My professor in my vector analysis course likes to always have a two part question where one part can be solved by stokes, greens or gauss theorem. They are pretty straight forward if you understand them and I find it that I have. The other part is where I take the same problem and solve it without the use of the theorem.
So here is an example I've solved for stokes, but I don't know how to solve without it.
We have a triangle with vertris (1,0,0) (0,1,0) (0,0,1).
solve the closed integral of [int] xy dx+yz dy+zx dz over that plane.
Attempt at solution
We know the plane equation is x+y+z=1
We also know [int] xy dx+yz dy+zx dz = [int] F dot dr, which is a closed line integral.
This is pretty much as far as I get..
What I need help with
I have a vague memory that I can devide C into parts of integration, but as I have just one integral and can only integrate over x,y or z seperatly.
Please give me a few hints so I can carry on with my understanding of calculus