Sketch the region R=T(S) in xy-space

[stvnseagal]

Homework Statement

Consider the rectangle S=[0,1]x[0,$$\pi$$/2]
Sketch the region R=T(S) in xy-space.

Homework Equations

T(r,$$\theta$$) = (rcos$$\theta$$,rsin$$\theta$$)

The Attempt at a Solution

how is the given a rectangle in polar coordinates? it seems to me to be a quarter circle

 

[Dick]

S is a 'rectangle' in cartesian [r,theta] space. The image of T in cartesian [x,y] space is, indeed, a quarter circle. If you regard T as a transformation between polar r-theta coordinates and cartesian x-y coordinates then it is really a quarter circle in both. I'll admit the difference is a bit confusing if you are used to thinking of r,theta as polar coordinates. If it helps try thinking of T(x,y)=(x*cos(y),x*sin(y)) both in cartesian coordinates. That really does take a rectangle into a quarter circle.

 

[tiny-tim]

Welcome to PF!

Hi stvnseagal! Welcome to PF!

I think the question is just trying to confuse you …

Technically, "rectangle" means all its angles are right-angles - which they are!

It's just that there's only three of them!

Don't worry! ​

 

[HallsofIvy]

It is a rectangle in "r, $\theta$" space, not in x,y space. Since the two vertices (0, 0) and (0, $\pi/2$) both have r= 0, T transforms both of them into the single point (0,0) in x,y space. (1, 0) in r, $\theta$ space is transformed into (1, 0) in xy space and (0, 1) in r, $\theta$ space is transformed into (0, 1) in xy space.

Tiny Tim, "technically" a rectangle has four vertices! As I said, the "rectangle" part on applies to r, $\theta$ space.