Monthly Archives: January 2013

teaching

Math 32B/H Midterm 1 Review Materials

I have prepared some review materials for the first midterm for Math 32B and Math 32B H. I wrote up some sample problems (here) and solutions (here). As usual, let me know if you find any errors or if an explanation is unclear.

teaching

Explanation of 16.5 #56


By popular demand, I present here an explanation of problem 16.5 #56 from the homework due on Monday. The problem gives the equation of the wave function for an electron in a hydrogen atom
\[
\psi_{S1}(\rho) = -\frac{1}{\sqrt{\pi a_0^3}} e^{-\rho / a_0}.
\]
Here, \(\rho\) is the distance from the origin, so that the equation for \(\psi_{1S}\) is in spherical coordinates. The probability that the electron is in some region \(W\) can be computed by integrating the function
\[
p(\rho) = \left|\psi_{S1}(\rho)\right|^2 = \frac{1}{\pi a_0^3} e^{-2 \rho / a_0}
\]
over the region \(W\). The problem asks you to compute the probability that an electron is found outside of the Bohr radius, \(a_0\). In this case, the region \(W\) is everything in \(\mathbf{R}^3\) outside of the ball of radius \(a_0\) centered at the origin. In polar coordinates, \(W\) is easily described by the following inequalities:
\[
\begin{align*}
a_0 \leq &\rho < \infty\\ 0 \leq &\varphi \leq \pi\\ 0 \leq &\theta \leq 2 \pi \end{align*} \] So the integral we must compute in (in spherical coordinates) \[ \int_0^{2\pi}\int_0^\pi\int_{a_0}^\infty \frac{1}{\pi a_0^3} e^{-2 \rho / a_0} \rho^2 \sin \varphi \, d\rho\, d\varphi \, d\theta. \] The \(\varphi\) and \(\theta\) integrals are straightforward to compute, but the \(\rho\) integral is a little bit tricky. I computed it using integration by parts (applied twice). Using this method, I got a solution of \[ \frac{5}{e^2} \] which agrees with the answer the book provides.

math teaching

Limit of Average Values of a Continuous Function


A recent homework assignment asked to prove the following fact: If \(f : B \to \mathbf{R}\) is continuous where \(B \subset \mathbf{R}^n\) is open, and \(x_0 \in B\) then
\[
\lim_{r \to 0} \frac{1}{V(B_r)} \int_{B_r} f(x)\, dV = f(x_0)
\]
where \(B_r\) is the ball of radius \(r\) centered at \(x_0\). The limit-and can be interpreted as the average value of \(f\) on \(B_r\). Informally, the claim says that the average value of a continuous function tends to the value of the function as we consider smaller and smaller balls containing \(x_0\).

There were some problems with the solutions I saw on the homework, so I wrote up a careful solution available here. In general, when trying to prove a fact such as this, a good starting point is to write the definitions of terms in the hypotheses and conclusion. In this case, you need to start with the definitions for continuity, limit, and volume in order to have a fighting chance of proving the statement.

expository math teaching

Integrability and Continuity


I just posted a short essay that defines a function with some curious properties. The function is defined by

\[f(x) = \begin{cases}\frac{1}{b} & x = \frac{a}{b}\in\mathbf{Q}\text{ in lowest terms}\\ \\0 & \text{if } x \in \mathbf{R}\setminus\mathbf{Q}.\end{cases}\]

In the essay I prove that

  1. \(f(x)\) is continuous on \(\mathbf{R}\setminus\mathbf{Q}\)
  2. \(f(x)\) is discontinuous on \(\mathbf{Q}\)
  3. \(f(x)\) is integrable, and in particular \(\int_0^1 f(x)\, dx = 0\)

At first blush, these properties may seem counterintuitive.

The theory of Lebesgue measure completely characterizes Riemann integrable functions: they are functions which are almost everywhere continuous. The above function is almost everywhere continuous because its set of discontinuities (\(\mathbf{Q}\)) is countably infinite, and therefore has Lebesgue measure \(0\).

music musings

Revel in Ravel’s Repetition

Bolero has been stuck in my head for the past couple days.

You can download a recording of Bolero here. Bolero was one of Maurice Ravel’s last compositions, and is of a very different nature from his earlier work. This Radiolab podcast postulates that Ravel suffered from frontotemporal dementia. The podcast draws a parallel between the end of Ravel’s life, and the life of biologist-turned-painter Anne Adams. See some of her paintings here.

teaching

HW1 Problem

I have written up an explanation of problem 51 from section 16.1 in the text. A PDF of the write-up is available here. Feel free to let me know if you have any questions or notice any errors.

teaching

Welcome to Math 32BH, Winter 2013

This quarter I will be TAing for Math 32BH which covers multivariable integral calculus. Since this is the honors section, the course emphasizes a rigorous development of the theory, as well as computation. Here is a link to the course website. This post and all others relating to my section of Math 32BH will be tagged M32BH.

I am currently planning on holding my office hours on Tuesdays after class (from 10 – 11 AM) in my office, MS 3915B. I am also available by appointment.

teaching

Welcome to Math 32B, Winter 2013

This quarter I will be TAing for Math 32B which covers multivariable integral calculus. It is my first time working on this particular course, although I fondly remember taking the equivalent class I took as an undergrad. Here is a link to the course website. This post and all others relating to my sections of Math 32B will be tagged M32B.

I am currently planning on holding official office hours on Tuesdays after class (from 1 – 2 PM) in my office, MS 3915B. I will generally be available after class on Thursdays as well from 1 – 1:45. I am also available by appointment.

adventures

Getty Villa

 

I went to the Getty Villa over the break with Alivia and my mom, which I really enjoyed. The museum is situated in a canyon above the beach at the edge of Malibu, and is made to look like, well, an Italian villa. Like the Getty Center, the grounds themselves are stunning. The main part of the museum surrounds a well-manicured courtyard garden, and radiating out from the center are breezeways to more pools and fountains. We were lucky to go on a drizzly morning which kept the masses away. For the first hour we were there, we essentially had the outdoor areas to ourselves.

The museum collection consists of art and artifacts from ancient Greece and Rome. The displays conjured memories of a humanities course I took as an undergrad that was a cornerstone of my college education. It was an absolute delight to see the artifacts in person!

adventures

Verdugo Mountains Hike

 

San Gabriel Mountains

 

This morning I went on a hike (written up here) through the Verdugo Mountains between Burbank and La Crescenta. The visibility was phenomenal, so I managed to snap a picture of the San Gabriel mountains in all their glory. From the peak, I could see all the way to Long Beach!