Stokes' theorem and unit vector

[Sami Lakka]

Homework Statement

Use Stokes' theorem to show that

\oint\ \hat{t}*ds = 0

Integration is done closed curve C and \hat{t} is a unit tangent vector to the curve C

Homework Equations

Stokes' theorem

\oint F* \hat{t}*ds = \int\int \hat{n}*curl(F)*ds

The Attempt at a Solution

Ok, this is really teasing me because I know that is probably simple. Could someone help please?

 

[Dick]

Your notation is pretty sloppy. If t=(tx,ty,tz) the result of your first integral is the vector whose first coordinate is the integral of tx*ds, second is integral ty*ds, and third integral tz*ds. You want to show all of those are zero. Apply Stokes to the constant vector field F=(1,0,0). What does that tell you? What other vector fields would be good to use?

 

[Sami Lakka]

Your notation is pretty sloppy. If t=(tx,ty,tz) the result of your first integral is the vector whose first coordinate is the integral of tx*ds, second is integral ty*ds, and third integral tz*ds. You want to show all of those are zero. Apply Stokes to the constant vector field F=(1,0,0). What does that tell you? What other vector fields would be good to use?

Yes, sorry about the notation, it is direct copy from the book that I'm studying. t is a unit vector with components (dx/ds, dy/ds, dz/ds) so after the multiplication the integral is taken from vector (dx, dy, dz)

 

[Sami Lakka]

Yes, sorry about the notation, it is direct copy from the book that I'm studying. t is a unit vector with components (dx/ds, dy/ds, dz/ds) so after the multiplication the integral is taken from vector (dx, dy, dz)

Ok, now I think I got it. I should use Stokes' theorem with F=1 (scalar). What bothered me was that I was all the time looking at the cross product in curl which is not defined for scalars. Of course the cross product is only a notation, not actual vector cross product.

 

[Dick]

Ok, now I think I got it. I should use Stokes' theorem with F=1 (scalar). What bothered me was that I was all the time looking at the cross product in curl which is not defined for scalars. Of course the cross product is only a notation, not actual vector cross product.

No! You can't use Stokes theorem on a scalar. Use it on the vector F=(1,0,0)=1*i+0*j+0*k.