Raising and lowering differential forms

[toqp]

Homework Statement

Calculate the contravariant components of the differential 1-form
$$\omega|_x = x^3 dx^1 - (x^2)^2 dx^3$$
that is raise it into $$\omega ^\#|_x$$

$$\eta ^{\mu\nu}(x)=diag(1,-1,-1,-1)$$

The Attempt at a Solution

I'm at lost here. I don't really understand how these differential forms work.

Can I just transfer the 1-form into an ordinary covector
$$\omega | _\nu=(0,x^3,0,-(x^2)^2)$$

and then raise it using
$$\eta ^{\mu\nu}\omega _\nu$$?

 

[mighty2000]

Yes, you can raise the differential 1-form using the metric tensor, as shown below:

Let's first define the differential 1-form as:
\omega = x^3 dx^1 - (x^2)^2 dx^3

Now, to raise it into a contravariant form, we can use the metric tensor:
\omega ^\# = \eta ^{\mu\nu} \omega _\nu

Substituting in the values for the metric tensor and the covector, we get:
\omega ^\# = (1)(x^3)dx^1 + (-1)(x^3)dx^2 + (-1)(-(x^2)^2)dx^3 + (-1)(0)dx^4

Simplifying, we get:
\omega ^\# = x^3 dx^1 - x^3 dx^2 + (x^2)^2 dx^3

Therefore, the contravariant components of the differential 1-form are:
\omega ^\# | _x = (x^3, -x^3, (x^2)^2, 0)