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Join The Computer-Based Math™ Education Forum

The computerbasedmath.org community has been growing steadily since the project first started in 2010. Several thousand of you have signed up to show your support, share your ideas, and help spread the word. The Computer-Based Math™ Education Summit has been a great tool for bringing the community together, but we wanted a central hub where the community can gather more than just once a year. So we've launched the The Computer-Based Math Education Forum. Whatever your background, join the conversation and share your experiences.
Michael Belcher, Marketing Project Coordinator, European Sales
Education & Academic

Falling Faster than the Speed of Sound

Earlier this month, on a nice day, Felix Baumgartner jumped from 39,045 meters, or 24.26 miles, above the Earth from a capsule lifted by a 334-foot-tall helium filled balloon (twice the height of Nelson's column and 2.5 times the diameter of the Hindenberg). Wolfram|Alpha tells us the jump was equivalent to a fall from 4.4 Mount Everests stacked on top of each other, or falling 93% of the length of a marathon. At 24.26 miles above the Earth, the atmosphere is very thin and cold, only about -14 degrees Fahrenheit on average. The temperature, unlike air pressure, does not change linearly with altitude at such heights. As Wolfram|Alpha shows us, it rises and falls depending on factors such as the decreased density of air with rising altitude, but also the absorption of UV light by the ozone layer. At 39 kilometers, the horizon is roughly 439 miles away. At this layer of the atmosphere, called the stratosphere, the air pressure is only 3.3 millibars, equivalent to 0.33% of the air pressure at sea level. To put it another way, the mass of the air above 39 kilometers is only 0.32851% of the total air mass. Given this knowledge, we know that 99.67% of the world's atmosphere lay beneath him. This information was important to Felix's goal to break the sound barrier in free fall because the rate of drag is directly related to air pressure. With less air around him, there would be less drag, and thus he could reach a higher maximum speed. Of course, this would require him to wear an oxygenated suit to allow him to breathe, in addition to keeping him warm.
Jason Martinez, Research Programmer, Wolfram|Alpha Scientific Content
Education & Academic

Calculating the Energy between Two Cubes

In my last blog post, we discussed 3D charge configurations that have sharp edges. Reader Rich Heart commented on it and asked whether Mathematica can calculate the force between two charged cubes, as done by Bengt Fornberg and Nick Hale and in the appendix of Lloyd N. Trefethen's book chapter. The answer to the question from the post is: Yes, we can; I mean, yes, Mathematica can. Actually, it is quite straightforward to treat a more general problem than two just-touching cubes of equal size: We can deal with two cubes of different edge lengths L1 L2 We can calculate the force for any separation X, where X is the distance between the two cube centers (including the case of penetrating cubes; think plasma) We will use a method that can be generalized to higher-dimensional cubes without having to do more nested integrals Instead of calculating the force between the two cubes, we will calculate the total electrostatic energy of the system of the two cubes. The force is then simply the negative gradient of the total energy with respect to X. The electrostatic energy (in appropriate units) is given by: (In the following calculations, we will skip the constant [with respect to X] prefactors Q1 L1-3 Q2 L2-3 or Q1 Q2 if not needed.) Approaching this integral head-on doing one integral after another is possible, but a very tedious and time-consuming operation. Instead, to avoid having to carry out a nested six-dimensional integral, we remember the Laplace transform of 1 / √s.
Michael Trott, Chief Scientist, Wolfram|Alpha Scientific Content
Education & Academic

3D Charges and Configurations with Sharp Edges

In my last blog post, we looked at various examples of electrostatic potentials and magnetostatic fields. We ended with a rectangular current loop. Electrostatic and magnetostatic potentials for squares, cubes, and cuboids typically contain only elementary functions, but the expressions themselves are often quite large compared with simple systems with radial symmetry. In the following, we will discuss some 3D charge configurations that have sharp edges. Let's start with a charged 2D rectangle in 3D space. Again, the potential is an elementary function that contains a few logarithms.
Michael Trott, Chief Scientist, Wolfram|Alpha Scientific Content
Education & Academic

On the Importance of Being Edgy—Electrostatic and Magnetostatic Problems with Sharp Edges

