Using Stokes' Theorem to Show F(r) is Conservative

[wesleyad]

Hi I've got this question that I've been stuck on a while now.. I am sure its really obvious but i can't see to get it:
Q: with the help of stokes's theorem, show that F(r) is conservative provided that nabla X F = 0.
nabla X F is the same as curl F?
Cheers.

 

[Galileo]

$curl F$ is sometimes written as $rot F$ or $\nabla \times F$. It's the same thing.

You either have to show that $\int F \cdot dr$ is independent of path,
or that F(r) can be written as the gradient of a function. (Depending on your definition).

Hint: independence of path is the same is $\oint F \cdot dr=0$ for any closed path.

 

[M98Ranger]

Hi there,

Yes, you are correct that nabla X F is the same as the curl of F. Stokes' Theorem states that for a vector field F and a surface S bounded by a curve C, the line integral of F along C is equal to the surface integral of the curl of F over S.

In other words, if we have a closed curve C bounding a surface S, then the line integral of F along C is equal to the surface integral of the curl of F over S. Mathematically, this can be written as:

∮C F(r) · dr = ∬S (curl F) · dS

Now, if we assume that nabla X F = 0, then this means that the curl of F is equal to zero. This implies that the surface integral of the curl of F over S is equal to zero. Therefore, using Stokes' Theorem, we can rewrite the equation as:

∮C F(r) · dr = 0

Since this holds for any closed curve C, we can conclude that the line integral of F along any closed curve is equal to zero. This is one of the conditions for a vector field to be conservative, as it means that the work done by the field is independent of the path taken. Therefore, we can say that F(r) is conservative.

I hope this helps clarify how you can use Stokes' Theorem to show that a vector field is conservative. Let me know if you have any other questions. Best of luck!