# Line integral over a Vector Field

#### 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.

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#### Dick

Homework Helper
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.
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] [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:

#### 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

Gold Member
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?

Gold Member

#### Smazmbazm

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

#### CAF123

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

"Line integral over a Vector Field"

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