I'm trying to show $n-1$-form $$ \iota_{S^{n-1}}^*(\sum_{i=1}^n(-1)^{i-1}x^idx^1\wedge \cdots \widehat{dx^i}\wedge \cdots dx^n) $$ is a nowhere vanishing differential form on $S^{n-1}$. For the case $n = 2,3$, using the coordinate representation $(\cos\theta,\sin\theta)$ and $(r\cos\varphi_1,r\sin\varphi_1\cos\varphi_2,r\sin\varphi_1\sin\varphi_2)$, after the painful computation, I got $d\theta$ for the first case and $\sin\varphi_1 d\varphi_1\wedge d\varphi_2$ for the second one. I suspect that only the term that does not contains $dr$ in the differential form survives. But I don't want to do this for general $n$. I believe there is a simpler way to show the above $n-1$ form on $S^{n-1}$ is nowhere vanishing differential form. Is there any way to show that? Is my method the simplest?
Note. I'm following the notation of spherical coordinate representation of $S^{n-1}$ here "spherical coordinate" section.
Yes, there's a simpler way. Stay in cartesian coordinates!
Note that if we consider the radial vector field $X=\sum x^i\dfrac{\partial}{\partial x^i}$, then your $(n-1)$-form can be written as (the restriction to the sphere of) $\iota_X (dx^1\wedge\dots\wedge dx^n)$. The last piece of the puzzle is ... what geometric relation does $X$ have to $S^{n-1}$ at the point $x\in S^{n-1}$?