How is Stoke's theorem applied in this situation?

[Pythagorean]

Homework Statement

Prove:

$$\int \hat{t} ds = 0$$

over C, a closed curve; t is the unit vector tangent to C.

Homework Equations

stoke's theorem

The Attempt at a Solution

My issue is that normally, stoke's theorem involves a vector function that we dot into the unit vector (t) (resulting in a scalar) and when we use stoke's theorem, we instead curl that vector function, then dot it into n, the surface normal vector.

However, in this situation, there's no dot product, so we're integrating a vector.

My first attempt was to use the diad product, but I feel like I'm being a bit cavalier with it. I'd just like a hint, but not a full solution so that I can think about it more.

 

[gabbagabbahey]

Try using an intermediate step: If you use Stoke's theorem directly for a vector $$\vec{c}$$ that's constant over space (and hence has zero curl ), you should be able to easily derive a useful corollary here.

 

[Dick]

gabbagabbahey is quite correct. If you want to see a dot product in there, the ith component of the integral around the curve of the tangent vector is the integral of the dot product of the tangent vector t with the ith basis vector.