Line integral over a Vector Field

[Smazmbazm]

Homework Statement

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)

Homework 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.

 

[Dick]

Homework Statement

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)

Homework 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.

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?

 

[Smazmbazm]

Here is the whole question

http://imgur.com/KGI28pJ

http://imgur.com/KGI28pJ

 

[klondike]

Here is the whole question

http://imgur.com/KGI28pJ

[PLAIN]http://imgur.com/KGI28pJ[/QUOTE]
Just follow the questions in sequence. Use Gradient theorem. Like Gauss's and Stoke's theorems,
they are fancy version of FTC in Calc.

 

[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...?

 

[CAF123]

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..?

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?

 

[Smazmbazm]

Yea ok, I figured that out. So the path I got, following http://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtI.aspx was r(t) = <t, t+1,3-t> and after evaluating the whole expression I got an answer of 5. Dunno if that's right because we aren't given answers to our questions for some silly reason -.-

 

[CAF123]

Yea ok, I figured that out. So the path I got, following http://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtI.aspx was r(t) = <t, t+1,3-t> and after evaluating the whole expression I got an answer of 5. Dunno if that's right because we aren't given answers to our questions for some silly reason -.-

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

 

[Smazmbazm]

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

 

[CAF123]

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

Exactly.