Why is Stokes Theorem failing me

multivariable-calculusvector analysis

I am really sorry if it's a stupid question, but I can't seem to get what I am doing wrong.

I have a final approaching and for the most part I feel like I understand Stoke's theorem (or at least why it's so convenient). But I am unable to find what's wrong with what I am doing.

Let $T$ be the triangle of vertices $(2, 0, 0)$, $(0, 2, 0)$ and $(0, 0, 2)$.
Compute the flux (surface integral) of the vector field $F (x, y, z) = (z, z, z)$ accross $T$.

I have worked out the problems using segments, and got the expected answer of $4$. I then tried to use Stokes theorem on it, but when I do that, I get zero.
From my understanding of stokes, it should apply here since:

  1. I have a vector field
  2. I have a surface and a boundary (plane with $x+y+z = 2$, and the triangle $T$)
  3. The boundary is simple and closed.

But when I work it out, it reduces to double integral over projection of T of $(-1,1,0)$ (Curl of $F$) and $(1,1,1)$ (Normal Vector). Which simply gives 0.

Best Answer

The exercise asks for the flux of the field $\mathbf{F}$ not the flux of the curl of $\mathbf{F}$. Therefore Stokes' Theorem is useless here. BTW, you are right, the flux of the curl of $\mathbf{F}$ accross $T$ is zero.

For the flux of the field, a direct computation is all you need: $$\iint_T \mathbf{F}\cdot d\mathbf{S}=\int_{x=0}^2\Big(\int_{y=0}^{2-x} (2-x-y,2-x-y,2-x-y)\cdot (1,1,1) dy\Big)dx.$$ Can you take it from here?

Related Question