Computing 1-form in spherical coordinates

Question

Given is the 1-form $w=x^1dx^1 + x^2dx^2 + x^3 dx^3$.

How do I calculate $w$ in spherical coordinates (How do I calculate the differentials $dx^i$)? Where:

$x^1 = r sin \theta cos \phi$

$x^2 = r sin \theta sin \phi$

$x^3 = r cos \theta$

In addition, how would I compute the following quantity?

$X = x^1 \frac{\partial}{\partial x ^ 2} - x^2 \frac{\partial}{\partial x ^ 1}$

As I understand, the $dx^i$ and $\frac{\partial}{\partial x ^ i}$ form a basis, so I should apply the coordinate transformation laws of tensors to compute those basis in spherical coordinates?

Note: Homework question.

Answer 1

Just differentiate it: $$dx^1 =\sin\theta\cos\phi\, dr\ +\ r\cos\theta\cos\phi\, d\theta \ - \ r\sin\theta\sin\phi\, d\phi$$ and so on.

Answer 2

Cheating:

\begin{align} x^1 dx^1 + x^2 dx^2 + x^3 dx^3 &= d\left(\frac 12 (x^1)^2 \right)+ d\left(\frac 12 (x^2)^2 \right)+d\left(\frac 12 (x^3)^2 \right) \\ &= \frac 12 d \left((x^1)^2+(x^2)^2+(x^3)^2 \right)\\ &=\frac 12 d (r^2) \\ &= rdr. \end{align}