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

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In summary, the given rectangle S=[0,1]x[0,\pi/2] is a quarter circle in both polar [r,theta] space and Cartesian [x,y] space when transformed by T(r,\theta) = (rcos\theta,rsin\theta). The confusion may arise from thinking of r,theta as polar coordinates, but T(x,y)=(x*cos(y),x*sin(y)) can also be thought of as transforming a rectangle into a quarter circle.
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Homework Statement



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

Homework Equations


T(r,[tex]\theta[/tex]) = (rcos[tex]\theta[/tex],rsin[tex]\theta[/tex])

The Attempt at a Solution


how is the given a rectangle in polar coordinates? it seems to me to be a quarter circle
 
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  • #2
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.
 
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  • #3
Welcome to PF!

Hi stvnseagal! Welcome to PF! :smile:

I think the question is just trying to confuse you …

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

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

:smile: Don't worry! :smile:
 
  • #4
It is a rectangle in "r, [itex]\theta[/itex]" space, not in x,y space. Since the two vertices (0, 0) and (0, [itex]\pi/2[/itex]) both have r= 0, T transforms both of them into the single point (0,0) in x,y space. (1, 0) in r, [itex]\theta[/itex] space is transformed into (1, 0) in xy space and (0, 1) in r, [itex]\theta[/itex] 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, [itex]\theta[/itex] space.
 

1. What does it mean to "sketch the region R=T(S) in xy-space?"

Sketching the region R=T(S) in xy-space means to visually represent the set of points that are mapped from the set S through the transformation T, in the Cartesian coordinate system. This allows for a better understanding of the relationship between the original set S and the transformed set R.

2. How do I determine the boundaries of R=T(S) when sketching in xy-space?

The boundaries of R=T(S) can be determined by analyzing the transformation T and the set S. For example, if T is a linear transformation, the boundaries of R=T(S) will be determined by the slope and intercept of the line. If T is a non-linear transformation, the boundaries may be determined by solving equations or inequalities involving the variables x and y.

3. Can I use a graphing calculator to sketch R=T(S) in xy-space?

Yes, a graphing calculator can be a useful tool for sketching R=T(S) in xy-space. Many graphing calculators have the ability to graph transformations and plot points, making it easier to visualize the region R=T(S).

4. What is the purpose of sketching R=T(S) in xy-space?

Sketching R=T(S) in xy-space allows for a better understanding of the relationship between the original set S and the transformed set R. It also helps in identifying patterns or trends in the transformation, which can be useful in further analysis or problem-solving.

5. Are there any rules or guidelines for sketching R=T(S) in xy-space?

There are no specific rules or guidelines for sketching R=T(S) in xy-space, as it will depend on the specific transformation and set being analyzed. However, it is important to accurately plot points and label the axes and any important points or boundaries to ensure a clear and accurate representation of R=T(S).

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