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I'm in need of dire help!
I have been given a topic; "polar functions."
I know how to do the calculus for it, I just don't know how to make a project off it.
For example, my friend is doing something on the serpinski gasket, showing that randomness does have a pattern; he is recreating an experiment.
I have wracked my brains and cannot think of anything. There are no 'experiments' I can do. It seems as if I can only summarize "polar functions are... bllah balh blah". What I want is something where i can analyse and work with the numbers.
My friend suggested working with its graphs, however, I only see myself doing surface level work if I work with the graphs.
If any of you have any suggestions on a mathematics paper (roughly 2000 words), please tell!
 
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Well, the first thing that comes to me with polar functions is always the applications. Maybe look at some novel ways of applying polar coordinates that use whatever calculus you're studying?
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Do Kapler's law, rose pedal graph or something : )
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On October 25 2007 16:00 evanthebouncy! wrote:
Do Kapler's law, rose pedal graph or something : )
This is actually a good idea. I think its "Kepler" but w/e. Look it up.
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Bill307
Canada9103 Posts
Polar coordinates (that's what you mean, right?) have numerous applications in physics, and anytime you need to work with something elliptical. Unfortunately, it's a bit difficult for me to come up with examples that are both interesting and not too advanced.
Here are some general problem areas I can think of where polar coordinates are useful:
o Elliptical motion in physics (e.g. due to gravity). Example: Kepler's Second Law.
o Elliptical objects. Example: derive parametric equations for a torus.
o Complex numbers. These are often expressed in the form:
r e^(i t) = r cos t + i r sin t
By the way, if you're looking for something a little more aesthetically interesting, you can look into Lissajous curves. In particular, if you're good at programming as well, you can write a program to plot the curves for you. This will allow you to experiment with as many parametric equations as you can imagine.
Lastly, although I don't recommend doing this for your project, because it's really much different from polar coordinates, you can take the equations and start putting the previous x and y coordinates into them. Now you can end up with chaotic behaviour that will trump your friend's "randomness" easily  . I did this during high school and wrote a program that allowed me to produce images like the following:
Example: define two sequences as follows:
Base cases:
x_1 = 0
y_1 = 0
Recursive cases:
x_(n+1) = y_n
y_(n+1) = sin( 3n ) * cos( sqrt(2)n - (x_n)^2 )
(notice we are not really using polar coordinates anymore, hence this is totally unrelated to your project =P)
(note: credit for these sequences goes to one of my highschool friends, who came up with them)
The following image shows the points (x_n, y_n) for n = 1 to 50,000, all coloured white for simplicity.
Pretty cool, huh?
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Also, I just remembered that the calc book I used (stewart) had a number of interesting problems and projects in each section.
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what exactly do you mean by polar functions?? a function in polar coordinates??
polar coordinates have numerous uses. in physics, anything with at least a partial central symetry is converted into polar/spherical coordinates.
the problem of a mass in a centraly symetrical potential seems like a good idea (kepler's problem is just a special case) can be interesting. you get a shitload of weird looking spirals.
another example would be the polar solution for the schrodinger equasion, l and m -quantization, legendre functions, spherical harmonics...
a 2-d harmonic oscilator leads to lissajous curves, like bill307 said. they are pretty easy to plot, look cool and have pretty wide-used aplications.
phase space trajectories of a sytem with a completely separated hamiltonian also use polar coordinates (angle/action coordiantes).
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ohh, lovley first image there Bill, but the thumbnail looks a lot better than the full version. 
I wish people asking about help like this would specify what level they are at, so you know what you can use to help them...
That said:
You could explain spherical symmetry in physics, and how you get conservation of angular momentum from it. Corresponding in x-y coordinates is translation invariance and momuntum conservation.
Also anything related to atomic physics will be crammed with sperical coordinates. (if that is what you are asking about). Maybe look at the schrödinger equation for a hydrogen atom (without the extra anisotropic corections) and solve it?
You could also write about the cute little trick with polar coordinates to calculate the intergral of a gaussian int e^(x^2)dx over entire real axis. Basically you look at the square of the integral, merge them into one integral of two variables x and y, do a change of variables to polar coordinates, and then you can find a primitive and you win.
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wow thanks so much everyone.
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You can talk about signals and systems, which is entirely based on complex numbers.
You have your Laplace transforms transforming your signals between time domain and frequency domain.
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Do some integrals and show the superiority of using polar coordinates when the domain is suitable.
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EPT![[image loading]](http://img81.imageshack.us/my.php?image=id22starlightcj0.png)
![[image loading]](http://img98.imageshack.us/my.php?image=graphnb6.png)
