It’s All Greek to Me

You know you’re an engineer when you recognize Greek letters on fraternities because you’ve seen them in your classes, not the other way around. When you get that deep into math and science, sometimes English just doesn’t cut it, but if the whole point was to increase the number of possible variables, why does each Greek letter stand for five different things? See below, with notes on which fields you’d commonly find each definition in.

Γ – gamma function (math), Poynting factor (thermodynamics)
Δ – change (math)
Θ – wavefunction (math, quantum mechanics)
Λ – deBroglie wavelength (thermodynamics)
Ξ – one of the thermodynamic ensembles (thermodynamics)
Π – product (math)
Σ – summation (math)
Φ – dissipation (heat and mass transfer, fluids)
Ψ – wavefunction (math, quantum mechanics)

Ω – domain of integration (math), angular velocity (general science), thermodynamic 

       ensemble (thermodynamics)

Not used (they look like their English counterparts): Α, Β, Ε, Ζ, Η, Ι, Κ, Μ, Ν, Ο, Ρ, Τ, Υ, Χ

Here's where things get real fun:
α – kinetic energy correction factor (fluids), angle (math), heat diffusivity (heat and mass
      transfer)
β – angle (math), coldness (=1/k_bT; thermodynamics)
γ – specific weight (general science), activity coefficient (heat and mass transfer,
      thermodynamics), heat capacity ratio (thermodynamics, chemistry)
δ – distance (general science), delta function (math), Kronecker delta (δ_ij) (fluids, heat and
      mass transfer, quantum mechanics)
ε – molar absorptivity (chemistry), electromotive force (physics), electron charge (also
      physics, general science), permittivity of free space (hey, look, physics again), strain
      (fluids), permutation tensor (ε_ijk) (math, fluids), energy (general science), effectiveness
      (thermodynamics)
ζ – dimensionless breakage time (fluids)
η – dynamic viscosity (fluids), efficiency (thermodynamics)
θ – angle (math, general science)
κ – bulk modulus (fluids)
λ – wavelength (physics, general science), eigenvalue (math), distance (general science),
      mean free path (thermodynamics)
μ – dynamic viscosity (fluids), growth rate (biology), coefficient of friction (physics), chemical
      potential (thermodynamics), moment (physics), shear modulus (fluids), reduced mass
      (chemistry)
ν – kinematic viscosity (fluids), frequency (thermodynamics), stoichiometric coefficient
      (chemistry), microstate (thermodynamics), frequency factor (chemistry)
ξ – extent of reaction (chemistry), complex function (math), mesh size (fluids),
      nondimensional length (probably fluids)
π – 3.1415926535897932384626 (life), Henry’s law coefficient (thermodynamics, chemistry),
      dimensionless group (ChemE)
ρ – density (general science), mass concentration (heat and mass transfer)
σ – surface tension (fluids), standard deviation (math), stress tensor (fluids), stress (fluids),
      particle diameter (chemistry, kinetics), symmetry number (kinetics), Steffan Boltzmann
      constant (heat and mass transfer), resistivity (physics)
τ – torque (fluids), time (general science), deviatoric stress (fluids)
φ – probability function (math), characteristic function (math), polar angle (math), volume
      fraction (general science), benzene group (chemistry), fugacity coefficient
      (thermodynamics)
χ – mole fraction (ChemE)
ψ – wavefunction (math, quantum mechanics)
ω – angular velocity (general science), degeneracy of microstates (thermodynamics)
Not used: ι (looks like i), ο (looks like o), υ (no idea what this is)

ε and σ win for most definitions off the top of my head (8 each), with μ coming in second with 7 and φ in third with 6. Totally not confusing at all.

$9.03

In an effort to apply the knowledge gained in my two introductory economics classes avoid giving the Cornell Store any of my parents’ hard-earned money, I bought all my books for this semester on Amazon or from Ithaca’s only cooperatively owned bookstore (Buffalo Street Books). Due to the Cornell Cartel’s Store’s manufacturer-suggested pricing, with added contribution to the Buy the Dean a Drink fund, I got all the textbooks I needed for my ChemE classes for less than the Cornell Store was charging to rent a used copy of the book.

