Jacobian transformation

koalamath · #1 $Old$ 11-19-2008, 05:08 AM

Suppose that the double integral over D of f(x,y)dA-1 where D is the disk $x^2+y^2_<9$ Now suppose E is the disk $x^2+y^2_<36$ and $g(x,y)=4f(x/2,y,2)$ What is the value of the double integral over E of g(x,y) dA?
Please help
thank you

shawsend · #2 $Old$ 11-20-2008, 05:32 AM

I'd like to make an attempt at this even though I don't think it's what you want:

Consider:

$\mathop\int\int\limits_{\hspace{-22pt}x^2+y^2=9} f(x,y)dA$

I'll use the following change of variables:

$u=\frac{x}{2},\quad v=\frac{y}{2}$

$x=2u,\quad v=2u$

Then:

$J=\left|\begin{array}{cc}2 & 0 \\ 0 & 2\end{array}\right|=4$

Substituting the values of u and v into the circle in terms of x and y I get the new domain in terms of u and v:

$u^2+v^2=9/4$

Thus:

$\mathop\int\int\limits_{\hspace{-22pt}x^2+y^2=9} f(x,y)dA=4\mathop\int\int\limits_{\hspace{-22pt}u^2+v^2=9/4} f(u,v)dA$

(Would be nice to have a real double and triple integral sign in here . . . )