How the compute the pullback of a differential form

[logarithmic]

If there's a function, $$\phi(t,u,v)=((v+t)u,t,u,v) =: (w,x,y,z)$$. How do I compute the pullback of dw: $$\phi^*(dw)$$?

I think what you do is: $$\phi^*(dw)=\frac{\partial}{\partial t}((v+t)u)dt+\frac{\partial}{\partial u}((v+t)u)du+\frac{\partial}{\partial v}((v+t)u)dv$$.

Is that correct?

This is using the formula for the pullback. A version can be found on wikipedia here: http://en.wikipedia.org/wiki/Differential_form#Pullback (Look at the 2nd last equation in the Pullback section).

Can someone explain how the composition in that formula works, and how it's used in my answer above? Here that composition would be $$dw(\phi(t,u,v))$$. What would that equal?

I think another reason the answer I gave works is because of the fact $$\phi^*(dw)=d(\phi^*(w))$$, and $$\phi^*(w)=w(\phi(t,u,v))$$. Somehow, this equals (v+t)u (but how?) and we then take the exterior derivative.

 

[logarithmic]

Any help please?