)
Green's Theorem, V1, updated 11/19. (Class notes)
Green's Theorem in Action, V1, updated 12/1. (Class notes -- some sections did not discuss Theorem 42). See the planimeter handout below!
Curl and Divergence, V1, updated 12/2. (Class notes)
Green's Theorem for Flux Integrals, V1, updated 12/3. (Class notes)
Graphing, level curves, & functions of 2 variables
Graphing, level curves, & functions of 2 variables
For a different view on the same material, check out Paul's Online Notes.
Two useful tools to visualise surfaces/graphs of functions of two variables are Desmos and Geogebra.
Left: the graph for the constraint-satisfaction problem from Q2 of Class Assignment 2.
Partial derivatives are everywhere
Partial derivatives are everywhere
To the right, I have the link to the GIF of a solution to the 2D wave equation, mentioned in class.
I also have a Geogebra link where you can visualise the partial derivatives as slopes on any surface, and visualise the tangent plane to any surface!
Differentials -- a small problem
Differentials -- a small problem
The Wikipedia page for differentials repeats a lot of what I said in class (unsuprisingly....my source was Wikipedia!). Notice two important quotes: "the differential represents a change in the linearization of a function" -- we saw this explicitly in class. And under "Approaches" to making differentials mathematically rigorous, "...[the approaches] have in common the idea of being quantitative, i.e., saying not just that a differential is infinitely small, but how small it is." We saw this too -- the differential dx is a variable to which we assign a value; in essence, saying "how small" it is.
If you're feeling brave, you can read about the hyperreal numbers on Wikipedia and their use in fixing the problems raised by infinitesimals. The Wikipedia page for "infinitesimals" also has gems such as "Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632." It was American mathematician Abraham Robinson in the 1960s who finally proved "infinitely small numbers" could be treated in the same logical and rigorous manner as real numbers, putting an end to centuries of difficulty grappling with the notion.
A Basic Idea of Machine Learning: Gradients
A Basic Idea of Machine Learning: Gradients
In this (extra curricular) example, I use gradients to predict the height of a child based on the height of their parents. This is exactly the method by which machine learning algorithms operate!
Vector Field Graphers
Vector Field Graphers
The Planimeter!
The Planimeter!
One amazing physical real-life use of Green's Theorem is the planimeter, an instrument that can measure the area of a region by tracing the boundary of that region. See The Planimeter Handout (V2 -- diagram typo fix) below!
Math 141 (Calculus I) - Winter/Summer '24
Math 141 (Calculus I) - Winter/Summer '24
General Information:
Office hours: MTWR 4:10-5PM. Other times by appointment only.
Materials: J. Stewart’s Calculus, 8th Ed. (Cengage Publishing, 2016) + class notes + an internet connected device.
How to Present Assignments to Maximise Marks, V3, updated 06/27.
Additional Material:
Assignment Solutions:
Class Assignment 1 Solutions
Take-Home Assignment 1 Solutions
Take-Home Assignment 2 Solutions
Class Assignment 2 Solutions
Take-Home Assignment 3 Solutions
Class Assignment 3 Solutions
Take-Home Assignment 4 Solutions
Exam Material:
Practice Final/Practice Final solutions (Part 1) and Q6 and Q7 (Part II)
Personal notes:
Limits and Continuity, V1, updated 06/25. (Class notes)
One Sided Limits, Infinite Limits, Limit Laws, V1, updated 06/26. (Class notes)
Limit Laws II, V1, updated 06/27. (Class notes)
Limit Laws III and Derivatives, V1, updated 07/01. (Class notes)
Derivatives and Derivative Rules, V1, updated 07/02. (Class notes)
Derivative Rules II and Trig Derivatives, V1, updated 07/03. (Class notes)
Chain Rule and Implicit Differentiation, V1, updated 07/08. (Class notes)
Related Rates and Review, V1, updated 07/09. (Class notes)
Maximums, Minimums, Critical Points, V1, updated 7/11. (Class notes)
Increasing/Decreasing, First/Second Derivative, and Concavity Tests, V1, updated 7/15. (Class notes)
Graphing and Optimisation, V1, updated 7/16. (Class notes -- only "Graphing" was covered, not "Optimisation")
Additional Graph Sketching Example!
