Who is Fubini?
I dig my calc III teacher. He’s awesome. But I wish he’d do what I wish all math teachers would do when they introduce a theorem or lemma or rule: tell us a little bit about the person responsible for it, especially if the theorem/lemma/rule is named after the dude.
Like today we talked a lot about Fubini’s Theorem. We used it in like three examples. I used it on the homework I did right after class.
All the while without knowing who the heck this Fubini guy was.
So I checked him out this afternoon. Guido Fubini was an Italian mathematician who lived from 1879 to 1943. He was pretty into geometry and calculus for most of his life and moved around in different professorships in Europe before accepting an invitation to teach at Princeton in 1939 (partially to get away from the Nazis; he was Jewish).
So what the heck is this theorem, anyway?
Well. Let’s just look at rectangular domains first (because that’s all we’ve learned so far, haha…we’re doing non-rectangular domains tomorrow). So let’s look at a pretty double integral to start.
(P.S. I’m loving this chapter on double integrals already, simply because it means I have to write more integral signs. I FREAKING LOVE THAT SYMBOL.)
Say some rectangular region R is defined by the intervals [a,b] x [c,d]. If a function of two variables z= f(x,y) is continuous over R, then we can write the volume of the solid that lies below the surface z = f(x,y) and above the rectangle R as:
Or:
Iterated integrals!
Cool? Cool. So what does Fubini’s Theorem state? Again, assuming z = f(x,y) is continuous over R and R is a rectangular region, Fubini’s Theorem allows us to switch the order of integration while still getting the same correct result at the end:
Which is pretty snazzy (there’s a few other statements in the theorem; I just chose this conclusion as the example to show here).
But what I found most interesting about this theorem is that while double integration has been around for quite a long time, this theorem was proved sometime during Fubini’s lifetime–sometime in the late 1800s or early 1900s. (I can’t find an exact date for it, but that’s mainly because my internet’s deciding to be a bitch right now). Which makes sense, I guess, considering there exist cases where this doesn’t hold and so it may not have been an “obvious” thing or may not have been easily provable…but still. Interesting passage of time before we got to this theorem.
HUZZAH CALCULUS!
DAMMIT CALCULUS III
(sung to The Police’s “Message in a Bottle”)
Takin’ calculus as a summer class, oh
Kinda nervous, too; afraid it’ll kick my ass, oh
Since it’s the last class in the calc trio
But it turns out it’s just calc I in R3, oh
I was scared I would not understand
I was scared I would not understand
But so far it’s a bunch of
So far, it’s just a bunch of
So far, it’s just a bunch of
Matrix operations, yeah
Matrix operations, yeah
A week has passed and I’ve learned so much
This stuff is fascinating to me
The actual math is quite easy
I can work out tangent planes but can’t draw them in 3-D
All my hyperboloids look like crap
All my hyperboloids look like crap
I hope they don’t ask me to
I hope they don’t ask me to
I hope they don’t ask me to
Draw one on a test, yeah
Draw one on a test, yeah
Oh, draw one on a test, yeah
Draw one on a test, yeah
Went to class today, I don’t believe what I saw
Many gradients and partials written on the wall
This seems simpler than calc II, I can tell
The hardest thing for me is writing that little del
Partial derivatives give me joy
Partial derivatives give me joy
I wish that this class wasn’t
I wish that this class wasn’t
I wish that this class wasn’t
Over in six weeks, yeah
Over in six weeks, yeah
Over in six weeks, oh
Over in six weeks, yeah
Calculate a vector length!
Calculate a vector length!
Calculate a vector length!
Calculate a vector length!
Etc.
Seriously though, I love this class.
Somebody needs to do this if it hasn’t been done yet
Imagine a creation story where the Cosmos gives us two brother gods: Integration and Differentiation. They are responsible for two components of the Universe.
Integration—”The Great Summer”—is in charge of unity and space (well, area, but let’s just go with space). He wields integral symbols as weapons and lives in the sky.
Differentiation—”The Great Changer”—is in charge of division and, of course, change. He’s able to take the smallest components of the universe (hence the “division” aspect) and create a degree of change in it*. He has armor made out of barbs tangent to his skin and lives in the earth.
