# Neighborhood about a simple closed curve, and changing coordinated for one forms

I've been struggling with the following two problems for while now, and i am grateful for any assistance or hint to solve them

1. let $\gamma$ be a simple smooth closed curve in R^2 of length L which is parametrized by arclength. Let $N_\epsilon$ ($\gamma$) denote the neighborhood of radius $\epsilon$ about $\gamma$. show that for sufficiently small $\epsilon$ the area of $N_\epsilon$ ($\gamma$) is exactly $2\epsilon$ times the length of $\gamma$. Hint: let n(s) denote the unit normal to $\gamma$ at $\gamma(s)$, and consider the map $T: \gamma$ x $( - \epsilon, \epsilon)$ --> $N_\epsilon$ ($\gamma$) where $T(\gamma(s), t) = \gamma(s) + tn(s)$. This map is a diffeomorphism for sufficiently small epsilon ( no need to prove this part)

2. Given the following 2 one forms on $R^3$ $w_1 = dx$ and $w_2 = dy - zdx$. Determine whether or not there exists coordinates (u, v, w) near the origin such that $w_1= du$, and $w_2 = dv$

its really easy to do that if it were vector fields not forms, but how do you apply this on forms?

Thank you very much in advance.

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Hint 1: show this first for a circle. Now locally every curve looks just like a circle. Hint 2: $d^2 = 0$ -> what does it tell you about $w_2$? – Marek Sep 11 '12 at 8:51
thank you! got them! – Jessica Sep 12 '12 at 15:26