(This is the first post in a three-part series about electrostatic and magnetostatic problems involving sharp edges.) Mathematica can do a lot of different computations. Easy and complicated ones, numeric and symbolic ones, applied and theoretical ones, small and large ones. All by carrying out a Mathematica program. Wolfram|Alpha too carries out a lot of computations (actually, tens of millions every day), all specified through free-form inputs, not Mathematica programs. Wolfram|Alpha is heavily based on Mathematica, and many of the mathematical calculations that Wolfram|Alpha carries out rely on the mathematical power of Mathematica. And while Wolfram|Alpha can carry out a vast amount of calculations, it cannot carry out all possible calculations, either because it does not (yet) know how to do a calculation or because the (underlying Mathematica) calculation would take a longer time than available through Wolfram|Alpha. So for a detailed investigation of a more complicated engineering, physics, or chemistry problem, having a copy of Mathematica handy is mandatory. But there is also the reverse relation between Mathematica and Wolfram|Alpha: Wolfram|Alpha's knowledge, especially its data knowledge, allows it to carry out investigations and calculations that can substantially increase the power of pure Mathematica. And all of this is because Wolfram|Alpha's knowledge is accessible through the WolframAlpha[] function within Mathematica.
Michael Trott, Chief Scientist, Wolfram|Alpha Scientific Content
Education & Academic

Society’s Changing Needs for Math Debate

In the "Society's Changing Needs for Math" session at the The Computer-Based Math (CBM) Education Summit 2011, Marcus du Sautoy, Paul Wilmott, Charles Fadel, and Tim Oates discussed their views in one of the summit's key sessions. There was a lot of energy for debate from our summit attendees, and we did not have the time to expand on every topic after each talk. Hopefully these bite-sized videos from our speakers will open up discussions to all. Have your say and leave your thoughts on the comment section of this post or on Computer-Based Math's YouTube Channel.
Jackie Tran, Courseware Author
Education & Academic

Wavelets and Their Application in Mathematica

What do computer animation, oil exploration, and the FBI's database of 30 million fingerprints have in common? Wavelet analysis. As of Version 8, wavelet analysis is an integral part of Mathematica. Wavelets themselves are short-lived wave-like oscillations. Taking the Morlet wavelet, for example, we can see that unlike sines and cosines, this wave-like oscillation is localized in the sense that it does not stretch out to infinity.
Samuel Chen, Product Manager, Technical Communication & Strategy
Education & Academic

Breaking Secret Codes with Mathematica

Mathematica can make you feel like a computational superman. Armed with that attitude and some schoolboy knowledge of cryptography, I turned my attention to cipher breaking this week, only to discover buried kryptonite. The weakness of ciphers (where you swap every occurrence of a particular letter in your message with the same different letter) is that they don't change the patterns of letters. The simplest attack that exploits this fact is frequency analysis. The most common letter in English is "e", and so it follows that the most common character in an encoded message (assuming the message is written in English) will correspond to "e". And so on through the alphabet. Mary Queen of Scots famously lost her head when Queen Elizabeth's spymaster broke Mary's cipher using frequency analysis. I figured that if sixteenth century spies could do it by hand, I should be able to automate it in Mathematica in about 10 minutes.
Jon McLoone, Director, Technical Communication & Strategy
Best of Blog

Automatic Physical Units in Mathematica

I just published a Mathematica package that provides an alternative, richer implementation of units and dimensional analysis than the built-in units package. You can get it here. Aside from being a really nice extension to Mathematica, it is also an interesting case study in adding a custom data "type" to Mathematica and extending the knowledge of the built-in functions to handle the new "type". First I have to explain the point by answering the question, "What's wrong with the built-in units package?" Well, there is nothing actually wrong with it, it just doesn't apply Mathematica's automation principles. It can convert between several hundred units and warn if a requested conversion is dimensionally inconsistent. But give it an input like... and it does nothing with it until you specify that you want the result in a specific unit. The core reason is that it doesn't teach the system, as a whole, anything about units, or even that the symbol "Meter" is any different than the symbol "x". All of the knowledge about units and Meter in particular is contained in the Convert command.
Jon McLoone, Director, Technical Communication & Strategy
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