Something to draw, maybe…?
*Yes, I know taking the derivative of a function does not cause the change measured. Just work with me here.
TWSB: Continuous, Continuous Everywhere but Not a Derivative to Find
WOAH, I’m pretty sure I’ve heard of this before but didn’t know what it was.
This is a Weierstrass function. Pretty, isn’t it?
Know what else is cool about it? While it’s continuous everywhere, it’s differentiable nowhere.
This challenged the notion that every continuous function was differentiable except on a small set of isolated points when it was discovered and published by Karl Weierstrass in 1872 (though some suggest that Riemann had made this discovery in 1871).
Haha, and I didn’t know this, but examples like this (functions that are continuous but not differentiable except only on a set of points of measure zero) are called monsters of real analysis.
That would be a FANTASTIC thrash metal band name. They could tour with Step Reckoner (the other fictional metal band I came up with, named after Leibniz’ calculating machine).
I LOVE THE PIGEONHOLE PRINCIPLE ASDFDHFHJDFJFLFSGHHH
So today is the last day of classes before finals. We spent today talking about the Pigeonhole Principle in Discrete Math. The Pigeonhole Principle is one of my favorite math-related things. I’ve done a blog on the actual Principle before, but while we were talking about it today I think I realized why the conclusions you can reach from the Principle seem so counter-intuitive to people (or at least why it was so to me when I first learned of it).
Let’s take a fairly simple example to demonstrate.
Suppose I have a group of 27 individuals. According to the Pigeonhole Principle, I can state that at least two of these people will have names that start with the same letter of the alphabet. I won’t go into why this is so (you can read my version of the explanation here on my previous PP post, linked above), but even if you’re pretty familiar with it, this still seems a little counter-intuitive, doesn’t it? You think, “wait, how can that possibly be a valid conclusion? There’s no way we can guarantee that!”
Where does this aversion to this conclusion come from?
Well, originally for me, I realized it stemmed from how I actually interpreted the conclusion itself. I always automatically interpreted the conclusion as claiming, in the case above: “there will be at least one name that starts with every letter of the alphabet in my group of 27 people.”
Which, of course, is not what the conclusion says at all. There is no claim made about the dispersion of the number of names per letter other than the fact that at least one letter will be the first letter in two names. I could have the case where I have a single name beginning with each letter A-Y, and then two names that begin with the letter Z. That still fits with the conclusion. However, I could also have NO names that start with a letter A-Y and have all 27 start with the letter Z. That’s valid, too. All the conclusion tells us is that at least one letter will begin two names. It doesn’t say that all the letters have to start a name (that is, it doesn’t say that all “pigeonholes” actually have to be utilized).
Now that I’ve typed that out, it seems like a really stupid reason for having trouble understanding the Principle, so it’s probably just me who has this issue. But anyway.
Isn’t this Principle COOL either way?!
ZOMG KLEIN AND KOLMOGOROV
(Again, sorry for spamming y’all with mathematicians.)
(Also, I’m using this as a distraction from the fact that I’m a complete moron who will never amount to anything ever.)
HOKAY.
SO.
Two badass math fellows were born on this day.
1. Felix Klein (isn’t that a badass name?)
Klein was a German mathematician born on this day in 1849. He is most famous for the non-orientable surface named after him: The Klein bottle!
Picture (from Wiki):
A Klein bottle is similar to a Mobius strip except unlike the Mobius, it is a closed manifold rather than an open one. And while a Mobius strip can be imbedded in 3-D space (which is why we can make them with just a piece of paper and some tape!), a Klein bottle cannot.*
2. Andrey Kolmogorov
Kolmogorov, a Soviet mathematician born on this day in 1903, was known for quite a lot of things, but for me the thing that jumps out is the nonparametric statistical test he had a hand in developing.
The Kolmogorov-Smirnov test is one used when your data just aren’t being normal but you still need to examine them somehow. In particular, the test is used when you want to determine if two datasets differ significantly. It gets around the issue of nonparametric-ness by making no assumption about the distribution of data in either set.
