Minimizing Quantity Part 4: Euler-Lagrange Equation

November 23, 2013

In the last three posts, I talked about paths that minimize time. In all those cases, it involves objects going through a path and minimizing certain quantities. But is there a single equation that covers all physical situations that involve finding the path that minimizes quantity? The answer is yes, and it is called the Euler Lagrange equation. Read the rest of this entry »

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Volume of n-Sphere

November 10, 2013

If you know calculus and the gamma function, you might as well use it to try and learn to get the volume of an n-sphere:


Minimizing Quantity Part 3: The Brachistochrone

October 30, 2013

Last post, I finished the post cryptically, mentioning that there is a certain path that an object falling will take less time to travel than any other. Now, one of the solutions, as I mentioned, is just a straight fall. That is only one of the solutions, but only if the endpoint is directly underneath the starting point. What if the endpoint is to the side, what curve would the fastest path follow?

brachsetup Read the rest of this entry »


Prime Divergence

December 12, 2012

Did you know that:

\sum_{n=prime}^{ \infty}=\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{prime_n}+...

is equal infinity? Now, some of you might be thinking, “Oh, I already knew that.” Well, then this post is not for you, move along. Others, though, might think, “hold on, isn’t adding anything up to infinity equals infinity?” Nope because the sum (1/2)+(1/4)+(1/8)+ etc for example is equal to 1. And in fact, all geometric sum (in case you don’t know, it goes like n+n^2+n^3+… etc) with a number less than one converge into a number that is not infinite.

Some others who has learned stuff from second semester calculus class might think “Really? It looks real slow.” Well, it is really slow, but as you shall see, it nevertheless does diverge, and demonstrating this will be the topic of this post.

This post is based on a section in the book “An Imaginary Tale” by Paul Nahin. The book talks about this, and I thought the mathematics was super neat, and so I decided to put it here.

The post is in three parts:

a)I will show a proof originally done by Nicole Oresme (1320-1382) on the divergence of the harmonic series, which is 1+(1/2)+(1/3)+…

b)Then I will change the Riemann zeta function, which is f(x)=1+(1/2^x)+(1/3^x)… into a product of primes, which was done by Leonard Euler (1707-1783).

c)And finally I will use the above results to prove that the sum of all 1/prime does diverge, which was proved by Euler. Read the rest of this entry »


Links, 03/31/10

March 31, 2010

Firstly, a new fascinating finding from Cassini spacecraft. Not only does Mimas look like a Death Star, in the infrared, it also looks like Pac Man eating a pellet. that just raises the coolness factor of the moon by a magnitude or two:

See why I love astronomy? There are always cool things like these being discovered. Of course, there are always cool things being discovered in science, but I find these personally to be much more exciting than all the other ones.

Secondly, a very surprising look to what would have happened to Apollo 13 astronauts if they would have failed in their reentry to Earth after the oxygen tank explosion:

(both stories a courtesy from Universe Today)

Finally, a very cool math video. There is no need for me to describe this one. Trust me, just watch and be amazed:

(from grrlscientist)


Math Club Stuff

November 5, 2009

In college, I was in a math club for a while. And being in a math club has its own charms, including having pi ices to chill my drink:

piice

Plus, I got to have a free mug with math equations on them!

Mathmug

Up there, you can see Maxwell’s equation of electromagnetism, circumference of a circle, planck’s quantum mechanic equation of energy, deBroglie’s equation of waves and momentum, Hubble’s law, and part of general relativity and Rydberg’s equation for hydrogen light emission.

mathmug1

Over here, you see Einstein’s energy mass equivalence, Euler’s formula, Schrodinger’s quatum mechanics wave equation, Dirac’s equation (which predicted antimatter!), Newton’s second law, Riemann’s zeta function, Fourier transform, and Boltzmann’s entropy equation. The part where you see zero, behind it is delta*S, which is path of least action.

Yeah, equation galore. Say, can you tell which one is which? I couldn’t tell what a third of them were until I looked them up. I mean, even though they did tell you the name of the equations, it is not like you know exactly what it means. And for around half of them, I wouldn’t even know how to use them. I mean really, what is that psi letter (the pitchfork looking letter, perhaps mark of the devil? ^_^) in Schrodinger’s? I know it stands for wavefunction, but that is not exactly helpful. Overall, being in the math club is a good deal. I get to have free pizzas and mug, while satisfying my math obsession.


Best-Math-Study-Ever

August 23, 2009

Title says it all: When Zombies Attack! Mathematical Modelling of an outbreak of Zombie Infection.

That is marketing genius I tell you. It might as well apply to other infections, although zombies are quiet different. I don’t understand any math in it, but the conclustion arrived is that unless aggressive actions are taken to quarantine and destroy the zombies, and done often, we are all doomed. Nice stuff to think about, eh?

hat tip neurotopia