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 = 7 in the first quadrant

Answer 1

Suppose S is a≤u≤b, A≤v≤B and let (u,v) → (r,θ) so then x=rcosθ, y=rsinθ … (i) 

We need 1≤r≤√3, 0≤θ≤π/2 so suppose r=f(u), θ=g(v) 

If f(a)=1 and f(b)=√3 we can set r=f(u)=3^{½(u−a)/(b−a)} so 1≤r≤√3 

If g(A)=0 and g(B)=π/2 we can set θ=g(v)=(π/2)(v−A)/(B−A) so 0≤θ≤π/2 

With (i) this r & θ define T