500 Word Response: Why Does the Aerospace Industry Use Old Stuff?

This is the first of what I'd like to make a new feature on this blog a "500 Word Response" (in reality maybe a little more or little less) to a question I hear asked about aerospace all the time. Today: why does the aerospace industry use old stuff? If you have a question you want answered, please leave a comment!

In pop culture, new 'high-tech' systems, with a vast number of bells-and-whistles, are often regarded as good and 'advanced' technology. Every time I see a Science Fiction movie it showcases all sorts of (CGI'd) aerospace tech with light-up touch screens and advanced robotics and networking. While this is fun to watch it is a solidly fantasy version of aerospace and bears no resemblance to what a real aerospace engineer will look for in a good system. Instead of looking for the most complicated possible design, a good aerospace engineer will choose the simplest, most reliable design which satisfies the requirements, typically given by the customer. If the customer wants the moon they'll get it, typically for a big price tag (F-35 anyone?) but otherwise 'extra features' and additional complexity are often and rightly seen as hindrances. The space industry takes this to the furthest extreme since you can't exactly go up and fix a malfunctioning spacecraft! So it is to be understood that any design decisions are going to be based on maximizing reliability and minimizing complexity.

So why all the old stuff? The answer is simple: heritage. How do you know, beyond any shadow of a doubt,that your systems is going to be reliable when it is flown on an actual mission? Because it has already proven itself! So long as the old system meets the requirements the argument from heritage will often (if not always) outweigh any concerns along the lines of 'but this is 1980s technology!' As with simplicity and reliability, the space industry makes this the biggest priority. That's why whenever I see someone making fun of an aerospace system which uses heritage components I know for a fact they have no real knowledge of what their talking about—heritage, i.e. old tech, is a blessing not a curse. Of course old tech is more limited than modern tech. Your smartphone could probably outperform many spacecraft computers, but can you guarantee your smartphone will work in space? Can you guarantee it will be reliable in the radiation environment? Can you guarantee there wont be some silly software or hardware error because of the unnecessary complexity? Probably not. As an aside there actually was a 'phonesat' movement a few years ago, but this came out of a new paradigm in the space industry called 'microspace', and would require a lot more words to adequately explain, and they experienced all the problems listed above.


Finally, the obvious question is how do aerospace systems improve if they only fly old tech. Again the answer is simple: new requirements are given which rule out the use of the old tech. Ultimately in any design it is the simplest system which meets all of the requirements. There is no such thing as skimping on demanded performance for the sake of simplicity—otherwise all spacecraft would be like Sputnik, a radio beacon with primary cells. Once the new tech is designed and successfully flown to meet more exacting requirements it too gains flight heritage and enters the pantheon of TRL 9 aerospace systems. 

Thoughts on LightSail

This is the first of what is likely going to be multiple posts on solar sails, and in particular my thoughts on using solar sails on CubeSats. Although there are many more basic starting points than what I'm going to talk about in this post, I'd like to dig in immediately and talk about the recent trouble with The Planetary Society's CubeSat LightSail. For those who haven't heard, LightSail is a CubeSat mission with the goal of showing solar sails can be deployed from a CubeSat in space with a secondary goal of making sure the extra inertia can be controlled by the ADCS (attitude determination and control system). Although I'm an unrepentant partisan for all things small sats and a big fan of Bill Nye, especially after his heavy promotion of science education during the past few years, I am simply unsure about this mission.