The liberal studies class I signed up for mainly because it fit in my schedule didn't have any books listed with the Cornell Store, but when I went to the first lecture, it turned out that we did indeed require books. Nine of them, in fact. We had the option of buying the books online from the bookstore and having them delivered to class, buying them from another source, or using the books on reserve at the library. I prefer to have my own copy of the texts, so that meant the bookstore or Amazon.

Because I obviously have nothing better to do with my time, I found all the books on Amazon and cataloged their titles, prices, and ISBNs. Then I compared the prices on Amazon to the bookstore prices. The bookstore sold the whole course pack at list prices for $137 while the books on Amazon cost about $115 (16% cheaper). However, Amazon would charge tax (about 8%) while the bookstore did not, and the bookstore would deliver to the lecture in time for me to do the first book reading, compared to five to eight days for an Amazon delivery. Furthermore, the books on Amazon were not uniformly 16% cheaper but ranged from list price to over 20% cheaper.

Naturally, what I ended up doing was buying the first four books we needed from the bookstore because they were all either list price or 15 cents cheaper on Amazon. I ordered the last five books from Amazon a few days later. I saved $9.03.

St. Patrick’s Day

The Google Doodle of the day tells me that it’s St. Patrick’s Day today, so what better way to celebrate than with some math problems?

1) A leprechaun is playing a game. He rolls a die, and every time his roll and the number opposite on the (normal, six-sided) die sum to seven, he drinks a quart of beer. After four rolls, what is the probability that the leprechaun has drunk a gallon of beer?

2) At the leprechaun bar, the probability that a leprechaun is forbidden another drink by the bartender is the same as the probability of flipping an infinite number of tails in a row with a fair coin. How many leprechauns go home sober?

3) Two leprechauns are playing ping pong with a ball with a coefficient of restitution of 0. The loser of each point drinks a pint of beer. What is the final score of the game, how long does the game take, and how drunk are the leprechauns at the end of the game?

Answers to come in my next post, and for the record, here’s the real reason St. Patrick’s Day is celebrated.

Happy Valentine’s Day

In the name of nonconformity and a lot of problem sets, my Valentine’s Day greetings are officially a day late. But it’s always a good time to say

and

and finally

What Does the Fox Say? and Other Musical Conundrums

First off, to clear up any possible confusion, the fox most definitely does not say “Ring-ding-ding-ding-dingeringeding,” “Fraka-kaka-kaka-kaka-kow,” or “A-hee-ahee ha-hee.” Foxes are animals. They do not talk. They do, however, make a variety of sounds, including high pitched barks, a type of scream/howl, and something called “gekkering.” See this site for a whole article about fox noises.

Next, if you've ever heard “Blowin’ In The Wind,” you might know that the song is absolutely filled with questions. However, to leave time to address several other songs, I’m only going to discuss the first question: “How many roads must a man walk down/Before you call him a man?” So, I thought about this, and decided that this man should probably be an expert walker. Studies have determined that it takes about 10,000 hours for a person to be an expert at something. If the average person walks 3.1 miles an hour, this man would need to walk 31,000 miles. Taking a few roads (non-interstate highways) in Ithaca as examples, road length varies from tenths of a mile to over ten miles. If a road is taken, on average, to be around 5 miles long, a man needs to walk down 6,200 roads before he can be called a man. If, however, this man wishes to walk down I-90, he only needs to walk down 9.994 roads (I-90 is 3101.77 miles long.)

Here’s one that’s right in the title: “Should I Stay or Should I Go?” Since I have no idea where you’re staying or going, I do not feel qualified to answer this question.

From the pep band folder: “What is Hip?” Well, the hip is a joint that connects the femur to the pelvis. It is a ball and socket joint, which allows for a large range of motion.

And the last one, also from the pep band folder, has been mentioned before on this blog: “Are You Gonna Be My Girl?” If you ask like that, the answer’s no.

Third Down, Five to Go

It’s third down, five to go. Do you
a) Try and run the ball. It’s only five yards and you might be able to get the ball through.
b) Attempt a pass. Again, since it’s only five yards you just need a short pass.
c) Give up and punt.
d) Change the unit of measurement to semesters and transition into talking about college.

Answer: d. I have survived yet another semester at Cornell. Not that I had much say in it, but here’s what I thought of the classes I was told to take this semester.

