# Another line integral question

• Jun 3rd 2011, 06:09 AM
Glitch
Another line integral question
The question
Consider the vector field F(x, y, z) = (yz + 1)i + xzj + (xy + $\displaystyle 2e^{2z}$)k.

Evaluate $\displaystyle \int_C{F.dr}$ along the path C from (0, 0, 0) to (1, 0, 0), following the helix (x, y, z) = (cos(t), sin(t), t) from (1, 0, 0) to (1, 0, 4$\displaystyle \pi$) and then the straight line from (1, 0, 4$\displaystyle \pi$) to (0, 0, 0).

My attempt
I first found a scalar potential $\displaystyle \phi$ with the property $\displaystyle \nabla{\phi}$ = F, which is $\displaystyle xyz + x + e^2z$

So the vector field is conservative. I noticed that the path is actually a closed loop, so if I'm not mistaken, I can apply the Fundamental Theorem of Line Integrals. Thus, the solution should be 0, right? However my book says -1. :/

Where have I gone wrong? Thanks.
• Jun 3rd 2011, 06:21 AM
Jester
Let $\displaystyle \phi = xyz + x + e^{2z}.$ Then the line integral is

$\displaystyle \left.\phi\right|_{p_1}^{p_2}$ where $\displaystyle p_1 = (0,0,0)$ and $\displaystyle p_2 = (1,0,0).$ When you substitute you get

$\displaystyle (0 + 0 + e^0) - (0 + 1 + e^0) = -1$.
• Jun 3rd 2011, 06:41 AM
Glitch
Why did you choose $\displaystyle P_2$ to be (1, 0, 0)? I thought we used the initial and end points of the path?
• Jun 3rd 2011, 06:48 AM
Jester
Because that's what the question said - from (0,0,0,) to (1,0,0)

($\displaystyle p_1 \to p_2$)
• Jun 3rd 2011, 07:12 AM
Glitch
Oh, oops, for some reason I thought it was a line from (0, 0, 0) to (0, 0, 1) then the helix then the line. >_<

That'd explain my confusion! Thanks!