Line integral over a Vector Field

  • Thread starter Smazmbazm
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  • #1
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Homework Statement



Given a vector field

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

Calculate the line integral

[itex]∫_{A}^{B}F\bullet dl[/itex]

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

[itex]∫_{A}^{B}(yz + 3x^{2})dx + xzdy + xydz[/itex]

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

[itex]∫_{0}^{1}(yz + 3x^{2})dx + ∫_{1}^{2}xzdx +∫_{3}^{2}xydz[/itex] ?

Thanks for any assistance.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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Homework Statement



Given a vector field

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

Calculate the line integral

[itex]∫_{A}^{B}F\bullet dl[/itex]

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

[itex]∫_{A}^{B}(yz + 3x^{2})dx + xzdy + xydz[/itex]

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

[itex]∫_{0}^{1}(yz + 3x^{2})dx + ∫_{1}^{2}xzdx +∫_{3}^{2}xydz[/itex] ?

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?
 
  • #4
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Here is the whole question

http://imgur.com/KGI28pJ

[PLAIN]http://imgur.com/KGI28pJ[/QUOTE] [Broken]
Just follow the questions in sequence. Use Gradient theorem. Like Gauss's and Stoke's theorems,
they are fancy version of FTC in Calc.
 
Last edited by a moderator:
  • #5
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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

[itex]∫_{C}F.ds[/itex]

[itex]∫^{B}_{A}F.dl[/itex]

and

[itex]∫_{C}F.dr[/itex]

what are the differences between all these...?
 
  • #6
CAF123
Gold Member
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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?
 
  • #8
CAF123
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  • #9
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Is that because it's a conservative field and calculations are therefore path independent?
 
  • #10
CAF123
Gold Member
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88
Is that because it's a conservative field and calculations are therefore path independent?
Exactly.
 

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