- #1

Z90E532

- 13

- 0

## Homework Statement

Let ##S ## be a cylinder defined by ##x^2 + y^2 = 1##, and given a parametrization ##f(x,y) = \left( \frac{x}{ \sqrt{x^2 + y^2}}, \frac{y}{ \sqrt{x^2 + y^2} },\ln \left(x^2+y^2\right) \right)## , where ##f: U \subset \mathbb R^2 \rightarrow \mathbb R^3 ## and ## U = \mathbb R ^2 /{(0,0)}##

1. Find a left inverse of ##f##

2. Show that ##f## is injective.

## Homework Equations

## The Attempt at a Solution

I'm not even sure where to begin with this, My professor has done a very poor job of explaining how he wants us to go about these problems, and the book doesn't help much at all. The first thing that confuses me is that ##S## isn't a cylinder, it's a circle.

In class, we did inverses of simple functions which you could fairly easily solve, such as ##f(x,y) = (\sin xy, y)##, where we would just set it equal like this: ##y' = y## and ##\arcsin x' = xy##, then we would have ##f^{-1} (x',y') = (\frac{\arcsin (x'y')}{y'},y')##.

Also regarding the second question, I'm not aware of anyway to show that a function is injective unless its derivate is a square matrix, in which case you can take the determinant of the derivative and see where it's nonzero.