Explain why the function $ \ f: [-\pi, \pi] \times [-\pi,\pi] \to \mathbb{R}^3 \ $ is continuous

Question

Explain why the function $ \ f: [-\pi, \pi] \times [-\pi,\pi] \to \mathbb{R}^3 \ $ defined by $ \ f(u,v)=((\cos u+2) \cos v, (\cos u+2) \sin v, \sin u ) \ $ is continuous.

Also find the range of $ \ f \ $ .

Answer:

$ \ f(u,v)=((\cos u+2) \cos v, (\cos u+2) \sin v, \sin u ) \ $ is continuous because each of its components functions $ \ (\cos u+2) \cos v , \ \ (\cos u+2) \sin v, \ \ \sin v \ $ are continuous on the domain $ [-\pi, \pi] \times [-\pi,\pi] $

But what are its range in $ \ \mathbb{R}^3 \ $ ?

Help me out

Answer 1

Hint: $[\cos(u)+2, \sin(u)]$ describes a circle centred at ...

And then you rotate ...