How can I show the $n$ form: $$i_{\bf X}\ dx^1 \wedge \cdots \wedge dx^{n+1}=\sum (-1)^{i-1} x^i \, dx^1 \wedge\cdots \wedge \widehat{dx^i} \wedge \cdots \wedge dx^{n+1}$$ is nowhere vanishing on the sphere $S^n$? The vector field $X$ is given by $$X=\sum_ix^i\frac{\partial}{\partial x^i}$$
I'm thinking whether it's correct to say that since for every point on $S^n$ there is at least one $x^i$ is not 0, then by linear independent, the n form must be non-zero everywhere.
Are you sure those are linearly independent as forms on the sphere? (Note that each $n$-form, without the coefficient, vanishes at certain points when restricted to the sphere.) Better, think about the geometry, as $X$ is the unit normal. What is the value of the original $(n+1)$- form evaluated on $X,v_1,\dots,v_{n}$ if the $v_i$ are a basis for the tangent space at a point?