What is the meaning of "nowhere vanishing form"?! I'm studying "An Introduction to Manifolds, Loring W. Tu". I have a problem with this section of it:

One should distinguish between a nonzero form and a nowhere-zero or nowhere- vanishing form. For example, $xdy$ is a nonzero form on $\mathbb{R}^2$, meaning that it is not identically zero. However, it is not nowhere-zero, because it vanishes on the y-axis. On the other hand, $dx$ and $dy$ are nowhere-zero 1-forms on $\mathbb{R}^2$.

My question is "Why $dx$ is a nowhere-zero form, while $dx(\frac{\partial}{\partial y}) = 0?$".

The form $dx$ is never zero at any point. on the other hand, $f(x, y) = x dy$  is zero for all $x = 0$. That is, $f(0, y) = 0$
A form is an object in its own right, and when we speak of the vanishing of the form, we are speaking of the form itself, not it's evaluation on a tangent vector.
By analogy, we say that the vector $5 \hat i$ is non zero, even if $ 5 \hat i \cdot \hat j = 0$. By analogy, the form $dx$ is also nonzero.