Optimisation, V1, updated 7/17. (Class notes) Extra: Example 69, Road Drilling
Antiderivatives, V1, updated 7/18. (Class notes)
Area, and the Fundamental Theorem of Calculus, V1, updated 7/22. (Class notes)
Integration, and the FTOC, V1, updated 7/23. (Class notes)
Substitution, V1, updated 7/24. (Class notes)
Extra resources
Extra resources
Optimisation: More examples below to help you practice
Optimisation: More examples below to help you practice
Optimisation problems are probably the most difficult problems we'll encounter in Calculus I. What makes them so difficult is that there is no "set process"/algorithm to always follow and arrive at the solution: frequently, the biggest hurdle in these problems is translating it into mathematical statements. Each of these problems will have a function we want to find the maximum or minimum of (one of the "unknowns"), subject to a fixed constraint (one of the "knowns").
In Example 64, one "unknown" is the field's Area, which we want to maximise. The area is a function of the lengths of the sides of the fence. The sides of the fence are subject to the constraint that the total fence length is 2400ft (one of the "knowns").
Paul's Online Notes are a fantastic resource to see examples of solving these problems. These two links above have 29 fully worked examples (all steps explained)!
Math 143 (Calculus III) - Fall '23, Summer '24, Winter/Spring '25
Math 143 (Calculus III) - Fall '23, Summer '24, Winter/Spring '25
General Information:
Office hours: 3:10-4PM, MTRF, in 25-211. Other times by appointment only.
Materials: J. Stewart’s Calculus, 8th Ed. (Cengage Publishing, 2016) + class notes + an internet connected device.
How to Present Work to Maximise Marks, V4, updated 09/25.
Personal Notes:
Introduction to sequences, V1, updated 3/31. (Class notes)
Intro to sequences II, V1, updated 4/1. (Class notes)
Determining Limits, V1, updated 4/3. (Class notes)
Introduction to series, V1, updated 4/6. (Class notes)
Geometric and Telescoping Series, and the Test for Divergence, V1, updated 4/7. (Class notes)
The Integral Test, V1, updated 4/8. (Class notes)
The Comparison Test(s), V1, updated 4/10. (Class notes)
Alternating Series, V1, updated 4/13. (Class notes)
The Ratio Test (plus absolute/conditional convergence), V1, updated 4/14. (Class notes)
The Root Test, and Strategy, V1, updated 4/16. (No notes -- blackboard only)
Power Series, V1, updated 4/18. (Class notes)
Power Series II, V1, updated 4/21. (Class notes) -- further examples of representing functions as power series can be found here (1), here (2), and here (3).
Taylor and Maclaurin Series, V1, updated 4/23. (Class notes)
Taylor and Maclaurin Series II, V1, updated 4/24. (Class notes)
Numerical Approximation of Series, V2, updated 5/1. (Class notes -- typo in Example 42 fixed)
Blackboard Examples, V1, updated 4/28. (No class notes)
Parametric Equations and Curves, V1, updated 5/3. (Class notes)
Parametric Curves, Areas, and Lengths, V2, updated 5/5. (Class notes)
Parametric Curves, Surface Areas, and Tangents, V1, updated 5/8. (Class notes)
Polar Coordinates and Polar Curves, V1, updated 5/11. (Class notes)
Polar Calculus, V1, updated 5/12. (Class notes) -- Desmos link for Example 62
Polar Calculus II, V1, updated 5/13. (Class notes) -- Desmos link for first example
Polar Calculus III, then Intro to Vectors, V1, updated 5/15. (Class notes) -- see link to the right for Paul's Online Notes, too.
Intro to Vectors II, V1, updated 5/17. (Class notes)
Intro to Vectors III, V1, updated 5/19. (Class notes)
The Dot Product, Angles, and Projections, V1, updated 5/20. (Class notes)
The Cross Product, V1, updated 5/22. (Class notes) -- note some material was presented on the blackboard too.
