1. ## 2-forms

Find the value of the 2-form $dx \ dy + 3dx \ dz$ on the oriented triangle with vertices $(0,0,0), \ (1,2,3), \ (1,4,0)$.

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 $xy$ plane?

Because $\text{flow} = A dy \ dx + B dz \ dx + C dx \ dy$

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

3. The answer is $-3 \frac{1}{2}$. 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?

4. I think I got it. Its $\left(\text{area of projection on xy axis} \right) + \left( 3 \times \text{area of projection on xz axis} \right)$.

I ended up getting $- \left( \frac{18}{4} - \frac{\sqrt{14}}{4} \right) \approx -3.5$

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