Bay 12 Games Forum

Please login or register.

Login with username, password and session length
Advanced search  
Pages: 1 ... 96 97 [98] 99 100 ... 173

Author Topic: Mathematics Help Thread  (Read 226977 times)

Skyrunner

  • Bay Watcher
  • ?!?!
    • View Profile
    • Portfolio
Re: Mathematics Help Thread
« Reply #1455 on: March 05, 2014, 09:15:25 pm »

Bah, it's only x4 error :P
Logged

bay12 lower boards IRC:irc.darkmyst.org @ #bay12lb
"Oh, they never lie. They dissemble, evade, prevaricate, confoud, confuse, distract, obscure, subtly misrepresent and willfully misunderstand with what often appears to be a positively gleeful relish ... but they never lie" -- Look To Windward

Pnx

  • Bay Watcher
    • View Profile
Re: Mathematics Help Thread
« Reply #1456 on: March 10, 2014, 07:02:03 pm »

Ok, integrating 1/(3+2x-x2)1/2 is giving me trouble, I think I have to get it into some kind of format that I can do trig substitution for it, but I'm having trouble doing that.
« Last Edit: March 10, 2014, 07:14:27 pm by Pnx »
Logged

MagmaMcFry

  • Bay Watcher
  • [EXISTS]
    • View Profile
Re: Mathematics Help Thread
« Reply #1457 on: March 10, 2014, 07:25:10 pm »

Ok, integrating 1/(3+2x-x2)1/2 is giving me trouble, I think I have to get it into some kind of format that I can do trig substitution for it, but I'm having trouble doing that.
1/sqrt(3+2x-x^2) = 1/sqrt(4-(x-1)^2). Now substitute x=2*sin(y)+1.

You'll get int(a, b, 1/sqrt(3+2x-x^2)) = int(arcsin((a-1)/2), arcsin((b-1)/2), 1), which means that the antiderivative of 1/(3+2x-x2) is arcsin((x-1)/2).
« Last Edit: March 10, 2014, 07:28:32 pm by MagmaMcFry »
Logged

Skyrunner

  • Bay Watcher
  • ?!?!
    • View Profile
    • Portfolio
Re: Mathematics Help Thread
« Reply #1458 on: March 10, 2014, 09:15:51 pm »

I finally signed up for AP Calculus BC in May @_@
Logged

bay12 lower boards IRC:irc.darkmyst.org @ #bay12lb
"Oh, they never lie. They dissemble, evade, prevaricate, confoud, confuse, distract, obscure, subtly misrepresent and willfully misunderstand with what often appears to be a positively gleeful relish ... but they never lie" -- Look To Windward

Pnx

  • Bay Watcher
    • View Profile
Re: Mathematics Help Thread
« Reply #1459 on: March 13, 2014, 01:40:25 am »

Firstly, before I forget to remember to actually say this: Thank you very much MagmaMcFry for your help.

Secondly, I've got a headache of a problem right now...

It's the integral of (2x2-1)/(4x-1)(x2+1). It's the whole partial fractions thing, and my answer seems to consistently wind up being nothing like what my book/wolfram tell me it's supposed to be. Which tells me I may be going about the whole thing wrong? Can anyone help?
Logged

Skyrunner

  • Bay Watcher
  • ?!?!
    • View Profile
    • Portfolio
Re: Mathematics Help Thread
« Reply #1460 on: March 13, 2014, 02:21:24 am »

Separate into A/(4x-1) + (Bx+C)/(x^2+1) form, then do the magic!
Logged

bay12 lower boards IRC:irc.darkmyst.org @ #bay12lb
"Oh, they never lie. They dissemble, evade, prevaricate, confoud, confuse, distract, obscure, subtly misrepresent and willfully misunderstand with what often appears to be a positively gleeful relish ... but they never lie" -- Look To Windward

Furtuka

  • Bay Watcher
  • High Priest of Mecha
    • View Profile
Re: Mathematics Help Thread
« Reply #1461 on: March 14, 2014, 06:55:17 pm »

I need some help with a multivariable calculus problem.

Spoiler: The instructions (click to show/hide)

Spoiler: What I did (click to show/hide)

Logged
It's FEF, not FEOF

Mego

  • Bay Watcher
  • [PREFSTRING:MADNESS]
    • View Profile
Re: Mathematics Help Thread
« Reply #1462 on: March 14, 2014, 08:03:58 pm »

Given that you're integrating over a hypersphere, converting to spherical coordinates, (x,y,z) -> (p, q, r), and then polar coordinates, (p, t) -> (u, v), is probably going to be easier and simpler.

