Maths anyone?

shep · #1 2006-03-16, 02:17 PM CST

f you know anything about maths -

I've just been examined on the topic of Double Integration/Change of Variable (and a few other 2nd year integration methods) and me and my friend are having a dispute about the Jacobian in the following example:

evaluate:

over the region enclosed by the 4 curves:
xy=2, xy=3, x^2=y and x^2=4y

I said the the Jacobian was y/(3x^2) but he got something different.
Any help would be much appreciated in solving this dispute.

BandC · #2 2006-03-16, 02:27 PM CST

The answer is 42.

shep · #3 2006-03-16, 02:43 PM CST

Hahaha - yeah...

Probably.
But seriously, can you do it?

Shakes · #4 2006-03-16, 03:16 PM CST

I'm rather embarrassed, as a former second year maths student myself, to say I can't remember how to do any of this kind of stuff! Seriously, it's so easy to forget 4 years later! That's obviously no use to you, but maybe u should avoid the mistakes I have and keep on top of this. Of course maths might be ur degree, whereas mine was computing science, in which case u'll have to be on top of this anyway. Best of luck, I'll be starting some maths revision in the near future!

BandC · #5 2006-03-16, 03:20 PM CST

I'm in the same boat as Shakes. I "used to" to be able to do it because I have an Electrical Engineering degree but I don't remember anything because I took Calculus like 10 years ago. I could probably do it if I revisited my Calculus book but the question is: Do I really want to?

shep · #6 2006-03-16, 03:46 PM CST

Haha, that seems to be the general consensus with maths. CompSci is my degree as well, but maths modules are easy credits for me so I can focus on the computing stuff more without having to worry too much about 'under-studying' for the other subjects and totally flunking.
What uni do you go to Shakes - I see you're a fellow scot..

jowah · #7 2006-03-16, 06:30 PM CST

Hmm...using a subsitution

u(x,y) = xy
v(x,y) = x^2/y

I'll arrive at a Jacobian determinant of -y/3x^2

It's 2:30 AM here and I'm trying to cure insomnia with calculus... errors are to be expected.

shep · #8 2006-03-16, 06:44 PM CST

Aye, that's what I thought...
But in the Integral to calculate the area, take the modulus of the Jacobian so we multiply (x^4)y by +y/(3x^2). Even though technically the Jacobian is (-)y/(3x^2).
Final answer of 19/3 by any chance???

Omega Blue · #9 2006-03-16, 09:55 PM CST

Bah. If you had given this thread the title of "Fun with calculus," I'd avoid it like the plague.

shep · #10 2006-03-17, 03:41 AM CST

C'mon - how can you not enjoy Calculus

Of all the categories of maths, it's certainly the easiest - A LOT easier than Analysis, and as for Algebra -***** Algebra up the nose with a sharp pointing implement...it properly suck ass.

bryancole · #11 2006-03-17, 04:26 AM CST

as Jowah suggests, use a substituion of

u(x,y) = xy
v(x,y) = x^2/y

then (x^4)*y -> (u^2)*v

and the limits become u=2..3 and v=1..4

The integral is transformed to the u,v space using the Jacobian determinant i.e.

integrate over (u^2)*v |J| du dv

where |J| is the Jacobian determinant. This is evaluate as

| dx/du dx/dv |
| dy/du dy/dv |

or

|J| = dx/du dy/dv - dx/dv dy/du = 1/(y -(x^2)/(y^2) - x 2x/y) = 1/(-(x^2)/y - 2(x^2)/y) = -y/3(x^2) = -1/(3v)

The integral is then simply over -(u^2)/3 du dv , with u=2..3, v=1..4

Final Answer = 1/9 * (3^3 - 2^3) * 3 = 19/3 , yup you got it right.

The Jacobian itself depends on your choice of substitution, so if you're doing a different substitution then you'd get a different Jacobian. Any substitution is valid provided it lets you get the right answer!

shep · #12 2006-03-17, 04:49 AM CST

Aye, that's exactly what I did - except I removed the 1/3 from the (u^2)/3 integral to make life a bit easier for me when preforming the actual integration.
Thanks folks - I didn't realise that maths would create such a response.

Shep.

Shakes · #13 2006-03-18, 06:23 PM CST

Quote:

Originally Posted by shep

What uni do you go to Shakes - I see you're a fellow scot..

I graduated from Aberdeen last year. The original plan was joint maths and compSci, but after two years of maths I decided I couldn't be bothered with it anymore. Besides I would have missed half my compSci courses in 3rd and 4th year.

What uni are you going to? Is it straight up Comp Sci ur doing? How u finding things so far?

shep · #14 2006-03-19, 01:20 PM CST

Quote:

Originally Posted by Shakes

...The original plan was joint maths and compSci, but after two years of maths I decided I couldn't be bothered with it anymore...

Exact same with me mate - i'm currently in 2nd year at Glasgow Uni, but after this year i'm dropping maths to do Software Engineering (if I get the grades that is...

)

The programming stuff is going pretty well and we get to make a website in XML this semester, so this year should be sound as long as I stay on top of the work - they've got us programming in Ada95 and a bit of Haskell which sucks, but i've just started learning Java for the Object Orientated stuff next year, so that should be pretty cool.

What languages do they throw at you lot?

w5set · #15 2006-03-19, 04:02 PM CST

Now I know who to contact to help with my next years Income taxes.
All this fancy high falutin math has even my friend Jethro Bodine amazed.
Good thing I have all of these fingers and toes to count with, but which toe/finger is the x and which is the y?
(Actually I'm just jealous--it looked like Greek (another pun) to me)
Guess I should have finished college instead of goofing off in the army.
My hat is off to y'all. "bowing in eastern direction"