1. Line integral

let vector field F be (z^2/x,z^2/y, 2zlnxy). Evaluate the line integral of F
Where C is the path of straight line segments fromP = (1; 2; 1) to Q = (4; 1; 7) to R = (5; 11; 7), and then back to P.

2. Re: Line integral

Hey apatite.

Can you show us what you tried? (Hint: Write down the definition of the line integral for one single segment to begin).

3. Re: Line integral

i get zero as final answer. I want to double check it. Thank you. Is my answer right? In this particular case i use find the partial function and use that to get the line integral.

4. Re: Line integral

Well, yes.

If you calculate the curl $\nabla \times \vec{F} = \left( \dfrac{\partial}{\partial x}, \dfrac{\partial}{\partial y}, \dfrac{\partial}{\partial z} \right) \times \vec{F}$ of this vector field, you will find that it equals zero and so the field is conservative. This means that it has path independence and regardless of the path you take, if you end up at the same place (P), the work done on it equals zero.

I do not get what you mean by using the 'partial function' to solve the line integral, do you mean you used the gradient function to solve for the curl?

To solve this line integral you have to parameterise the function and then solve $\int_{t_0}^{t_1} \vec{F}(t) \times \vec{r} (t) dt$