I’d go into how, but it’s late and I have to get up early tomorrow and if I start talking about stats I won’t shut up for days, so make me promise I’ll do a blog on the K-S test later.
But yeah. Two awesome math dudes today! Woo!
*That doesn’t stop people from making Klein-esque models. If you want a Klein bottle model, go here.
Mathed Memes
HAHAHA MY LIFE IS COMPLETE!
That parabola sure can shake its minimum.
Edit: holy freaking crap, I laughed for like ten minutes at this:
THE SLOPE! THE SLOPE! THE SLOPE IS ON FIRE!
(Yeah, I’ve pretty much given up on my titles.)
So here’s a question that you may or may not have pondered: when we write the slope-intercept equation for a line, the m in y= mx + b is our slope, right?
Why the heck do we denote it with “m”?
There’s quite a range of theories.
According to Pat from Pats’blog, the word “slope” itself is derived from the Latin root slupan for “slip.” Which makes sense when you think of what the slope actually is.
A common myth is that Descartes first used m because it was the first letter of some French word related to slope, but according to a bunch of people who speak French (and we should probably trust them about their language) the appropriate word for slope is “pente.”
Pat digs up some info from Jeff Miller, who claims that the earliest use of m dates back to 1844 when Brit Matthew O’Brien wrote “A Treatise on Plane Co-Ordinate Geometry” and Irish George Salmon published “A Treatise on Conic Sections.”
Another possibility was pointed out by John Conway, who suggested that m could stand for “modulus of slope.”
But in the end, no one’s really sure exactly when and why we got to using m for slope. I’m sure there are a fair number of mathematical symbols we use that don’t have a clear origin, but I know I’ve never really thought about m for slope before. I guess that’s because when I first learned y = mx + b I always thought m was appropriate because if you follow the trace of the letter the slope changes a whole bunch.
I was a dumb kid.
Will Will will Will’s will to Will?
Today I present The Stages of Claudia’s Reaction to a Math Test
Right before the test: I freaking love calculus! I totally know this stuff.
Looking over the problems: LET’S DO THIS!
Doing the problems: What’s a plus sign?
Right after handing it in: Crap. That went badly.
10 minutes after handing it in: I suck I suck I suck I suck I suck
Rest of the day: WHY AM I SO BAD AT EVERYTHING I LOVE?
Next day: I’ve disappointed myself.
Following day: I’ve disappointed the gods of calculus.
Following day: I’ve disappointed everyone.
All next week: I am a worthless soul who can’t do anything right. Why do I even bother, it’s not like I’m smart enough for any of this. [insert obnoxious amount of pointless angst]
Getting the test back: Oh, an A. Okay.
This has seriously happened three times this semester. Still trying to shake that damn math test anxiety that’s been following me since high school.
The math part of my brain (that ITTY BITTY LITTLE TROOPER) is internalizing some substantial portion of this awesome stuff. Why can’t the rest of my brain figure that out?
I never have this problem with stats. More proof that at least for me, stats and math are quite different things.
N-N-N-NAPIER
I should just change my (semi-)weekly science blogs to “In This Blog Claudia Blah Blahs about a Mathematician” because that’s pretty much what I do weekly anyway.
(It’s ‘cause of that damn birthdays site, man.)
Today’s feature: John Napier of Scotland!
Yeah, he was a cool dude. Did some stuff, you know, just a few small things like DISCOVERING LOGARITHMS.
Napier studied math as a hobby (his main focus was theology) but, wisely, turned more towards math upon discovering logarithms and subsequently publishing a book about them in 1614. He created tons of calculating tables that were used to make calculations involving e much easier. He also invented an abacus-like device that could be used to quickly calculate products and quotients of numbers. This tool was called Napier’s bones because it involved the use of 10 long rods printed with numbers. The rods, back in the day, were made of ivory and thus looked like long bones.
He also did work with decimal notation, refining previous notational standards set in place by Simon Stevin.
Despite natural logs being my natural enemy (HA GET IT no seriously my brain cannot handle them), I’ve gotta admit that discovering freaking logarithms is pretty damn snazzy.
Not “discovering calculus” snazzy, but snazzy nonetheless.
END!
N-N-N-NAPIER
