Mathematical Methods

Assignment 10
Separation of variables, part 2
Due on April 30

Your final homework assignment covers separation of variables in three dimensions (using Cartesian coordinates) and separation of variables in spherical polar coordinates with an axis of symmetry.

Assignment 9
Separation of variables
Due on April 23

Two problems on separation of variables in PDEs with two variables. There is also an optional exercise about solutions of the wave equation in three dimensions (You don't need to hand in the optional problem, but it is instructive to work through!).

Assignment 8
Legendre polynomials, Method of Frobenius
Due on April 9

Exercises involving Legendre polynomials of the first kind, Legendre series representations of functions, and generalized power series solutions of ordinary differential equations using the Method of Frobenius.

Assignment 7
Series Solutions of ODEs
Due on March 29

This assignment will help you practice working out Maclaurin series solutions of differential equations, and introduce you to Taylor series solutions written as expansions around points other than 0.

Assignment 6
Vector Calculus
Due on March 17

A few exercises involving the divergence, curl, and Laplacian in orthogonal coordinate systems; the Divergence Theorem; and Stokes's Theorem.

Assignment 5
Complex Fourier Series, Periods other than 2\(\pi\)
Due on February 26

Now you will determine the complex Fourier Series representations of a few simple functions, and the Fourier Series for periods other than 2\(\pi\). In the last problem you will work out the odd extension of a function from \(0 < x < L \) to \(0 < x < 2L \), and determine the sine series coefficients for the example from our first lecture.

Assignment 4
Basic Fourier Series
Due on February 17

This assignment covers some of the basic (but essential!) integrals needed for evaluating Fourier Series. You will also compute the Fourier Series representation of three simple functions on the interval \(-\pi \leq x \leq \pi\). Be sure to carefully read the statement at the top of assignment!

Assignment 3
Orthogonal Coordinates, Velocity and Acceleration
Due on February 8

In this assignment you will work out the velocity and acceleration vectors in spherical polar coordinates, derive an equation used in our discussion of celestial mechanics, and apply the methods we developed for converting a vector from Cartesian coordinates to another OCS.

Assignment 2
Spherical Polar Coordinates
Due on February 1

In class we learned how to determine the scale factors and basis vectors for any orthogonal coordinate system. For this assignment you will apply this to spherical polar coordinates, which are useful in situations where a system has spherical symmetry.

Assignment 1
Review of Vectors and Multivariable Calculus
Due on January 22

This assignment is a review of some basic material on vectors and multivariable calculus. It's due on Friday of the first week of class. Please email if you have any questions!

A Warning

Never, ever hand in an assignment that has been copied from a solutions manual or a source you found online. You won't learn anything that way, and it will earn you a grade of zero for that assignment. If it happens more than once it will be reported to the Department Chair and the Dean. Consider yourself warned. Click here to see the College of Arts and Sciences Statement on Academic Integrity.

These are the handwritten notes that I work from during lectures. Sometimes I will post them in advance when they include material that isn't in the textbook. The notes are pretty neat and should be more or less searchable. The diagrams are definitely cleaner than the ones I draw during class, and there are usually some worked examples we don't get to. Additional examples and discussion can be found in the next section.

Series Solutions and Special Functions

Many of the ODEs you encounter in physics can be solved by assuming that the solution has a power series expansion. These notes cover the basics of this method, its application to a few equations that frequently appear in physics, and the properties of the resulting “special functions” (like Legendre polynomials and Besssel functions).

Vector Calculus

Notes on the gradient, divergence, curl, and Laplacian in general orthogonal coordinate systems, their fundamental theorems, and a brief review of surface and volume integrals.

Fourier Series

Lecture notes covering real and complex Fourier Series, Parseval's Theorem, intervals with length \(2L\), and even and odd extensions of functions. Most of this material is in the textbook but the presentation may not be the same.

Series: A Brief Review

A few important facts about Taylor series, Mclaurin series, and what it means to write down an infinite series representation of a function.

Orthogonal Coordinate Systems
Includes material not covered in the textbook

These notes cover the definition and properties of Orthogonal Coordinate Systems, and how to translate from one OCS to another. A fun application is the orbital mechanics of a satellite. Setting up the problem in spherical polar coordinates makes it much easier to show that angular momentum is conserved or that orbits are ellipses.

This section contains supplementary notes with extra discussion, examples, and applications. This is also where I'll post Mathematica notebooks.

Method of Frobenius: Examples

Here are a few examples of ODEs solved by assuming a generalized power series.

How Will We Use Legendre Series in Other Courses?

In class I mentioned that you will see a lot of Legendre series in upper level courses like E&M or Quantum Mechanics — they show up whenever we solve equations involving the Laplacian in spherical polar coordinates. This Mathematica notebook works through an E&M example: the electrostatic potential inside and outside a hollow, insulating sphere.

Legendre Polynomials: What They Are, How We Use Them

These notes review Legendre's equation and its series solutions. The Legendre polynomials of the first kind have special properties that allow us to use them like building blocks in the same we constructed Fourier series with sines and cosines.