Furtuka

  • Bay Watcher
  • High Priest of Mecha
    • View Profile
Re: Mathematics Help Thread
« Reply #1463 on: March 14, 2014, 08:32:37 pm »

I don't quite understand what you mean? As in, I don't see how to get to the polar parts from what I have :(

am I missing something obvious? my brain is a bit fried right now from a lack of sleep
« Last Edit: March 14, 2014, 08:36:07 pm by Furtuka »
Logged
It's FEF, not FEOF

MagmaMcFry

  • Bay Watcher
  • [EXISTS]
    • View Profile
Re: Mathematics Help Thread
« Reply #1464 on: March 14, 2014, 08:37:12 pm »

I need some help with a multivariable calculus problem.

Spoiler: The instructions (click to show/hide)

Spoiler: What I did (click to show/hide)


Your objective is to find the volume V of that four-dimensional sphere (let's call it B4, the set of all points (x, y, z, t) with x²+y²+z²+t²<=rad²), by integrating 1 over its contents. To do this, you'll first want to find a nice parametrization of all the points in that sphere. For any valid parametrization p from any parameter space A to B4, the volume of the sphere is given by V = intA{det(Dp(x1, x2, x3, x4))}. (*)

The first and most obvious parametrization you could use is the canonical parametrization
    p1: B4 -> B4; (x, y, z, t) |-> (x, y, z, t).
In this case, det(Dp1) = 1, but B4 isn't the most Fubini-friendly set you could integrate over. Let's try and make our parameter space more rectangular, shall we?

So let's try a different parametrization using double polar coordinates:
    p2: B2' x [0, 2*pi] x [0, 2*pi] -> B4; (r, u, s, v) |-> (r*cos(s), r*sin(s), u*cos(v), u*sin(v)).
Here B2' is the set of all points (r, u) with r>=0 and u>=0 and r²+u²=rad² (it's a quarter-disk). Here, det(Dp2) = r*u, but our parameter space is much more comfortable to work with. It's still stupid, because r and u are still not independent.

To fix this, let's try a third parametrization based on the previous one:
    p3: [0, rad]x[0, pi/2]x[0, 2*pi]x[0, 2*pi] -> B4; (p, q, s, v) |-> (p*cos(q)*cos(s), p*cos(q)*sin(s), p*sin(q)*cos(v), p*sin(q)*sin(v)).
Note that this is very similar to the previous parametrization, we just replaced r and u by p*cos(q) and p*sin(q) respectively. Now our parametrization space is rectangular! Now det(Dp3) = p³*sin(q)*cos(q) = p³*sin(2*q)/2, which is not bad as far as Jacobians go. We'll go with this one to integrate over the hypersphere.

So, let's insert p3 into the formula (*) from above:

V=intA{det(Dp3(p, q, s, v))}
=int(p³*sin(2*q)/2, p=0..rad, q=0..pi/2, s=0..2*pi, v=0..2*pi)
=pi²*rad4/2.
Logged

Pnx

  • Bay Watcher
    • View Profile
Re: Mathematics Help Thread
« Reply #1465 on: March 16, 2014, 02:18:15 pm »

So I've got this one extra credit fluid pressure problem I'm struggling with.

I have an elliptical tank with a height of 8m and a width of 4m.

I have a paddle in the shape of the bottom right corner of an elipse that has a height of 1m and a width of 1.5m (these are the dimensions of the paddle, not the elipse). The verticle flat edge is centred in the tank, and the horizontal edge is 2m from the bottom of the tank.

The fluid density is 12,000 N/m3.

I'm really not sure how to set this up as an integral due to the depth changing horizontally as well as vertically. Can anyone help me out?
Logged

MagmaMcFry

  • Bay Watcher
  • [EXISTS]
    • View Profile
Re: Mathematics Help Thread
« Reply #1466 on: March 16, 2014, 08:37:11 pm »

What is it you're trying to find out?
Logged

Pnx

  • Bay Watcher
    • View Profile
Re: Mathematics Help Thread
« Reply #1467 on: March 16, 2014, 09:03:41 pm »

Oh shoot, I should probably say that. I'm trying to find the force exerted on the paddle. Which is a function of the distance from the surface, given the way this changes I'm having a hard time finding how to set up an integral.
Logged

MagmaMcFry

  • Bay Watcher
  • [EXISTS]
    • View Profile
Re: Mathematics Help Thread
« Reply #1468 on: March 17, 2014, 06:06:11 am »

I have good news for you: Buoyancy is independent of fluid pressure (as long as the fluid is incompressible), and the buoyant force is exactly the negative of the gravitational force that would appear if the paddle were made of the fluid instead. So all you need to do is find out the volume of the paddle, multiply it by the fluid's density, then multiply it by -g.
Logged

Zrk2

  • Bay Watcher
  • Emperor of the Damned
    • View Profile
Re: Mathematics Help Thread
« Reply #1469 on: March 17, 2014, 07:50:36 am »

Yeah, I find a lot of Fluids and Thermodynamics is like that. It looks really fucking scary until you know what you're talking about, and then all the actual math is pretty simple.
Logged
He's just keeping up with the Cardassians.
Pages: 1 ... 96 97 [98] 99 100 ... 173