Another line integral question

**Glitch** · June 3rd 2011, 06:09 AM

The question
Consider the vector field F(x, y, z) = (yz + 1)i + xzj + (xy + $2e^{2z}$ )k.

Evaluate $\int_C{F.dr}$ 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 $\pi$ ) and then the straight line from (1, 0, 4 $\pi$ ) to (0, 0, 0).

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

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.

**Danny** · June 3rd 2011, 06:21 AM

Let $\phi = xyz + x + e^{2z}.$ Then the line integral is

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

$(0 + 0 + e^0) - (0 + 1 + e^0) = -1$ .

**Glitch** · June 3rd 2011, 06:41 AM

Why did you choose $P_2$ to be (1, 0, 0)? I thought we used the initial and end points of the path?

**Danny** · June 3rd 2011, 06:48 AM

Because that's what the question said - from (0,0,0,) to (1,0,0)

( $p_1 \to p_2$ )

**Glitch** · June 3rd 2011, 07:12 AM

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!

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