Very Nice Java Applets

I’m always on the look out for nice java applets (or flash, or javascript, or whatever) that help visualize tough math concepts. The best applets are not merely gimmick, but truly make the mathematics more accessible and invite you to explore, predict, and play with the concepts.

Most recently, my dad pointed me to this website.  At this site, you’ll find well over one hundred very nice applets that show things I haven’t seen around the web before. For instance, play around with your favorite polyhedra in 3d, then let the applet truncate or stellate the polyhedron, all with nice animation. And of course, you’ll want to mess with all the settings on the drop down menus, too! Additionally, you can see a mobius strip, klein bottle, cross cap, or a torus. Also, I was just recently blogging about complex functions. Some of these applets allow you to visualize with animation and color various complex functions. Having just completed a grad class in differential equations, I was also interested in these ODE applets. You may also have fun with fractals, like this one.

There are lots of other nice applet websites out there. As you know, I’m a huge fan of GeoGebra (free, open source, able to be run online without installation, easy to use, powerful). Someday maybe I’ll post more extensively about its merits. In the meantime, you can check it out here.  Many of these applets use GeoGebra. This site has a lot of nice applets too.  And here. This one’s rather elementary, but fun for educators. These are great, too. These are all sites I’ve used in class. Like I said, I think they’re more than just ‘cool’–they really do help elucidate hard-to-understand topics.

Feel free to share other applet sites you like.

Functions of a Complex Variable

My precalculus classes just finished a unit on polar coordinates and complex numbers. When I teach about complex numbers,  I mention functions of a complex variable in passing, but we don’t really give it much thought. We do complex arithmetic and that’s all; that is, problems like these:

Evaluate.

$\frac{i^5(2-i)}{1+3i}$

$\left(2+2i\sqrt{3}\right)^{10}$

$\sqrt[4]{40.5+40.5i\sqrt{3}}$

In our precalculus class, we also understand how to plot complex numbers. Complex numbers must be plotted on a two-dimensional plane because complex numbers are…well…two dimensional! The real number line has no place for them. For instance, we represent the complex number $w=2-3i$ as the point $(2,-3)$

But we don’t ask questions about complex functions. This is sad! Because functions of a complex variable are fairly accessible.  That is, we want to consider functions like

$f(z)=z+1$

$f(z)=z+4i$

$f(z)=3z$

$f(z)=iz$

$f(z)=z^2$

The first thing you’ll notice is that I’ve written these functions in terms of $z$, to indicate that they take complex arguments and (possibly) return complex values. Here’s where the problem comes. Take for instance, $f(x)=x+1$. We’re used to visualizing it this way:

Notice we’re wired (because of schooling, perhaps) to understand the $x$ coordinate as being the “input” to the function and the $y$ coordinate as being the “output” from the function. Now, think about $f(z)=z+1$ where $z$ is complex. Do you see the problem? Remember, complex numbers are two-dimensional.  A function $f(z)$ that operates on complex numbers has two-dimensional input and two-dimensional output. For example, take the function $f(z)=z+1$. If we try putting a few complex numbers into the function for $z$, what happens? If $z=-2+4i$, then $f(z)=-1+4i$. Geometrically, what is happening to a complex number on the complex plane when we apply $f(z)$? If you thought “the function translates complex numbers to the right by one unit in the complex plane,” you’re right!

I’ve built another Geogebra applet to help you visualize this kind of function. Make sure you use it with purpose, rather than just dragging things around randomly. Try making predictions about what will happen before revealing the result. Read the directions.

Have fun, and I hope you learn something about complex functions! I’m sure to post more on them someday. There’s a lot more to say.

Heat Conduction in a Rod

I’m currently taking a grad class in differential equations. I just had a homework problem that asked about the heat conduction in a long, thin rod. It inspired me to create another GeoGebra applet, and I thought I’d share. The math might be a bit inaccessible, but the results are fairly straightforward.

