Finding Jacobian for Homework Statement

[roam]

Homework Statement

Here's my question:

[PLAIN]http://img140.imageshack.us/img140/1500/89319562.gif

The Attempt at a Solution

(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 entries in that matrix?

 

[jtyler05si]

Part A looks correct to me.

 

[jtyler05si]

I believe for part B.

Use x=r*cos(theta) and y=r*sin(theta)

there for set up your jacobian as [(dx/dr),(dx/dtheta)];[(dy/dr),(dy/dtheta)]

solve and you will get the proper conversion. You will know when you get it.

sorry for the lack of proper notation, etc.
I put the matrix notation as I would in a TI version calculator. semi colon means next row.

 

[Je m'appelle]

Yeah, part A is correct.
On part B, shouldn't the Jacobian be:

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

Where

$$x = rcos\theta$$

$$y = rsin\theta$$

Now just plug-in that in:

$$\int \int_R f(x,y) dxdy = \int \int_S f(rcos\theta,rsin\theta)\begin {bmatrix} {\partial (x,y) \over{\partial (r,\theta)} \end {bmatrix} drd\theta$$

 

[jtyler05si]

Yeah, part A is correct.

On part B, shouldn't the Jacobian be:

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

Where

$$x = rcos\theta$$

$$y = rsin\theta$$

Now just plug-in that in:

$$\int \int_R f(x,y) dxdy = \int \int_S f(rcos\theta,rsin\theta)\begin{bmatrix} {\partial (x,y) \\ {\partial (r,\theta)} \end{bmatrix} drd\theta$$

Exactly.

Again sorry for my lack of correct notation and such.

 

[roam]

Thanks guys. I found the Jacobian to be $$J=r$$. 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?

 

[The Chaz]

... 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?

A common strategy to show that a function (let's say f(x), for instance) is 1-1 ("injective", "invertible")is to:
1) ASSUME that f(a) = f(b) (i.e. you got the same mapping/output/image)
2) Show that a (MUST) = b

e.g.
g(x) = x^3.
Assume g(a) = g(b). That means a^3 = b^3.
Taking the cube root of both sides gives a = b. g(x) is 1-1.

h(x) = x^2
Assume h(a) = h(b). That means that a^2 = b^2.
But taking the square root (or using the "square root property" as some call it) gives
a = +/- b. Two possibilities, so we can't say that a "MUST" = b.
h(x) is not 1-1

 

[roam]

A common strategy to show that a function (let's say f(x), for instance) is 1-1 ("injective", "invertible")is to:
1) ASSUME that f(a) = f(b) (i.e. you got the same mapping/output/image)
2) Show that a (MUST) = b

e.g.
g(x) = x^3.
Assume g(a) = g(b). That means a^3 = b^3.
Taking the cube root of both sides gives a = b. g(x) is 1-1.

h(x) = x^2
Assume h(a) = h(b). That means that a^2 = b^2.
But taking the square root (or using the "square root property" as some call it) gives
a = +/- b. Two possibilities, so we can't say that a "MUST" = b.
h(x) is not 1-1

I know this definition. What I don't understand is, how to apply it to this particular problem. Do we just say since the Jacobian is "J=r", then any value we substitute into r is itself, thus every element in the domain has a distinct and unique image under this transformation? I doubt this is correct though...

 

[The Chaz]

$$x = rcos\theta$$

$$y = rsin\theta$$

This is the mapping, not J. Basically, you're trying 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

 

[roam]

This is the mapping, not J. Basically, you're trying 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

I see. But how do we need to ensure this for every single point on the unit circle?

 

[lanedance]

bit hard to follow exactly where you are, but here's some comments

The inverse function theorem shows if at a given point, a continously differntiable function has a non-zero jacobian determinant the function is invertible near p.

As your jacobian is non-zero everywhere but the origin it is invertible everywhere except the origin.

It is only left to select the domain such that the given mapping is one to one. Hence why [0,2pi) is usually used.

 

[lanedance]

If you can't see why its ono to one around a circle of raidus r, try plotting the inverse function & it should be clear

 

[roam]

If you can't see why its ono to one around a circle of raidus r, try plotting the inverse function & it should be clear

The question asks "prove that the mapping used is 1 to 1 on region R". I see what you mean, but I'm confused because I don't know what to write down in order to "prove" that the mapping is 1 to 1. What should I write that will be sufficient as a proof?

 

[lanedance]

so the non-zero Jacobian guarantees you are locally invertible & 1:1.

Now you want to show the functions are 1:1 on the global domain, have a look at the functions each way:
$$(r,\theta) \to (x(r,\theta), y(r,\theta))$$
$$(x,y) \to (r(x,y), \theta(x,y))$$

then to show the function is globally 1:1, can you show that for a given $(r,\theta)$ there is a unique solution (x,y) in the given domain and vice versa

 

[roam]

so the non-zero Jacobian guarantees you are locally invertible & 1:1.

Now you want to show the functions are 1:1 on the global domain, have a look at the functions each way:
$$(r,\theta) \to (x(r,\theta), y(r,\theta))$$
$$(x,y) \to (r(x,y), \theta(x,y))$$

then to show the function is globally 1:1, can you show that for a given $(r,\theta)$ there is a unique solution (x,y) in the given domain and vice versa

Does this show that for any given $$(r,\theta)$$ there is a uniqe (x,y):

$$(r,\theta) \to (x(r,\theta), y(r,\theta))=((r cos \theta), (r sin \theta))$$

By the way, they told us one way to do this problem is to produce the inverse mapping. How do we do that?