First, it seems only marginally interesting at best. In the Low Earth Orbit (LEO) they have selected for the CubeSat, atmospheric drag will far outweigh solar radiation pressure by many orders of magnitude. This means even if they manage to deploy the solar sail it will be useless as a propulsion system, instead doing the opposite and pulling the CubeSat down even faster than normal by increasing the drag force (proportional to area) on the CubeSat and sail. This isn't necessarily a useless thing however, as there's actually a certain niche for deorbiting devices! This is because those of us in the small sat world tend to love our LEO orbits and not want to clutter them with debris, so we've all agreed that every CubeSat has to burn up in the atmosphere at the end of its life. Having a deorbiter allows you to maintain a higher altitude LEO while still meeting this requirement so it's actually useful! But that's not what the purpose of the sail on LightSail is, nor what it is continually billed as in the media. It's instead billed to be a solar sail, a la Clarke's concept, so why not simply select an orbit which will allow the sail to actually function as a propulsion system? Although there could be many reasons why TPS didn't do this (e.g. what if the sail doesn't deploy and LightSail itself becomes space debris in a higher altitude orbit?), it doesn't make sense to actually go through the entire design process just to do a marginally interesting technology demonstration. Now that I think of it, this could just as easily been done on a series of suborbital parabolic flights. Despite these qualms however, this thing was selected, designed, constructed, and presumably passed some sort of reviews, although recent issues make me think these must have been minimal.

Now let's get to LightSail's current situation: which is that it's just recovered after experiencing a software failure. Basically because LightSail beacons telemetry data every 15 seconds for some unknown reason and because the flight software stores these beacons for some even more unknown reason, the on-board computer (OBC) experienced a system-wide software failure and became frozen. The excuse for this was that the flight software was unpatched by some commercial distributor, an excuse I don't believe I've ever heard before, doubtless because it is so weak. It is unbelievable that you would pass Flight Readiness Review without having convincing validation of your software and without running it for at least a few days in simulation! Most spacecraft have specially written flight software for exactly this reason and suffering a failure because of not having a patch (which will now be a recurring issue if you cannot uplink the required changes) sounds more like an excuse for a computer game not working than a spacecraft failing in flight. Ridiculous.

To make matters worse, a second critical design flaw--one that can't be fixed by telemetry--prevented TPS from power cycling to overcome the issue: they have no analog reset! Weird stuff happens to flight software all the time and frequently you just have to power cycle and resume where you left off. But how do you know to do this? The standard solution is to have an element in the Electrical Power System (EPS) which constantly pings the OBC and, if it fails to receive a response to the ping for a certain amount of time (typically hours or days), initiates an analog reset into the deployment state by pulling bus power and firing up a timer circuit which in turn initiates the boot phase for the OBC (alternatively the timer can be left out or the pinger can be internal to the OBC board and simply reboot as soon as a freeze is detected). Evidently TPS didn't include this basic component and so they tried, unsuccessfully, to telemeter a commanded reset knowing their OBC was frozen! This is beyond words for me. Of course they failed and simply had to shrug and say they hoped a single-event-upset would reboot them. It is beyond fortunate that this occurred in a short amount of time since their entire mission could have easily failed otherwise.

The now inexplicably long wait time of 28 days to deploy the sail, itself bizarre and simply asking for something like this to interfere with their primary objective, has been thoughtfully revised to be as soon as they are sure they've regained control of the spacecraft. All this drama and they've not even attempted their rather trivial threshold objective. Moreover, if the sail deployment mechanism design is anything like their OBC or EPS I am not so sure they will achieve even the small token of having deployed a sail in space. I'm sad to say LightSail is increasingly looking to me like a total waste of time and money. Moreover the publicity it has garnered due to Nye's celebrity, publicity which could have been used to help bring small sats more into the mainstream if they had only selected a good mission and diligently designed it, will doubtlessly arm those of my colleagues who still clutch the dying belief that CubeSats can only ever be toys with a high failure rate with yet more ammunition in reviews, publications, and formal and informal discussions informing (or rather misinforming) the attitude of our field in general.

So far LightSail has been a huge disappointment for me and I wish Bill Nye of all people could have given a little more thought to exactly what he was doing. I hope they are successful. I hope they can successfully deploy the sail and even swing the spacecraft around a bit with the sail deployed, but no matter the outcome I ultimately think LightSail will go down as a rather marginal and unremarkable piece of small sat history.

