No where vanishing exact $1$-form on compact manifold. I found several answers on the following question : 

Does there exists a no where vanishing exact $1$-form on a compact manifold without boundary?

All answer says that certainly not. But I cannot understand use of the fact that Boundary of manifold is empty . These answers emphasis on the fact that manifold is compact.
Is it still true that for a compact manifold with boundary there does not exist a no where vanishing exact $1$-form?
 A: The other two examples provide examples showing that you need to assume your manifold is boundaryless.  I want to show where the "usual" proof in the boundaryless case breaks down.  So, here is the usual proof.

Exactness means $\omega = df$ for some smooth $f:M\rightarrow \mathbb{R}$.  Because $M$ is compact, there is a $p\in M$ for which $f(p)$ is an absolute maximum.
We claim that $d_p f = 0$, so that $\omega = df$ is not non-vanishing.  To show this, we pick $v\in T_p M$ and want to show that $(d_p f) v = 0$.  To that end, let $\gamma:(-\epsilon, \epsilon)\rightarrow M$ be a smooth curve with $\gamma(0) = p$ and $\gamma'(0) = v$.
We want to show that $(d_p f) v = 0$, or, said another way, that $\frac{d}{dt}|_{t=0} f(\gamma(t)) = 0$.  By definition of derivative, we need to show that $$\lim_{h\rightarrow 0} \frac{f(\gamma(h)) - f(\gamma(0))}{h} = 0.$$
First, the numerator is $f(\gamma(h) - f(p) \leq 0$ since $f(p)$ is a maximum of $f$.  It follows that $\lim_{h\rightarrow 0^+} \frac{f(\gamma(h))- f(p)}{h} \leq 0$ and that $\lim_{h\rightarrow 0^-} \frac{ f(\gamma(h)) - f(p)}{h} \geq 0$.
By assumption, $\lim_{h\rightarrow 0} \frac{f(\gamma(h)) - f(p)}{h}$ exists, so the left and right hand limits must match.  Since one is non-negative and the other is non-positive, the conclusion is that $\lim_{h\rightarrow 0} \frac{f(\gamma(h)) - f(p)}{h} = d_p f v = 0$.  $\square$

$ \ $
So, where does this break down if $p\in \partial M$?  Well, tangent vectors on the boundary are defined differently: you only need $\gamma$ to have a domain which is $(-\epsilon, 0]$ or $[0,\epsilon)$.
If we're in the case that the domain of $\gamma$ is $[0,\infty)$, then the one sided limit $\lim_{h\rightarrow 0^-} \frac{f(\gamma(h)) - f(p)}{h}$ no longer makes sense:  $\gamma(h)$ doesn't make sense for negative $h$, so $f(\gamma(h))$ is meaningless.
Likewise, if the domain of $\gamma$ is $(-\epsilon, 0]$, then $\lim_{h\rightarrow 0^+} \frac{f(\gamma(h)) - f(p)}{h}$ no longer makes sense.
Thus, in either case, we lose one of the two inequalities.  Without both, the proof no longer works to force $(d_p f) v = 0$.  (And the other two answers show there is no way to "fix" the proof to handle the case where $p$ is a boundary point.)
A: On the compact manifold with boundary $[0,1]$, the differential $1$-form $dx$ is nowhere vanishing and exact. It is the exterior derivative of $x$.
A: Consider the cylinder, $S^1 \times [0, 1]$, parameterized by $\theta, t$. 
The everywhere nonzero form $dt$ is exact (it's the differential of the function $t$ that gives the "height" coordinate).
So the answer is "no". 
