
2forms
Find the value of the 2form $\displaystyle dx \ dy + 3dx \ dz $ on the oriented triangle with vertices $\displaystyle (0,0,0), \ (1,2,3), \ (1,4,0) $.
I drew the projection of the triangle on each of the coordinate planes. Now what do I do? Do I compute the area of the triangle on the $\displaystyle xy $ plane?
Because $\displaystyle \text{flow} = A dy \ dx + B dz \ dx + C dx \ dy $

Is the basically the sum of the areas in the projected planes?

The answer is $\displaystyle 3 \frac{1}{2} $. And if it is a 2 form then it has to be assigned a number to a oriented surface. Maybe I should draw a prism?

I think I got it. Its $\displaystyle \left(\text{area of projection on xy axis} \right) + \left( 3 \times \text{area of projection on xz axis} \right) $.
I ended up getting $\displaystyle  \left( \frac{18}{4}  \frac{\sqrt{14}}{4} \right) \approx 3.5 $

But if the surface is not a triangle then you can't do this right?