Series Solutions of ODEs: Examples

Here are three examples of ODEs solved by assuming a Mclaurin series solution. Try to work them out yourselves, then compare what you did with my solutions.

Visualizing Gradients

A Mathematica notebook to help us visualize how the gradient of a function is related to the direction in which the function most rapidly increases.

Even and Odd Extensions

In class we talked about even and odd extensions of a function, and how we can use them to build a Fourier series representation out of a different set of building blocks. For example, instead of using sines and cosines that fit a whole number of periods into \([0,2L]\), we can use the odd extension of a function to build a Fourier series out of sines that fit a whole or half number of periods onto that interval. This Mathematica notebook works out a few examples, including the Fourier sine series for the odd extension of the function \(4\,x - 4\,x^2\) on \(0\leq x \leq 1\). Using sines that fit a whole or half number of periods onto this interval, we can build a Fourier series representation out of the sort of normal modes we'd get on a clamped string. When each one oscillates with its natural frequency we see the behavior you'd expect for a vibrating string.

A Fourier Example from the Homework

This Mathematica notebook works out the Fourier Series representation of one of the functions from the Homework, and shows that it does all the things we talked about in class.

Another Fourier Series Example

In physics we often encounter functions \(f(x)\) that are defined piecewise. This means that you can't write down one expression that works for all values of \(x\) (usually because the function or its derivatives are discontinuous). A common misconception when first learning about Fourier Series is that you work out a different series for each part of a piecewise function. But this is not the case: a piecewise function on \([-\pi,\pi]\) is represented by just one Fourier Series. This short note reviews what we mean by a piecewise function and how to handle integrals in which they appear, then works through a detailed example of a Fourier Series for a piecewise function. You should read through this before working on Homework 4!

Adding Up Sines and Cosines

This is a short Mathematica notebook that explores adding up sines and cosines to assemble a function on the interval \(-\pi \leq x \leq \pi\). The last cell is interactive: Once you evaluate a window with sliders will appear that lets you adjust the coefficients of the sine and cosine terms in the sum.

Velocity and Acceleration in Spherical Polar Coordinates

This Mathematica notebook derives the unit vectors and scale factors for SPC, and then works out the components of the velocity and acceleration.

Velocity and Acceleration in 3-D Parabolic Coordinates

This is another example reviewing the calculation of velocity and acceleration in an orthogonal coordinate system. The same results are obtained in the Mathematica notebook linked further down the page. This does the calculations by hand and adds a little more discussion.

Position, Velocity, and Acceleration in CPC

In class we talked about how to express an object's position, velocity, and acceleration in a general OCS. These notes take a detailed look at this in Cylindrical Polar Coordinates. After working out \(\vec{r{}}\), \(\vec{v{}}\), and \(\vec{a{}}\) I briefly talk about why you might want to use an OCS other than Cartesian coordinates, and then look at an example where CPC makes it easier to write out Newton's Second Law.

Spherical Polar Coordinates and Unit Vectors

As part of homework 2 you derived expressions for \(\hat{r\,}, \hat{\theta\,},\) and \(\hat{\phi\,} \) that you will need to use on homework 3. Since you may not get your graded assignments back before you start working on the new assignment, I wrote out a short note with the info you need.

Orthogonal Coordinate Systems - Summary

A shorter set of notes reviewing orthogonal coordinate systems and how to derive the scale factors and unit vectors.

Parabolic Coordinates with Mathematica

This is a Mathematica notebook that explores some aspects of the parabolic coordinates we worked with in class. It draws the coordinate grid, works out the scale factors and unit vectors, examines how the unit vectors change from point-to-point in the plane, and calculates both the velocity and acceleration.

A Brief Introduction to Mathematica

Once you've downloaded and installed Mathematica you should have a look at the brief introduction in the link above. It shows some basic operations and should help you take the system for a test drive. For a more in-depth introduction I strongly encourage you to work through the first few sections of An Elementary Introduction to the Wolfram Language. Here are some more resources for getting started with Mathematica:

Hands-On Start to Mathematica
Fast Introduction for Math Students
Wolfram U

That last link offers in-depth tutorials and courses in a wide variety of subjects. If you're feeling adventurous you can also open the “Wolfram Documentation” window from the Help menu and browse some of the topics there.

Download and Install Mathematica

From time to time we will use the technical computing system Mathematica to visualize complicated functions, automate certain calculations, and explore concepts from class. Follow these instructions to download and register a copy of Mathematica.

Differentials

The differential of a function describes how it changes when its argument changes by an infinitesimal amount. Not all calc sections cover this in the same way, so these notes review differentials of scalar and vector functions of one or more variables. You may find the useful when working on the first homework assignment.

Line Integrals

This is a basic review of line integrals -- what they are, how to evaluate them, etc. It may be useful if you're a little rusty on this topic. The file is big (about 22 MB) because of the various plots. Let me know if you find any typos or mistakes and I will post a corrected version.