Honors Physical Chemistry I: Mainly, I enjoyed complaining about Mathematica and the Schrodinger equation. Other than that, it was interesting to approach chemistry from a more mathematical standpoint. We modeled chemical bonds as harmonic oscillators, discovered that orbitals are a lie, and complained some more about Mathematica.

Mass and Energy Balances: I didn’t want to dislike this class. I don’t think anybody goes into a class wanting to hate it. Though I didn't end up hating this class, I didn't love it either. I’d seen a lot of the material in Intro to ChemE last fall, but I didn’t mind that. Without going into too much detail, the class wasn’t structured in a way that let people learn and figure things out for themselves. Other than that, the material was interesting, and I thought the homework problems and projects were good. [Minus the due dates for the projects. The first project was due the morning after a p-chem prelim. The second project was scheduled such that my group ended up working on it for seven hours straight on a weekend. The final project was due during study break. By the time we thought to question whether that was even allowed, we’d been working for five or six hours and were “almost done.”]

Linear Algebra for Engineers: Out of the three math courses I’ve taken, I think this was the easiest. Part of it was due to the professor; he allowed prelim corrections and his prelims were very straightforward. It turns out that linear algebra and p-chem share a fair amount in common, to the dismay of the chemical engineers. Once was enough.

History of Science in Europe I: As it turns out, my liberal studies class required me to both read a lot and write essays. But I stuck with it because it fit in my schedule and I was actually interested in the topic. The class provided background information for how modern science (all the things I use in my other classes) developed. More work than I expected/was looking for in a liberal studies class, but I consider it worth it in the end. Though I don’t think I’ll be taking History of Science in Europe II. My excuse: it doesn't fit in my schedule.

Day Hiking: After nine years of mostly suffering through public school physical education, I can say that PE in college is much, much better. I took hiking just to get off campus once a week and tramp through mud, get bitten by bugs, and scratched by branches.

Double Trouble

I have come to the conclusion that although there are four academic classes listed in my Cornell student account, I’m really just taking the same class four times. [I am aware that the title of this post is “Double Trouble,” however, “Quadruple Trouble” doesn't rhyme, plus my liberal studies class is a little different. Mass and Energy Balances also has slightly different material, but I take it with all the same people as p-chem and linear algebra (to a lesser extent), so all three might as well be the same class.]

For example, we spent a good portion of the beginning of linear algebra talking about linear transformations because any linear transformation can be written as a matrix that’s multiplied by the vector or object to be transformed, and linear algebra is all about matrices. Meanwhile, in p-chem we were introduced to operators. Some operators are linear. Here’s how to tell if a transformation or operator is linear:

From linear algebra:

Definition: A linear transformation from Rⁿ to R^m is any function T(x) with two properties:

                1) T(u+v) = T(u) + T(v)

                2) T(cu) = cT(u)

From p-chem:

Linear operators

                Â[c₁f₁(x) + c₂f₂(x)] = c₁ Âf₁(x) + c₂ Âf₂(x)

Recently in linear algebra, we were thinking of functions as vectors in the context of inner products and related material.

Been there, done that.

Our linear algebra professor also made this note about inner products.

Don't forget the complex conjugate

And then last week we started our day off in physical chemistry learning about quantum spin and were introduced to the symmetrization and antisymmetrization operators. These operators have eigenvalues of 1 and -1 and work with the permutation operator to form the spin portion of the wavefunction for electrons. We went on to Mass and Energy Balances, had lunch, and ended the day in linear algebra, where the following happened:

We were writing quadratic functions as a vector x multiplied by a matrix A which was multiplied by x.  That’s not too difficult, but in order to apply a theorem we’d been working on, A needed to be symmetric.  Magically, if you call this symmetric matrix B and write B as (A+A^T)/2 + (A-A^T)/2, B will be symmetric.

The professor told us that the first matrix (A+A^T)/2 was symmetric, so naturally I made a comment to my friend about how (A-A^T)/2 should be called antisymmetric, in light of what we’d done earlier that morning in p-chem. I was completely kidding, but the next words out of the professor’s mouth were something along the lines of “and the second matrix is antisymmetric.” I’m not sure of his exact phrasing because all the ChemEs in linear algebra were too busy laughing, groaning, and/or crying in despair. ChemE is taking over our lives.

