1. ## The Jacobian

Here's my question:

(a)

$\int^{2 \pi}_0 \int^1_0 [r^2(cos^2\theta + sin^2\theta)]rdr d\theta$

$\int^{2 \pi}_0 \int^1_0 r^3 dr d \theta$ $= \int^{2 \pi}_0 \frac{r^4}{a} |^1_0 d \theta$

$=\frac{1}{4} \theta |^{2 \pi}_0 = \frac{\pi}{2}$

Is this correct?

(b) So, is the Jacobian for this problem given by the following?

$
J(x,y)= \frac{\partial(r,\theta)}{\partial (x,y)}=\begin{bmatrix} {\partial r\over \partial x} & {\partial r\over \partial y} \\ {\partial \theta\over \partial x} & {\partial \theta\over \partial y} \end{bmatrix}
$

If so, how can I obtain the four partials in the matrix?

2. The answer for (a) looks correct to me.

When you're changing to polar coordinates, you need to use the Jacobian. It is r, as you have used in (a).

To calculate the Jacobian, you use the same idea as you have written, but the numerator and denominator should be switched. So

$
\frac{\partial(x,y)}{\partial (r,\theta)}=\begin{bmatrix} {\partial x\over \partial r} & {\partial x\over \partial \theta} \\ {\partial y\over \partial r} & {\partial y\over \partial \theta} \end{bmatrix}
$

Using $x = r\cos\theta$ and $y = r\sin\theta$, the rest will follow.

3. first part of your problem is correct
for secondone see attachment

4. Originally Posted by demode
Here's my question:

(a)

$\int^{2 \pi}_0 \int^1_0 [r^2(cos^2\theta + sin^2\theta)]rdr d\theta$

$\int^{2 \pi}_0 \int^1_0 r^3 dr d \theta$ $= \int^{2 \pi}_0 \frac{r^4}{a} |^1_0 d \theta$

$=\frac{1}{4} \theta |^{2 \pi}_0 = \frac{\pi}{2}$

Is this correct?

(b) So, is the Jacobian for this problem given by the following?

$
J(x,y)= \frac{\partial(r,\theta)}{\partial (x,y)}=\begin{bmatrix} {\partial r\over \partial x} & {\partial r\over \partial y} \\ {\partial \theta\over \partial x} & {\partial \theta\over \partial y} \end{bmatrix}
$

If so, how can I obtain the four partials in the matrix?
The other responders have shown that. I want to point out that your use of " $r drd\theta$" already has the Jacobian in it! If u(x,y) and v(x,y) are a change of variable then $dx dy= \frac{\partial (u, v)}{\partial (x, y)} dudv$. Saying that $dx dy= r dr d\theta$ is the same as saying $\frac{\partial (r, \theta)}{\partial (x, y)}= r$

5. I followed Slovakiamaths' working and the Jacobian is $J=r$. So to answer part (b), the only point interior to R at which the Jacobian equals zero is (0,0), right?

Furthermore, the question asks: "Prove that the mapping used is 1 to 1 on region R excluding the boundary points, and excluding the origin. Note: one way to do this is to produce the inverse mapping." I know that a mapping is one to one if distinct points in the $r, \theta$ plane have distinct images in the xy-plane. But in this case how do I need to prove it?

6. Okay, my mistake. Here's the mapping

$x = rcos\theta$
$y = rsin\theta$

I don't know how to show it's one to one. I have to show that every point on the unit circle (except (1,0) ) is the mapping of exactly one angle in the interval 0 to 2π. But I don't know how to check that for all the points. Can anyone help?

7. Are you aware that the sine and cosine are 1-1 on the interval [0,2pi)?