Question

Tips on finding the appropriate equations for a transformation "T" using given bounds for a region R in the xy-plane?

Answer 1

I'm at a loss on how to do this...an example from the book is:

R is bounded by y=2x-1, y=2x+1, y=1-x, y=3-x. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes.

Answer 2

is this related to calculus

Answer 3

yes - part of "change of variables in multiple integrals"

Answer 4

calc 3

Answer 5

first lets graph the region R
https://www.desmos.com/calculator/nhjjlqitio

Answer 6

thats a parallelogram, and we want to turn it into a rectangle

Answer 7

yes, understood

Answer 8

y=2x-1, y=2x+1, y=1-x, y=3-x

Let's rewrite these questions

y-2x = -1
y -2x =1
y+x = 1
y + x= 3

this suggests the change of variables
u = y -2x
v = y + x

Answer 9

wow, that was much simpler than I thought it would be

Answer 10

finding the change of variables, that is

Answer 11

Now the lines in R correspond to u = -1, u = 1 , v = 1, v = 3

Answer 12

in the uv plane

Answer 13

we still have to find the transformation T

Answer 14

Yes, the change of variables was really all I needed help on. It seemed so arbitrary and didn't really know what to represent u and v with

Answer 15

oh ok :)

Answer 16

thank you so much! :)