Edit: As of June 7 2015, the CubeSat is once again failing to communicate with the earth with the batteries in a suspected safe mode. The current cause of the error is unknown.

Edit 2: As of June 8 2015 the CubeSat has recovered from the second failure and the boom deployment motor shows a record record of it moving to deploy the sail. Media and LightSail personnel have somewhat disingenuously reported this as "the sail is deploying" when in truth we have no images to make sure the deployment is proceeding correctly. Hopefully they are correct in assuming that just because the motor is turning the sail is correctly deploying.

Edit 3: It seems as if they have an image of the sail and it deployed successfully, a major relief to all of us. I still find their attitude laughable: "LightSail Test Mission Declared Success; First Image Complete"--a heading which has to be the most arrogant way I've yet seen of announcing a spacecraft has achieved its rather tawdry threshold objectives. It is unclear whether or not this means this spacecraft will now be left to rot until it reenters since most good missions typically have more than one objective, but given that the CubeSat now has a giant reentry device attached to it I can't see much else they can do before it reenters. 

Thank You

Before the next proper post, I'd like to take a few lines to thank those who took a look at the previous posting on Eagleworks. I've been keeping this blog on-and-off for about a year now with I think scarcely anyone looking at it but me. I don't mind that, since it's a labor of love and not a quest for fame, but it's nice to log on to my email and see a fellow left a comment saying "good article" and then see that over thirty people viewed the page in one day! Anyone's a sucker for a little praise and I'm flattered so many more than the usual amount of folks stopped to consider my opinion. Thanks to you guys and now on to more rocket science!

A New low for 'Eagleworks' and Space Journalism

Well folks, it's back like a bad dream. Another summer, another blizzard of media hype for a crackpot propulsion concept out of the self-named 'Eagleworks' lab at NASA's Johnson Space Center. Around this time last year I posted a rant about how they were using the conference structure to effectively 'publish' a paper without peer-review, the paper in question being hosted on an AIAA server and being typeset as an AIAA paper, all while the AIAA claimed to have 'not published' the seriously flawed work. The central outlandish claim was that they had found an object (I do not glorify it with the title 'thruster') which produced thrust without using any propellant--a 'reactionless' drive. The true fact that any such device would contradict the basic laws of physics was also widely reported, but instead of using this knowledge to come to the correct conclusion that the measurements were flawed, the media instead decided to pan the conservation of momentum, an unorthodox but evidently acceptable target for tabloid hatred. 

Of course even a cursory review of the conference paper revealed gaping flaws. Among many others was that measurements of micronewton thrusts were performed at atmospheric pressures! For those who are not engineers, a micronewton is a very small amount of force. If you softly blow on a piece of paper, you are still generating about 10-100 MILLInewtons of force (1 millinewton = 1000 micronewtons), and so the idea of performing such sensitive measurements under atmospheric pressure is laughable. This is likely the reason Eagleworks, in true crackpot fashion, obfuscated this fact by changing their abstract to falsely say the tests were performed in vacuum conditions when they weren't. 

Although little more than a quick laugh in the real space community, Eagleworks' nonsense infected social media for a couple of weeks. When I came to Purdue University last fall, I even found undergraduates (in Engineering) who told me their greatest ambition was to work for Eagleworks and Harold White--a dismal thing to hear from what is supposedly a top Aerospace engineering school. And now it's back in what has become the third episode in this saga of ridiculous claims and media hype. 

The newest episode is the most pathetic yet. Faced with the challenge of coming up with yet another piece of garbage to float around the ocean of public discourse and evidently failing to come up with anything original, Eagleworks has instead decided to sew together, in what resembles a Frankenstein's Monster of propulsion claims, their two previous pieces of nonsense regarding 'warp drive' and their 'reactionless' drive--that's right folks, a reactionless warp drive! Instead of even dignifying those of us who work on legitimate propulsion research with something resembling what a real researcher might do, e.g. submit a paper to a conference or give a talk, they've decided to spit in the face of the entire community by posting an announcement on an online forum.

