Can somebody answer this for me:
Ok so I need to know the answer to this and I have no clue how to do it. If anyone on this can answer this for me I will be very grateful and impressed. I think you need to do a transformation to answer it; a jacobian transformation.
Evaluate ∫ ∫D y^2/x dA where D is bounded by the graphs of y = x^2, y=x^2/2, x=y^2, x=y^2/2..
2006-08-11 01:21:18 · 4 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
This is for doug.....I am not sure if the email system on yahoo is working. I could really use your help. If you can....email me at Monica.Gorny@uconn.edu
2006-08-11 02:18:30 · update #1
Well, based on the nature of the problem, let's introduce new variables
u=x^2/y and
v=y^2/x
Then the region is bounded by u=1, u=1/2, v=1, and v=1/2 and the integrand is v. The only issue is the nature of the Jacobian for this transformation; d(x,y)/d(u,v).
There are two ways of doing this: one is to solve for x and y interms of u and v and taking the determinant of the matrix of partial derivatives.
The other is to find d(u,v)/d(x,y), find the reciprocal, and write the result in terms of u and v. I'll do this second way.
The partial derivatives are du/dx=2x/y; du/dy=-x^2/y^2; dv/dx=-y^2/x^2; and dv/dy=2y/x, so
d(u,v)/d(x,y)=4-1=3, which gives
d(x,y)/d(u,v)=1/3.
Thus, you will integrate (1/3)v over the region v=1/2 to v=1 and u=1/2 to u=1. I get an answer of 1/16.
Just for fun, do the Jacobian the other way also.
Added later: Aaack! I hate it when I get those limits messed up! Rats!
2006-08-11 02:01:46 · answer #1 · answered by mathematician 7 · 3⤊ 0⤋
You just asked this question last night!!!:
http://answers.yahoo.com/question/index;_ylt=AmHFRRZeydz3Xs4zJ0qqTY7sy6IX?qid=20060810200503AAm9Ph7
and I gave the correct solution there. Mathematician's solution is also correct, except that he mixed up the bounds that you get -- his bounds should be u=1 to u=2 and v=1 to v=2. Then you get the answer I got, 1/2, which is the correct answer. You can also verify this by splitting your integral into 3 integrals, without doing a transformation.
This is standard first-year of college multivariable calculus. There is no need to talk about Sturm-Liouville transforms.
2006-08-11 10:00:46 · answer #2 · answered by mathbear77 2 · 0⤊ 0⤋
A Jacobi Transform?? IIRC that's a special case of a finite Sturm-Liouville Transform.
Things like that are normally 1'st or 2'nd year graduate school problems.
What kind of math class are you taking?
e-mail me.
Doug
2006-08-11 08:53:00 · answer #3 · answered by doug_donaghue 7 · 0⤊ 2⤋
Jacobian : I will solve it
2006-08-11 09:26:33 · answer #4 · answered by Jatta 2 · 0⤊ 0⤋