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.