My question is to verify Stokes Theorem. I manage to do using Stokes Theorem $\iint_R \nabla\times\vec{F}\cdot d\textbf{S}$ and got my answer $7/6$
but I dont know how to do the direct line integral method which is $$\oint_c \vec{F} \cdot \vec{dr}$$
Given $F(x,y,z)=(z^2)i + (x^2)j + (y)k$
and the surface is taken to be positively oriented triangle which lying in the plane $x+y+z=1$ and bounded by the coordinates planes x=0,y=0 and z=0
Could someone please help me to guide me because I'm just started to learn vector calculus.So if you show some working I will be much appreciated.
Thank You
Parametrize. There are 3 sides to the triangle. Along the first side, say, in the plane $y=0$, we have $x+z=1$. Note that we should integrate along a positively-oriented direction - in this case, counter-clockwise. Then let $x=t$, $z=1-t$ for $t \in [0,1]$. You then have $d\mathbf{r} = (1,0,-1) dt$ and $\mathbf{F}=[(1-t)^2,t^2,0]$. Therefore, $\mathbf{F} \cdot d\mathbf{r} = (1-t)^2 dt$ and the integral over this leg of the triangle is
$$\int_0^1 dt \, (1-t)^2 = \frac13$$
Now repeat for the other two sides, keeping the counter-clockwise direction:
Side 2: plane $z=0$, $x=1-t$, $y=t$
Side 3: plane $x=0$, $y=1-t$, $z=t$
The contribution from Side 2 is $1/3$ and from Side 3 is $1/2$, so you may verify that the line integral is indeed $7/6$.