Calculating Work Using Green's Theorem

[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?

Homework Statement

"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

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

Homework Equations

As per above/below

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 can't seem to find the next "step". Id be grateful if anyone could tell me if I am on the right track, and maybe show me where to go on the next step?

Thanks in advance

 

[gabbagabbahey]

"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

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

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:

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

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

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

is the same as

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

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?

 

[farso]

Hi

Thanks for the speedy reply!

Looking at it again and the definition of greens theorem I am not entirely sure why I chose to try and use it.

Are you suggesting that I do it as:

$$\oint \nabla \theta (x,y,z) dr =$$$$\int_{0}^{1} \nabla \theta (x,y,z) dr_1$$ + $$\int_{0}^{1} \nabla \theta (x,y,z) dr_2$$

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 $$\nabla \theta (x,y,z)$$?