# Homework Help: Calculating when a covector is zero.

1. Oct 1, 2011

### Kreizhn

1. The problem statement, all variables and given/known data
Hopefully an easy question. Let M be an n-dimensional smooth manifold and $f: M \to \mathbb R$ a smooth function. Assume the covector field df is given in some coordinate basis as
$$df = \sum_{i=1}^n a_i dx^i$$.
Find all points $p \in M$ such that the $df_p = 0$

3. The attempt at a solution
I'm pretty sure all that needs to be done is find all point $p \in M$ such that $a_i(p) = 0$ for all i=1,..n. However, I just want to make sure nothing tricky is going on here that I'm missing. My reasoning is that if there is an $a_i(p) \neq 0$ then
$$a_i(p) dx^i \left(\left. \frac{\partial }{\partial x^i}\right|_p \right) = a^i(p) \neq 0$$

Are there other tricky cases that I might not be seeing? I can imagine that for fixed vector fields there are points where the differential could yield zero, but I take the statement $df_p = 0$ to mean that the 1-form is identically the zero map at that point.