Consider a solid rod of some kind of uniform material (maybe aluminum or cast iron). Say it’s 20 cm long. And say, for whatever reason, the temperature at a given place in the bar is initially given by this distribution:

Notice the bar is 70 degrees at each end and 50 degrees in the middle. (This is arbitrary…I just picked this distribution, just for fun.)

Now, let’s say the sides of the bar are insulated, and we just apply heat to the ends. If we maintain a temperature of 10 degrees at one end and 50 degrees at the other end, after a long time, we would expect the temperature throughout the bar to be evenly distributed, ranging from 10 degrees to 50 degrees. It would look something like this:

Now, the question is, what is the temperature throughout the bar after 1 second? It should be pretty close to the original distribution still, right? Right. What about after 10 seconds? 30 seconds? 30 days? Eventually it will look like the above distribution. That’s why we call this the “steady-state” distribution.

Here’s what the temperature throughout the bar looks like after 30 seconds, for instance.

Notice that the temperature distribution is still very similar to the initial distribution, but that the ends are changing temperature. This will happen more and more over time.

The applet I constructed lets you change everything about this situation: the length of the bar, the type of material, the temperature we apply to each end, and the initial temperature distribution. Like I said, the math is a bit nasty, but the results are intuitive, I hope. If you want to see some more of the math, feel free to do some reading!

Really Fun Limit Problem (revisited)

I posted this problem a few weeks ago:

The figure shows a fixed circle $C_1$ with equation $\left(x-1\right)^2+y^2=1$ and a shrinking circle $C_2$ with radius $r$ and center the origin (in red). $P$ is the point $(0,r)$, $Q$ is the upper point of intersection of the two circles, and $R$ is the point of intersection of the line $PQ$ and the $x$-axis. What happens to $R$ as $C_2$ shrinks, that is, as $r\rightarrow 0^{+}$?

And I also posted this applet, so you could investigate it yourself.

Now, I post the solution. Initially, I thought that the $x$-coordinate of $R$ would go off to infinity or zero, I wasn’t sure which. The equation for $C_2$ is $x^2+y^2=r^2$. Solving for the intersection of the two circles, $Q$, we find it has coordinates

$Q=\left(\frac{r^2}{2},\sqrt{r^2-\frac{r^4}{4}}\right)$

Remembering that $P=(0,r)$, we now find the equation of line $PQ$ in point-slope form.

$y-r=\left(\frac{r-\sqrt{r^2-\frac{r^4}{4}}}{0-\frac{r^2}{2}}\right)(x-0)$

Now, we seek to find the coordinates of $R$, the $x$-intercept of the line. Letting $y=0$ in the above equation, we solve for $x$ and find

$x=\frac{\frac{r^2}{2}}{1-\sqrt{1-\frac{r^2}{4}}}$

We now take the limit of $x$ as $r\rightarrow 0$.

$\displaystyle\lim_{r\rightarrow 0}\frac{\frac{r^2}{2}}{1-\sqrt{1-\frac{r^2}{4}}}$

If we try to evaluate this limit by plugging in 0, we get an indeterminant form 0/0. We can either use L’hopital’s Rule or evaluate it numerically. Either way, we find.

$\displaystyle\lim_{r\rightarrow 0}x=4$

I have to say, this result surprised me. Like I said, I expected this limit to evaluate to 0 or infinity–but 4?? I had to get a bit more of an intuitive understanding, so I built that Geogebra applet.

Also, I was told afterward, by the student who brought this to me, that there’s a straight-forward geometric way of deriving the answer, based on the observation that point Q and point R and the point (2,0) (0,2) form an isosceles triangle for all values of r. The observation isn’t trivial. Can you prove it’s true? Once you do realize this fact, though, the above result is clear.

We Love Vectors!

In our IB Precalculus class, we are in the middle of our Vectors and Parametric Equations unit. It’s one of my favorites. And it just so happens that Dave Richeson just created some fantastic java applets with Geogebra that let users play around with properties of vectors. He blogged about it today, here. All of you should go check it out.

Thanks, Dr. Richeson!