Platonic Friends

The most useless math class I have ever taken was my seventh grade pre-algebra class. We’d usually show up to class and begin with about five minutes of taking notes. Extra difficult topics, such as multistep single variable algebraic manipulations like 2x+3=7 – you have to subtract three from both sides, and then also divide by two – may have taken ten minutes. After copying down the notes, we’d do half a dozen practice problems that all required the same steps to get the answer.

Then we’d get riddle worksheets. If you've never seen these, you've missed out. Basically, the answer to each problem corresponds to a letter. When you get all the letters, you put them in a certain order and spell out the answer to a riddle given at the top of the page. The riddles were almost universally terrible. Awful humor aside, the other problem with these worksheets was that since so many letters were needed, and each letter corresponded to a problem, each problem required the computational power of a broken abacus.

Compare this to college, where upon seeing that your math homework for the week is a single problem, you should get very, very suspicious. When you go look up the problem in the book, you find that it has twenty-seven parts that all build on each other, meaning that when you get the first part off by a few hundredths, by the time you finish you've concluded that after the soccer ball was kicked, it reached a maximum velocity of 2,793 miles per hour. Right . . .

So you go back to part one, where you read: A rocket is launched at an angle 37° to the vertical. It reaches escape velocity 3.492 minutes after liftoff. For the first 57 seconds, it has a variable acceleration of 17x+89.43, where x is the distance above the ground of the center of mass of the rocket in inches. The rocket is 1 meter long with an initial payload of 20 pounds. Each second, 0.748 pounds of fuel burns off. Assume the rocket is a cylinder with the fuel uniformly distributed in a smaller cylinder inside the rocket. . . . (Three pages later) Find the escape velocity of the rocket and what color it is.

Compare both of these to my tenth grade geometry class, which was neither as useless as pre-algebra nor as ridiculous as multivariable calculus, but still had a few quirks of its own. Foremost among those was my geometry teacher’s use of the overhead projector. By that, I don’t mean that she taught using PowerPoint slides from a projector mounted on the ceiling. She had one of those, but she preferred to use a 1950s projector with one of those super hot bulbs and an arm that had to be adjusted to focus the image. She would then project her transparencies (circa 1980) straight onto the board, which would then make a bright spot that would a) blind you and b) make it impossible to copy the notes from that spot.

All of this is to lead into my favorite geometry lesson. It has nothing to do with riddles, confusing homework, or even the overhead projector. It has everything to do with the five Platonic solids. Which, in case you were wondering, are the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron.  (Platonic solids are polyhedrons with faces that are congruent convex polygons; the same number of faces have to meet at each vertex.)  As it happens, we have just enough of those magnetic bars and balls to make all five Platonic solids, with the dodecahedron stellated for structural reasons.

Clockwise from the large one: stellated dodecahedron, cube, tetrahedron,
octahedron, icosahedron

My personal favorite Platonic solid is the icosahedron. What makes my particular model even better is that

it glows in the dark.

On the Other End of Spring Semester

After suffering through putting a lot of effort into Intro to ChemE, multivariable calculus, and Honors General Chemistry fall semester, I found spring semester more relaxed, especially since I only ended up taking two engineering classes.  I dropped engineering stats a couple weeks into the semester for multiple reasons, one being that it was in a somewhat inconvenient time slot.  This way I got to eat lunch on Mondays.

Out of my remaining classes, one was Intro to Microeconomics, which didn't require a whole lot of outside work, and another was my writing seminar, which I just didn't care about mainly involved reading.  That left differential equations and my intro to computing class – I took MATLAB.  I wouldn't say either class was easy, but compared to Schrodinger’s equation and triple integrals in spherical coordinates, spring semester on the engineering front was definitely less ridiculous mind bending.

Here’s the rundown:

Introduction to Microeconomics: Since I had AP physics credit and didn't need to struggle through force diagrams and circuits again, I decided to get started on fulfilling my liberal studies requirements. The way this class was run spring semester, instead of problem sets we had to do online quizzes; that combined with the fact that I’d taken half a year of economics during my senior year of high school meant that this class wasn't too high on my I’m-going-to-fail-this-class-if-I-don’t-start-studying-yesterday list.  The material was interesting, and I did still learn some things, especially about consumer/firm theory and market structures.

