- prometheus25
- |
- Exalted Mythic Member
Old school Bungie, born and raised,
In the Septagon is where I spend most of my days.
Relaxin', maxin', posting all cool,
Talking about Halo, life and some school.
Got in one little argument, and the mods got scared,
they said "You're gonna get banned and your member title'll be bare!"
Posted by: Alt account1
I read the OP, I don't understand.
Someone explain the fact in short? I don't need to know why, just tell me the damn fact.
Basically, what is happening is that a surface is being made. The surface is a square with infinitely long sides (think of it more philosophically than literally). The height of the surface above the ground of zero at any point (x,y) is determined by e^-(x^2 + y^2). Here's a small portion of this surface.
What integrals do is they find the area below a curve, from point a to b. In this instance, the bounds are from negative infinity to infinity, which means you are finding the area beneath the entirety of the curve. By having a double integral (that's what dx and dy imply), you are finding the volume under the infinitely large square, or the entirety of the volume.
X and Y (and Z) coordinates (what is referred to as the "Cartesian coordinate system" isn't the only way to represent space. It could also be done in polar coordinates, where a position is identified by a radius (usually from the 0,0 origin) and angle (from the x axis counter-clockwise). The same space of an infinitely wide and tall square to a circle of infinite radius (again, "philosophically" the same thing) spun about the 0,0 point a full 360 degrees (though we use radians. It's simpler mathematically. 360 degrees = 2pi radians)
When you switch coordinates, the variables need to be adjusted. See in the OP where e^1(x^2+y^2) switches to e^-r^2? Think of the Pythagorean Theorem. x^2 + y^2 = z^2. z is the equivalent of a radius (Are you familiar with the Unit Circle), so this Theorem is tasked in this scenario as x^2 + y^2 = r^2, which you see in the OP.
Solving integrals is a bit complicated for those unfamiliar with it. Once a certain amount of familiarity is acquired with them, it's mostly pattern recognition. At this point you're just going to have to trust me/us on the issue.
So the solution here solves out to be pi. It is then assumed that since the two integrals at the beginning are so similar, you could treat them as if they are equivalent to one of them squared. This means that The square root of one of them squared, lim(a->oo)int(-a,a)e^(-x^2)dx, is equal to the square root of the answer, equating to sqrt(pi).
I have some issues with this proof, even though it mathematically is sound. They can be seen in the document linked in this post.