From the media hype however, you'd think this amateur, non NASA affiliated forum might as well have been a major aerospace journal! Left and right there were claims that "NASA has gotten one step closer to warp drive" or that they had now somehow validated their "reactionless" drive. Perhaps the most widely cited is this post on the unaffiliated nasaspaceflight.com. The conceited and almost offensively wrong article is simply too bad to not address, and actually I found that the main arguments against Eagleworks' nonsense can be summarized nicely by rebutting the ridiculous claims made in this post. The rest of this article will just be quotes from their article with anything anyone with a real background in physics would say following the quote

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Thrust measurements of the EM Drive defy classical physics’ expectations that such a closed (microwave) cavity should be unusable for space propulsion because of the law of conservation of momentum.

So they must be wrong. Classical physics in the classical regime is not debatable. Momentum conservation in any measurable regime is not debatable. If you find your measurements disagree with momentum conservation, they are wrong. Period. 

Last summer, NASA Eagleworks – an advanced propulsion research group led by Dr. Harold “Sonny” White at the Johnson Space Center (JSC) – made waves throughout the scientific and technical communities when the group presented their test results on July 28-30, 2014, at the 50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference in Cleveland, Ohio.

And it was the worst piece of nominally serious work in propulsion I've ever seen. It was a laughing stock within the major university laboratory where I was an engineer and even the AIAA did not want to take responsibility for it, denying it was ever published by them.

Those results related to experimental testing of an EM Drive – a concept that originated around 2001 when a small UK company, Satellite Propulsion Research Ltd (SPR), under Roger J. Shawyer, started a Research and Development (R&D) program.

And it was a crackpot idea which the British Government foolishly spent money trying to develop to no avail. Shawyer is not an engineer and his unpublished 'paper' on this concept is ridiculous. 

This lack of expulsion of propellant from the drive was met with initial skepticism within the scientific community because this lack of propellant expulsion would leave nothing to balance the change in the spacecraft’s momentum if it were able to accelerate.

This is strictly speaking wrong. EM waves carry momentum, and so they could in principle be used for propulsion, but the amount of momentum they carry is so tiny compared with the power required to produce them that it is an extremely inefficient way to do business. This is called a Photon Rocket and has been used for years as a way to understand the limitations of propulsion systems. 

However, in 2010, Prof. Juan Yang in China began publishing about her research into EM Drive technology, culminating in her 2012 paper reporting higher input power (2.5kW) and tested thrust (720mN) levels of an EM Drive.

Her paper is also garbage, published in a low-quality Chinese journal, and has a multitude of elementary errors. If you even take the numbers cited in this quote and calculate the thrust-to-power ratio, you will find it exceeds that of a photon rocket, equivalent to getting energy from nowhere, an absurd and obviously false conclusion. 

Dr. White proposed that the EM Drive’s thrust was due to the Quantum Vacuum (the quantum state with the lowest possible energy) behaving like propellant ions behave in a MagnetoHydroDynamics drive (a method electrifying propellant and then directing it with magnetic fields to push a spacecraft in the opposite direction) for spacecraft propulsion.

And as usual Dr. White is speaking nonsensical technobabble. There are no ions in the qft vacuum, and the charged particles which are produced disappear so fast that they could never even feel an applied electric field much less be accelerated through one. The qft vacuum is also, like everything else in the universe, relativistic, which is to say you can't really push against it. For White's claim to be true, all of physics starting with Newton would have to be wrong. There is so much evidence to the contrary, it is appealing to suggest that it is instead Eagleworks' shoddy laboratory skills which has produced this erroneous result.  

This model was also met with criticism in the scientific community because the Quantum Vacuum cannot be ionized and is understood to be “frame-less” – meaning you cannot “push” against it, as required for momentum.

oh wow, they actual get something right. So now you should conclude that since relativity is known to be correct, this explanation is garbage. 

