Question

A region R in the xy-plane is given. 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. (Let u play the role of r and v the role of θ. Enter your answers as a comma-separated list of equations.) R lies between the circles x2 + y2 = 1 and x2 + y2 = 8 in the first quadrant

Answer 1

Answer:

Step-by-step explanation:

given that a region R in the xy-plane is given. 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.

Let u play the role of r and v the role of θ.

R lies between the circles

$x^2 + y^2 = 1$and $x^2 + y^2 = 8$ in the first quadrant

We know that in parametric form in polar coordinates (r,t)

$x=rcost : y = rsint$ for a circle

Here instead of r and t we use u and v.

Also since only in I quadrant, we get v can take values only between 0 and pi/2 both inclusive

u being radius varies from 1 to square root of 8

So region R would be

$x = ucos v\\\\y = usin v\\0\leq v\leq \frac{\pi}{2} \\1\leq u\leq 2\sqrt{2}$

represents region R