# Homework Help: Differential geometry:2-form computation on a pair of tangent vectors

1. Jun 16, 2009

### phymatast

1. The problem statement, all variables and given/known data

I am given a one form Psi = zdx -xydy and two vectors v(1,1,-2) and w(-2,1,1) both tangent vectors of R3 at point P(2,-1,0).

I am asked to find dPsi(v,w).

2. Relevant equations

Lie bracket?

3. The attempt at a solution

I know how to computer Psi(v) at p but this is a 1-form.
I am not sure how to compute the 2 form though.

I was able to find dPsi:

dPsi = d(z)^dx +d(-xy)^dy (^== wedge product)

= -ydxdy -dxdz

Now I have looked it up in the Internet. The only thing I found was dPsi of two vector fields. But here I just have 2 vectors.
I am a bit lost.

Any hints?

Thanks

2. Jun 16, 2009

### phymatast

I have actually also found this formula on wikipedia:

(A^B)(v,w)p = A(vp)B(wp) -A(wp)B(vp)

So I tried to rewrite my two form as a wedge of two 1-forms:

A = dx
B= -ydy -dz

Finally i applied the formula and I got
A(vp)=1
B(wp)=1
A(wp)=-2
B(vp)=4

And

dPsi(v,w) = (A^B)(v,w) = 9

I am not sure if my work is correct...
I would appreciate a good enlightment..