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.
Stokes' theorem is a mathematical theorem that relates the surface integral of a vector field over a closed surface to the line integral of the same vector field around the boundary of the surface. It is also known as the generalized Stokes' theorem or the Kelvin-Stokes theorem.
A unit vector is a vector with a magnitude of 1, often used to indicate a direction in space. It is typically represented by a lowercase letter with a caret (^) symbol above it, such as ̂v.
In Stokes' theorem, the surface and line integrals involve the dot product of the vector field with the unit normal vector to the surface or the unit tangent vector to the curve. This allows for the calculation of the flux or circulation of the vector field in a specific direction.
Stokes' theorem is a fundamental tool in vector calculus and is used in many areas of science and engineering, including fluid mechanics, electromagnetism, and differential geometry. It allows for the simplification of complicated surface and line integrals and helps in solving problems related to flux and circulation of vector fields.
Yes, Stokes' theorem can be applied in both two and three dimensions. In two dimensions, it is known as the Green's theorem, and in three dimensions, it is known as the Kelvin-Stokes theorem. However, the underlying principles and equations remain the same in both cases.