American Voices: Monumental America: My second and last first year writing seminar.  I liked the discussions and content of this writing seminar better than the one I took fall semester.  However, my last writing seminar probably did more to improve my literary analysis in my writing.  But I’m an engineer, and among other things, I've been told to write entirely in passive voice before.

Differential Equations for Engineers: From what I can tell, a large difference between engineering math classes and math classes for other people is the proofs.  In over eighty math lectures, the closest thing I saw to a proof was a “proof.”  It went something like this: “So this leads to this, which we’re going to assume is true, and then you should believe this because I say it’s true, which means that this is the answer we’re looking for.  Now let’s do an example.”  Not much else to say about this class except that I now know how to separate variables and the professor I had takes some getting used to (including his inability to actually press down hard enough with the chalk on the chalkboard to make visible lines).

Introduction to Computing with MATLAB: or something like that. Everyone just calls it MATLAB. Definitely an intro class, since I did fine without having coded a single program before taking the class. My favorite projects were the ones with graphics (including a MATLAB valentine and several fractals), but MATLAB blackjack was pretty amusing as well.

And lastly,

Basic Rock Climbing: I've said it before, but this is the best PE class I've taken. I technically don’t have to take any more PE classes to graduate, but I've already jumped into cold bodies of water and climbed out of windows. A little more exercise couldn't hurt. Right?  So I enrolled in hiking for next semester.

Math and Music

It’s said that there’s a connection between the logical realm of math and the more abstract land of music. Scientists have done studies analyzing the rhythmic and structural underlyings of music to try and understand how they help to enhance math skills. They've got the research. I have the real life experience. Now presenting: How to become a math genius in three easy steps with the three Bs: Bach, Brahms, and Beethoven. That time in MATLAB I was reminded of pep band. Which isn't that hard, because a lot of things remind me of pep band.

Anyway, it was the last MATLAB lecture of the semester, and we were finishing up with sorting methods. We’d just done recursion, so the last sorting method we were learning (after insertion and bubble sort) was the merge sort. To start our discussion of the merge sort, the professor asked us if it would be easier to straight up sort a thousand items or merge two sets of five hundred items that had already been sorted. Well, this one time in band . . .

Actually, it’s been a lot of times in band. So the pep band “folders” are stacks of half sheets of music shoved into a pocket-type holder made of construction paper and/or duct tape. Normally, people keep their music alphabetized so they can find it more easily. However, during our busier events (hockey), most people don’t have time to put their music away between sets, so after the game there’s the folder, and then there’s the giant pile of music outside the folder . . . that needs to be sorted.

At this point, you have three options: one, take each piece of music and go through your folder to find out where it goes; two, alphabetize the music outside of your folder and then combine (hint: merge) the two piles; and three, ignore the giant stack of un-alphabetized music by shoving it into your folder and regretting it the next time you have to find a piece of music. Well, if you ask me, options one and two sound a lot like the insertion and merge sorts.

Speaking from experience, if around half your folder is sitting in a heap waiting to be put back into the folder, alphabetize it first so you don’t have to flip through the a’s seven billion times. (Merge sort) Otherwise, just sort each song in the order you find it. (Insertion sort)

And that is how math (okay, computer science, which needs math) is like music.

Pi Day

Pi day was . . . last Thursday. I had raspberry pie to celebrate. I also spent two and a half hours at RPCC for dinner, but that’s another story.

If you don’t know me or haven’t been reading my blog, you might not know that I like math. Unfortunately, I don’t have any math planned for today. Fortunately (or also unfortunately), I can just make it up as I go.

So, here goes. The closest fractional approximation for pi with both numerator and denominator under one hundred is 22/7. This comes out to be 3.142857, repeating all six decimal places. Sometime in middle school, I learned that every fraction can be written as a terminating or repeating decimal. Conversely, any repeating decimal can be written as a fraction (being able to write terminating decimals as fractions is fairly obvious).

Since then, I have probably gotten way too much amusement dividing fractions out mentally until they repeat. My personal favorite is 1/7, which happens to be the fractional leftovers of 22/7. Which brings us back to pi day.

