Finding the expression of a one form in a chart.

Question

Given a one form on a manifold the formula I was given for finding its expression in a given coordinate chart is very strange and I dont understand it. I would appreciate if someone could give me a source, or explain here how I am supposed to find a one form in a chart. For example here is an exercise,

Let $f: S^n \cap \{{x \gt 0}\}\to \mathbb{R}^n:(x_0,...x_n)\mapsto (x_1,...x_n)$. be a coordinate chart and let $i:S^n\to\mathbb{R}^{n+1}$ be the inclusion. How can I calculate the one-form $i^*dx_j$ for any $j$ in the chart $f$?

How do I approach this?

Answer 1

You take $$(f^{-1})^*i^*dx_j = (i\circ f^{-1})^*dx_j.$$ Of course, $i\circ f^{-1}:f^{-1}(S^2\cap\{x>0\})\to\mathbb{R}^{n+1}$ is given by $$(x_1,\ldots,x_n)\longmapsto\left(\sqrt{1-(x_1^2+\ldots+x_n^2)},x_1,\ldots,x_n\right).$$ Now we have to compute the pullback of $dx_j$ by this map (which we will call $\phi$ from now on). To do this you can proceed as follows:

1. Find $\phi_*\partial_{x_i}$. You can compute it knowing that $$\phi_*\partial_{x_i} = \left.\frac{d}{dt}\right|_{t=0}\phi(x+t\partial_{x_i}).$$
2. Compute $\phi^*dx_j$ using the fact that $$(\phi^*dx_j)(\partial_{x_i}) = dx_j(\phi_*\partial_{x_i}).$$