The tests reported by Dr. White’s team in July 2014 were not conducted in a vacuum, and none of the tests reported by Prof. Yang in China or Mr. Shawyer in the UK were conducted in a vacuum either.

you wouldn't know this from their abstract, but yes, this is likely to be one source of error. On a side note, not knowing you have to test microthrust devices in vacuum is the sort of abysmal ignorance that would be expected of a freshman in the lower half of his or her physics 101 class, not the leader of a NASA "research" group. 

The scientific community met these NASA tests with skepticism and a number of physicists proposed that the measured thrust force in the US, UK, and China tests was more likely due to (external to the EM Drive cavity) natural thermal convection currents arising from microwave heating (internal to the EM Drive cavity).

To just name one source of possible error (probably the most egregious). This DOES NOT mean this is the ONLY source of error. Proper microthrust measurements take months if not years for new devices. If you get the result that your device violates known physics, you should check your apparatus because there is a mistake. If the device is not in vacuum you are not measuring microthrust, you are measuring small perturbations on the device due to the environment. 

However, Paul March, an engineer at NASA Eagleworks, recently reported in NASASpaceFlight.com’s forum (on a thread now over 500,000 views) that NASA has successfully tested their EM Drive in a hard vacuum – the first time any organization has reported such a successful test.

Could you imagine if this was true? "Hey guys, I just found out all of physics starting from Newton must be wrong! Should we try and retake our measurements and, if we find they're good, submit our revolutionary findings to a peer-reviewed journal?" "Nah, let's just post something on an online forum, it's basically the same thing."

To this end, NASA Eagleworks has now nullified the prevailing hypothesis that thrust measurements were due to thermal convection.

No they haven't. A couple of idiots have just posted some piece of crackpot garbage on a forum. If this were a real lab we'd like to see some kind of data first. Knowing that Eagleworks disingenuously makes things up or obfuscates their flawed procedures to support their absurd claims however, I would not trust anything they produce. 

A community of enthusiasts, engineers, and scientists on several continents joined forces on the NASASpaceflight.com EM Drive forumto thoroughly examine the experiments and discuss theories of operation of the EM Drive.

I didn't wade through the >100 page sewer that was that forum post, but I'm guessing everything was as hilarious and wrong as White's bizarre claims. I also like how we're now giving "enthusiasts"--i.e. amateurs--the same amount of say in this as engineers and scientists. Though if these were Eagleworks' "engineers" and "scientists" the amateurs might actually be better than them. 

This synergy between NASASpaceflight.com contributors and NASA has resulted in several contributions to the body of knowledge about the EM Drive. 

Yep, advanced propulsion is officially in the toilet. Of course they don't name a single contribution made by the amateur forum posters, although if they had it wouldn't matter since everything involved in this refuse is about as legitimate as Douglas Adams' Impossibility Drive--and not half as entertaining. 

The NASASpaceflight.com group has given consideration to whether the experimental measurements of thrust force were the result of an artifact. Despite considerable effort within the NASASpaceflight.com forum to dismiss the reported thrust as an artifact, the EM Drive results have yet to be falsified.

Probably because these are amateurs and not professional engineers. If you want to really claim acceptance you should submit to peer-review (whence the wonderful phrase "peer-review or it didn't happen"). From how the article sounds it seems like a lot of the amateurs had the good sense to understand Eagleworks' claims are nonsense though, so that makes me feel a little better. Lastly, it's not the amateurs' (or anyone else's) responsibility to "falsify" the Eagleworks nonsense, it is Eagleworks' job to prove it. If they are doing ridiculous things like testing for microthrust without using a vacuum it is exceedingly likely their results are garbage because they are obviously incompetent. 

After consistent reports of thrust measurements from EM Drive experiments in the US, UK, and China – at thrust levels several thousand times in excess of a photon rocket, and now under hard vacuum conditions – the question of where the thrust is coming from deserves serious inquiry.