Well, pi sounds like pie, which is like cake, which leads me to another fun mathematical problem. Suppose you have a square cake with icing on its top and all four sides. Suppose you also have ten friends who all want slices of cake with not only an equal amount of cake but also an equal amount of icing. Suppose any leftover cake will be smashed into your hair by angry friends who are mad that they didn't get more cake (this will also happen in the case of unequal slices). One cake, eleven people – how do you slice this cake to avoid washing dessert out of your hair?

It turns out that by dividing the perimeter of the cake by the number of slices you need, marking out these divisions on the cake perimeter, and cutting from each mark to the center, every slice will have the same amount of cake and icing. Seen from the top, each slice can be thought of as a triangle (or two triangles, for pieces spanning the corners). All the slices have the same height, as well as the same base (sum of the two bases for corners), since the bases are equal portions of the perimeter.

Therefore, since the area of a triangle is half of the base times the height, every person gets the same amount of cake and icing. Other solutions include buying a circular cake (you’d naturally divide a circular cake in exactly the same method as described above, but for some reason people like cutting square cakes into square/rectangular pieces), tossing the cake into a blender and distributing portions by weight, or getting rid of three friends (cutting eight equal pieces is a lot more intuitive than cutting eleven equal pieces).

I read about the solution (the first one) to the cake problem in the book Why Do Buses Come in Threes? (if you couldn't tell, the last three solutions are mine). This is one of my favorite books, and it discusses how math is involved in everyday life. Yes, I read books about math in my free time.  Credit to the authors for compiling the mathematical solution, but also feel free to use any of mine.

Excel did my Math Homework

We started the week in differential equations talking about Euler’s method of approximating the value of a function given the derivative and an initial condition.  Anyone who has suffered through been fortunate enough to learn about Euler’s approximations knows that it’s a pain in the neck, and inaccurate, to use Euler’s method to approximate the function once you get around 0.00001 units or so away from your starting point.  (It may be slightly more accurate than that, but math books like to have students do problems like “using a change in x of 0.01 and starting from 0, use Euler’s method to approximate f(x) at x=1.”  As you’ll see in a moment, not fun.)

So Euler’s method says that given y' = f(x,y), y_n+1 = y_n + (x_n+1-x_n) * f(x_n,y_n).  This basically assumes that the slope at the left-hand endpoint of the interval is the slope over the whole interval, and breaks up a function into linear pieces.

To do one of these problems by hand, you start with the given point and calculate the slope at that point.  Since the slope is the change in y over change in x, the slope multiplied by the change in x equals the change in y.  The change in y can then be added to the original value of y to get the new value of y.  The tediousness of the process arises because the smaller the change in x is, the more accurate the approximation.  But the smaller the change in x is, the more calculations it takes to get any significant movement along the graph, so after finding the new y, there should be sentences that say, “And repeat.  A lot.”

I was not in the mood to sit around punching seven hundred numbers into my calculator, so I figured I’d have Excel do the calculations for me.

Homework, done. #WhatILearnedInChemE

The Wheels on the Bus

The other day in the car, I was thinking about absorbers and strippers - in the engineering/ChemE sense, not the red light district bar sense.  That led to a contemplation on distillation columns (can you tell that ChemE has already started taking over my life?) and for whatever reason I thought that the rhythm of “L over V” (the slope of an operating line for a distillation column when doing McCabe Thiele analysis is “flow rate of liquid over flow rate of vapor,” or “L over V”) sounded a lot like “round and round” in “The Wheels on the Bus.”  Without further ado, I present “The Slope on the Graph” with its inspiration.

The slope on the graph                                       The wheels on the bus
is L over V                                                          go round and round
L over V                                                             round and round
L over V                                                             round and round
The slope on the graph                                      The wheels on the bus
is L over V                                                         go round and round
For distillation columns                                     All through the town

And I’m not done with that song yet.  The other night I realized that I’d soon be taking the bus back to Cornell.  After some research, I found that a bus wheel has a radius of about 20 inches, for a circumference of 40π inches.  The trip back to Cornell is around 325 miles, and so, the wheels of the bus have to go round 163,866 times for me to get back to Cornell. (For the record, the wheels of a car would have to go round 218,488 times on the same trip – car tires have a radius of about 15 inches.)
Just working to bring math into your everyday lives.