Finding the Normal Vector for a Triangle in the Plane Using Stoke's Theorem

[aaronfue]

Homework Statement

Let C be the oriented triangle lying in the plane 2x+y+z=4. Evaluate ∫
$_{C}$ F*dr where F(x,y,z)=y²i + z - xk.

Homework Equations

I will be using ∫_C∫curl F $\bullet$ $\vec{n}$ to solve this problem. But when I'm trying to find $\vec{n}$ using -g_xi - g_yj + k, I get

Using g(x,y)= -2x -y + 4
$\vec{n}$ = < -2, -1, 1>

Did I do that correctly? Will the k just be 1 or 4?

My final answer was 4/3?

 

[Dick]

Homework Statement

Let C be the oriented triangle lying in the plane 2x+y+z=4. Evaluate ∫
$_{C}$ F*dr where F(x,y,z)=y²i + z - xk.

Homework Equations

I will be using ∫_C∫curl F $\bullet$ $\vec{n}$ to solve this problem. But when I'm trying to find $\vec{n}$ using -g_xi - g_yj + k, I get

Using g(x,y)= -2x -y + 4
$\vec{n}$ = < -2, -1, 1>

Did I do that correctly? Will the k just be 1 or 4?

My final answer was 4/3?

But gx=(-2) so -gx=2. Why are you writing <-2,-1,1>? Shouldn't it be <2,1,1>?