Line integral over a Vector Field

1. Jun 11, 2013

Smazmbazm

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

Given a vector field

$F(x,y,z) = (yz + 3x^{2})\hat{i} + xz\hat{j} + xy\hat{k}$

Calculate the line integral

$∫_{A}^{B}F\bullet dl$

where A = (0,1,3) and B = (1,2,2)

2. Relevant equations

Right, first of all, what is dl ? I've gone over all my course notes and don't see dl anywhere. I managed to find on this site that dl = (dx, dy, dz). Is that the case?

If so, my attempt at the solution is

$∫_{A}^{B}(yz + 3x^{2})dx + xzdy + xydz$

Now what? I'm not really sure how to set the intervals from those 2 points in space. Am I supposed to do something like

$∫_{0}^{1}(yz + 3x^{2})dx + ∫_{1}^{2}xzdx +∫_{3}^{2}xydz$ ?

Thanks for any assistance.

2. Jun 11, 2013

Dick

No, dl isn't equal to (dx,dy,dz) in general. dl depends on the path you are integrating along. They didn't give you a path. Any idea why not?

3. Jun 11, 2013

Smazmbazm

Here is the whole question

http://imgur.com/KGI28pJ

http://imgur.com/KGI28pJ

4. Jun 11, 2013

klondike

Last edited by a moderator: May 6, 2017
5. Jun 11, 2013

Smazmbazm

Sure. But I don't understand what dl is in this question. How does one determine dl and also what are the limits of integration..? What I find confusing is distinguishing between what dl, ds, and dr all mean in various contexts. Such as

$∫_{C}F.ds$

$∫^{B}_{A}F.dl$

and

$∫_{C}F.dr$

what are the differences between all these...?

6. Jun 11, 2013

CAF123

You get $\vec{dl}$ from the parametrisation of your path. So what path are you going to take? Why do you think the question has not specified a path?

7. Jun 11, 2013

Smazmbazm

8. Jun 11, 2013

CAF123

Yes, that's right, but what justifies using the straight line parametrisation?

9. Jun 11, 2013

Smazmbazm

Is that because it's a conservative field and calculations are therefore path independent?

10. Jun 11, 2013

Exactly.