No it doesn't. It deserves to be mocked and shamed. It's an absurd degree of professional incompetence and ignorance coupled with conceited media showboating which makes me weep for humanity. All of these 'experiments' are riddled with basic flaws and have produced widely divergent results, which are inconsistent not only with all physics but with each other. There was nothing in the claims of Shawyer, Guido Fetta, the Chinese (I don't remember their names and honestly don't care), and Eageworks the first time. There still isn't anything except a new low for NASA's biggest embarrassment and yet another insult to those of us who put in a lot of effort and sacrifice a lot of time to try and produce real propulsion devices. 

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The article goes on for a while after this, but honestly I'm so tired and so depressed that I have to stop here, shut down my computer, and walk to the liquor store to buy a bottle of Bourbon. After actually reading this drivel I'm going to have to spend the rest of the night drinking and weeping for my nation's sick and failing national civilian space agency. 

To the hope of better times ahead...

Mach Number Versus Speed and Velocity

I've just a quick post today clearing up a simple misunderstanding I see all the time in amateur/layperson discussions of AEROnautics (taking a break from spaceflight for just a little bit). This is the difference between the mach number and speed/velocity of an aircraft. Quite frequently people confuse mach number for speed--e.g. talking about how fast or agile an aircraft is by talking about what mach number it can fly at--when the truth is a bit more complicated than this.

While the meaning of an aircraft (or rocket's) speed is clear to most people, its mach number is often treated as an equivalent substitute. Saying you're moving supersonically is a euphemism for going fast. But the mach number of an aircraft--and which side of the sonic boundary it's on-- is equal not to just the the speed of the aircraft but rather the speed of the aircraft divided by the local speed of sound. As an equation this is written

\begin{equation}
M = V/a
\end{equation}
where M is the mach number, V is the speed, and a is the speed of sound. At first glance, especially to those new to aerospace, this may seem pointless; the speed of sound is constant, right? Well, actually no, it isn't. The speed of sound in a fluid varies as a function of three parameters: (1) the temperature of a fluid, (2) something called the specific gas constant of a fluid and (3) something called the specific heat ratio of the fluid. The first parameter has to do with the actual state of the fluid, while the second two only have to do with what kind of fluid is used and therefore are true constants for the atmosphere; at least to a good approximation.

Most people know that as you go further upward into the atmosphere, the pressure decreases. This is because there's less air molecules the further you go from the surface of the earth and thus there's less of a force per unit area pushing on you. What many people forget is that temperature too changes, as can be seen in the image below.

In everyday, low altitude flying, this change in temperature is not enough to significantly change the speed of sound, and thus in principle the mach number might be okay to use for a rough speed comparison, but there are actually only very few low altitude planes where the mach number matters, mostly because the speed of sound is so much faster than the aircraft at sealevel. At higher altitudes the speed of sound drops off. At still higher altitudes it grows again, drops off again, and finally grows strongly as you exit the planet. Thus the altitude an aircraft is flying at strongly influences the meaning of its mach number with respect to its velocity! 

A Boeing 747, for instance, cruises at roughly 20-30,000 ft--safely within relatively sealevel like conditions. Its mach number will be a good approximate indication of how fast it would be going at sealevel, but an SR-71 cruises at 85,000 ft, where the local speed of sound is almost 10% less! This means the mach number of the SR-71 would be greater than its mach number at the same speed at sealevel. There may even be flight regimes where the aircraft is operating supersonically in the altitude it was designed for and subsonically in a lower altitude. 

Other than speed, what is the mach number good for? Well, all sorts of things. In compressible fluids, like air, the equations governing aerodynamic and thermodynamic behavior are strongly related to the mach number, and the way they affect the structure and forces on an aircraft change considerably as it breaks the sound barrier. For instance, the heating which takes place behind a supersonic shockwave is significant, and thus an aircraft flying supersonically must have a well designed nosecone if it is to withstand this heating (since the nose is for all intents and purposes behind a supersonic shockwave as the plane is in flight). 