Lines and Planes I, V1, updated 5/23. (Class notes)
Lines and Planes II, V1, updated 5/27. (Class notes)
Perpendicular distance, and Introduction to Vector Functions, V1, updated 5/30. (Class notes)
Vector Functions II, V1, updated 6/2. (Class notes)
Vector Functions III, and Arc Length, V1, updated 6/4. (Class notes)
Reparameterisation with respect to arc length, and Curvature, V1, updated 6/5. (Class notes)
The course is now finished -- good luck in your final!
Assignments:
Take-Home Assignment 1/THA1 Solutions, V1, updated 4/10.
Take-Home Assignment 2/THA2 Solutions, V1, updated 4/17.
Class Assignment 1 Solutions, V1, updated 4/22.
Take-Home Assignment 3/THA3 Solutions, V1.
Class Assignment 2 Solutions, V1, updated 5/12.
Take-Home Assignment 4/THA4 Solutions, V1, updated 5/21.
Take-Home Assignment 5/THA5 Solutions, V1, updated 5/29.
Class Assignment 3 Solutions, V1, updated 6/2.
Exam material
Practice Midterm/Practice Midterm Solutions/Midterm Guide, V1, updated 4/28.
Midterm Bingo, V1, updated 5/1.
Practice Final/Practice Final Solutions/Final Guide, V2, updated 6/4.
Final Bingo, V1, updated 6/6. (Not played in class in Spring 2025)
The block-stacking problem:
The block-stacking problem:
"Place N identical rigid rectangular blocks in a stable stack on a table edge in such a way as to maximize the overhang".
Using arbitrarily many blocks we can make an overhang of arbitrary size! This follows from the divergence of the Harmonic series.
Getting a little extra from a power series
Getting a little extra from a power series
Abel's Theorem tells us about the behaviour of a function and its reflection as a power series, right at the edge of the interval of convergence. If G(x) = your power series (centred at 0), with radius of convergence 1, and the series at x = 1 converges, then G(x) is left continuous at x = 1.
This fully justifies why the alternating harmonic series converges to ln 2. See another "picture proof" below:
Conditionally convergent series can be anything they want to be
Conditionally convergent series can be anything they want to be
The order of summands in a finite sum does not affect the result: 1 + 2 + 3 = 3 + 2 + 1. This changes if the sum is infinite! The Riemann Series Theorem says " if an infinite series of real numbers is conditionally convergent, then its terms can be rearranged so that the new series converges to any real number, or diverges".
For example, 1 − 1 + 1/2 − 1/2 + 1/3 − 1/3 + ... = 0, (and it's conditionally convergent) but rearranging,
1 + 1/2 − 1 + 1/3 + 1/4 − 1/2 + ...
= 1 - 1/2 + 1/3 - 1/4 + ... = ln 2, as this is the alternating harmonic series!
Without justification I stated the above curve cannot be parameterised by fractions of polynomials, or sin/cos/tan. The below (technical) article delves into the details (see towards the end):
The fastest way to family drama
The fastest way to family drama
A brachistochrone curve, or "curve of fastest descent" is the one lying on the plane between a point A and a lower point B (where B is not directly below A), on which a bead slides frictionlessly under gravity from A to B in the shortest time. The problem of determining what this curve is was given by Johann Bernoulli in 1696.
In the following year, 5 mathematicians independently proved the solution is a cycloid: Newton, Jakob Bernoulli (though his brother Johann tried to pass off Jakob's solution as his own), Leibniz, von Tschirnhaus, and de l'Hôpital.
Check out the below video, where VSauce explains more about the curve and builds one with Adam Savage:
While at Oxford, I co-organised a seven-poster series called Mathematicians A-Z; 26 female and nonbinary mathematicians and their historical contributions. The first poster (A for Agnesi and B for Boole) can be found to the right:
(Right: the Witch of Agnesi as an explicit curve.)
Graphing curves
Graphing curves
Using Method 2: Redraw, we can see how the polar Rose Curve is created from drawing r = sin(4*\theta) on perpendicular "r-\theta" axes.
Also check out this 3-d surface plotter from Desmos:
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