Path Integral of Triangle: Parameterization & Solution Explanation

[Kuma]

Homework Statement

The problem asks:

find the integral of gamma F.ds where F(x,y,z) = (e^z, e^y, x+y).
gamma being a triangle with vertices:

(1,0,0) (0,1,0) (0,0,1) going in a counterclockwise direction

Homework Equations

The Attempt at a Solution

So I'm not even sure if what I'm doing is right. This is my attempt to parameterize

C1: x = t, y = 1-t, z = 0 for 0<t<1
C2: x = 2-t, y = 0, z = 1-t for 1<t<2
C3: x = 0, y = t-2, z = 3-t for 2<t<3

And to find the integral it's

F(c(t))*c'(t) dt

is that right?

 

[Some Pig]

It should be:
$C_1:\vec r=(1-t,t,0),0\le t\le1.$
$C_2:\vec r=(0,1-t,t),0\le t\le1.$
$C_3:\vec r=(t,0,1-t),0\le t\le1.$

 

[ehild]

Make a drawing, and show which lines are C1,C2,C3.

ehild