So next time someone quotes a mach number at you when talking about how fast an aircraft is going, ask them if it is a high altitude or low altitude aircraft. A large mach number for a high altitude aircraft may not correspond to a hugely greater speed than a low (or even subsonic) mach number for a lower altitude aircraft. 

A Mathematical Vocabulary for Modern Engineering

My adviser and I sat down today for one of our typical meetings. After we had talked about some of my new results and discussed a paper (not mine, thankfully) which he referred to simply as "a mess", the discussion shifted. He started asking me about my background and what sort of math and physics courses I had taken while an undergraduate. We have similar backgrounds in a way--both physicists turned engineers, both interested in the more scientific side of engineering, and both less persnickety about obtaining rigorous numerical results in any given calculation than we are about making sure our concept is right. But one huge difference in our backgrounds is where we were educated. I went to an American public university while he was educated in the Soviet Union. I graduated just over a year ago, having the full blessing (and curse) of powerful computing tools at my side, while most of his computations were done with pencil, paper and a lot of careful thought. I knew the main point in his asking me about my education was to try and estimate the skill level of the students in a course he is currently teaching in plasma physics, and his realization that many American engineering students today are woefully uneducated or undereducated in the physics and math they need to understand and solve modern "high-tech" engineering problems.

He asked me how many semesters of general physics were typically included in our curriculum, I said three. He was clearly disappointed and said "there should be five." He then asked me what sort of math most engineers take. I said "three semester of calculus, one of linear algebra, and one of differential equations."

Him: Any probability or statistics?

Me: Not required.

Him: What about PDEs? How many semesters of ODEs?

Me: PDEs are also optional and only one semester of ODEs.

Him: Complex variables?

Me: A few weeks in some math methods class if at all.

He frowned again. "when I was an undergraduate we had two semesters of both PDEs and ODEs, and another for complex variables." My heart sank with his as I too realized how little most American engineers are taught about math. I chose not to inform him of how little regard most of them even have for math--useful math like series expansions, special functions, and complex integration; not just obscure modern mathematics, which can only rarely be used to gain useful results. He told me some students in the class couldn't even understand how to compute the average of a function, that others had no idea what Bayesian probability meant, that still others had no idea what the divergence or curl of a vector field was. This meant he would have to tailor his discussion of concepts involving these basics to a level far below that of a graduate course, something that myself as a student enrolled in the course found disappointing. It is a sad state of affairs.

On my way home I started thinking about how this could be changed. There's certainly no shortage of stuff to do on this count, but I thought that the first step might be establishing a basic mathematical vocabulary for modern engineering. This would be a list of mathematical topics to serve as an educational guide for people who want to be good engineers (as opposed to marginal, unqualified, or simply bad engineers) and would include actually "advanced" topics rather than the basic topics of calculus, power series, rudimentary linear algebra and ODE theory--which are typically taught in upper-division engineering math courses, but in reality should be assumed knowledge from freshman courses and high school. I've decided to list these topics and post them here. If I get enough time I might try to find good resources for learning what you need to know about the topics I've listed, but for now you're on your own. In order to be complete, I'll go ahead and list the basic topics under "Basic Math" but hopefully everyone with an engineering degree has at least seen those. The highly recommended (but not typically taught) subjects are classified as "Mid-Level Math"; these are math topics which can be learnt once the courses in Basic Math are completed and should be the bulk of the higher level math knowledge of the typical good engineer. Finally comes "High-Level Math" which may not be extremely useful for every engineer but is a good idea to learn to get more of a math background and is essential for some of the more advanced parts of engineering.


Basic Math (currently required at most schools):

--High School Math (basic algebra, geometry, trigonometry, and "precalculus", e.g. analytic geometry and some properties of functions.)

--Calculus (3 semesters worth hopefully with an introduction to vectors and multivariate functions).

--Linear Algebra including the concept of a vector space.

--Ordinary Differential Equations.

--Rudimentary probability and statistics (even this is not always required however)


Mid-Level Math (optional or not taught at most schools but needed frequently):

--Calculus-based probability and statistics.

