Homework Help: Line Integral Problem

1. Apr 26, 2010

farso

Hi everyone.

I am going through examples for maths exams and am unsure on the final part of a question I am attempting so hoping you may help me?

1. The problem statement, all variables and given/known data

"Let C be the closed, piecewise smooth curve comprising individual curves C1 and C2
defined by r1 = (x, x2, 1) and r2 = (x,+√x, 1), respectively, with 0 ≤ x ≤ 1, see
Figure 1. Evaluate the work done by the vector field ∇ on a particle moving around
curve C once in the anticockwise direction, i.e. directly compute the integral

$$\oint \nabla \theta (x,y,z) dr$$"

2. Relevant equations

As per above/below

3. The attempt at a solution

$$\theta (x,y,z) = x^2z^2+3yz+2x$$

hence

$$\nabla \theta (x,y,z) = (2xz^2+2, 3z, 2zx^2+3y)$$

So, using green's theorem (I think this is correct)

$$\oint \nabla \theta (x,y,z) dr$$

is the same as

$$\int_{y=x^2}^{y=\sqrt{x}}\int_0^1 \nabla \theta (x,y,z) dxdy$$

I think this is correct, but cant seem to find the next "step". Id be grateful if anyone could tell me if Im on the right track, and maybe show me where to go on the next step?

2. Apr 26, 2010

gabbagabbahey

Maybe I'm misinterpreting the question, as I can't see Figure 1, but to me, the work done by the vector field $\mathbf{\nabla}\theta$ would involve a dot product, and would be a scalar quantity:

$$W=\oint \mathbf{\nabla}\theta\cdot d\textbf{r}$$

This makes no sense....Where exactly did you use Green's theorem and how?

Instead, just break the path integral into two sections (over $C_1$ and $C_2$) and integrate it directly... what is $d\textbf{r}$ for the first section? How about the second?

3. Apr 26, 2010

farso

Hi

$$\oint \nabla \theta (x,y,z) dr =$$$$\int_{0}^{1} \nabla \theta (x,y,z) dr_1$$ + $$\int_{0}^{1} \nabla \theta (x,y,z) dr_2$$
where $$dr_1 = (1,x^{2},0)$$ and $$dr_2 = (1, -(1/2)x^{-1/2}, 0)$$
So... If this is correct, do I just do the dot product of that with $$\nabla \theta (x,y,z)$$?