Finding a Left Inverse for a Cylinder: Proving Injectivity of a Parametrization

[Z90E532]

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.

 

[Mark44]

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 R², you are correct: the equation $x^2 + y^2 = 1$ represents a circle. In R³, though, the same equation represents a cylinder whose central axis lies along the z-axis. Your function f is a map from R² to R³.

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 derivative is a square matrix, in which case you can take the determinant of the derivative and see where it's nonzero.

What's the definition of a left inverse that you are using?

 

[Z90E532]

What's the definition of a left inverse that you are using?

The standard one: http://mathworld.wolfram.com/LeftInverse.html

I see what you're saying about $S$, I didn't realize that at first.