--Partial differential equations, including eigenfunction expansions and special functions

--Linear operator theory

--Detailed theory of function spaces, Fourier theory, integral transforms, and completeness

--Complex variable theory

--Detailed vector calculus theory

--Basic numerical analysis

--Basic optimization techniques


High-Level Math (typically not offered for engineers):

--Tensors and tensor calculus

--Applied group theory (do not learn this from a pure math book. Instead look for a physics book)

--Differential forms and curvilinear spaces

--Stochastic processes and measure theory (again making sure it is not from a pure math book)

--Classical and modern theory of the Calculus of Variations (could really be "Mid-Level")

--Topology and mathematical analysis (needed mostly in control theory)

--Numerical methods for ODEs and PDEs (CFD or other computational fields)


Is this a lot? Well, yes. Unfortunately it doesn't change the fact that modern engineering problems typically deal with math from at least the "Mid-Level" category, if not the "High-Level" category (I recently was surprised to find the group at Purdue which focuses on tracking Space Debris using hard-core measure theory to construct their models of the debris field.) High school, as you see, barely teaches you the most rudimentary aspects of the most basic mathematics which you will use as an engineer. The math courses you are forced to take in college cover less than a third of what you will probably have to use and less than a quarter of what you might have to know to be a top engineer in your field. To circumvent this failure of the typical engineering curriculum you will have to put in quite a bit of elbow grease and more than a few nights of voluntary studying. But if you are patient, dedicated, and reasonably intelligent you will find the benefits to understanding real engineering mathematics are enormous. With a definite and extensive mathematical vocabulary we will make the first step toward building much more widespread mathematical literacy in engineering, but the hard parts are yet to come.

Good night and good luck, folks.

The Worst News a Teacher Can Ever Hear

No, it's not that one of your courses was cancelled, or that you made a mistake during the lecture which caused everyone to fail the exam, or even that your favorite student has dropped out and decided to study philosophy instead.

The worst news a teacher can ever hear is that one of their students has been killed and sadly I've just heard that news.

The first class I ever taught was a physics class for the Humboldt County chapter of Upward Bound. It was the summer of 2011 and I was getting ready to pack my bags and head down to the Bay Area to finish my undergrad at UC Berkeley. Before I left I wanted to give something back to a community I knew I would never live in again and I chose Upward Bound because it has the noble goal of empowering disenfranchised and minority youth to receive educations and help their communities. As a poor kid myself, who left high school early and began my college career at a community college, I empathized with the kids who were in this program and decided I would offer them the physics class I wish I had in high school. For five weeks we talked about everything from Newtonian mechanics and falling bodies to quantum weirdness and the hunt for elementary particles, and I was seriously impressed with how these kids--poor Whites and Native Americans who never had the benefits of good high school classes--took to the material. I remember the summer of 2011 far more vividly than a lot of my time in Humboldt, and I think I'm safe in saying it was one of the most fulfilling things I've done so far.

All these memories have flooded back to me in a bitter sweet way as I learned one of my favorite students was killed in a car accident back in September. She wasn't always the one with the answer, but she always had a smile, a good attitude, and never once complained or whined about me or the course. After I finished teaching she 'Friended' me on Facebook and I had the opportunity of following her accomplishments. She graduated high school against all odds and even began attending the community college I started at. I desperately hoped to one day see a status that she had accepted a scholarship to attend UC Berkeley and telling her of all the things she could do there. Now the only updates I will ever see are condolences and friends saying "we miss you."

To say I'm tremendously sad, upset, and in grief is an understatement. I never expected to see news this sad so few years after I taught that course and I'm at somewhat of a loss for words. The only thing I can think to say is the Hoopa Nation has lost one of its finest young women and my once pristine memories of the wonderful summer of 2011 are now tinged with a sadness I can never remove.

My thoughts are with her family and her friends.

I'm sorry for the personal post, but the this sad and unexpected news really affected me. More Rocket Science to come soon!