Change of variables - transformations

[wakingrufus]

im working on these, and I am supposed to find the image of a set under a given transformation. can someone please explain to me a good way of doing this?

 

[turin]

Since no one else has attempted an answer, I will give it a try. Just keep in mind that I am not an expert. I will be using A textbook by Stewart called "Calculus: Early Vectors, Preliminary Edition Volume III"

The transformation is defined as a set of n equations for n coordinates. If you are going from cartesian coordinates, x_i, to some other coordinates, q_j, then the transformation is of the form:

q_j = f_j({x_i})

That is, the new coordinates can be expressed as functions of the old coordinates.

To find an image of a set, I'm assuming that you want to find the image of some bounded region, my text suggests to find the images of the boundary curves. So, You would have equations for the boundary curves in the cartesian coordinates, and you plug those into the functions, f_j({x_i}). This will give you the images of the boundary curves, and then the region containes inside these images is the image of the set. I don't know how you should know that the images of the boundary curves form a closed curve, but it seems intuitively reasonable.

I can give you an example out of my text if what I have described is in fact what you wanted.

 

[wakingrufus]

ok thanks. i think I got it now. my test was a bit hard to swallow on this section. "calculus: fourth edition" by James stewart. my teacher isn't good so i have to learn from the text :(