The discussion revolves around applying Stokes' theorem to demonstrate that the line integral of a unit tangent vector over a closed curve is zero. The original poster expresses confusion regarding the notation and the application of the theorem, specifically in relation to scalar and vector fields.
Some participants have provided guidance on the correct application of Stokes' theorem, emphasizing the need to use a vector field rather than a scalar. Multiple interpretations of the problem are being explored, particularly regarding the notation and the components of the tangent vector.
There is mention of notation issues and confusion regarding the definition of the curl in relation to scalars and vectors. The original poster's reference to the textbook notation is noted as a source of misunderstanding.
Dick said: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 said: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 said: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.