# All line integrals in the world are zero...?

Using Stokes' theorem, the line integral of a vector field gives a surface integral of the curl of the vector field, and after that, if we apply Gauss' divergence theorem in that, it gives a volume integral of the divergence of the curl of that vector field. But we know the divergence of the curl of a vector field is $0$. … so how can it be possible? Where is the mistake??

• Take a 3D object, say a solid ball. Its boundary is the surface, i.e. a sphere. Gauss relates those two integrals. Stokes relates the integral over that the sphere and its boundary curve. But, what is the boundary curve of the surface of a sphere? There isn't one! The exact same thing happens if you start with a torus. Misner, Thoren & Wheeler repeatedly state: boundary of a boundary is zero They do mean it in the context of differential forms, but it does apply geometrically as well. If it didn't we would run into problems of the type you describe! Oct 8, 2015 at 15:31
• But in case of a solid cylinder it has two circular surfaces. In those surfaces is the problem true? Oct 8, 2015 at 15:40
• No. The surface of a solid cylinder has three parts. The bottom disk, the top disk, and the surface of that cylinder. The two circles (=the common boundaries of those three parts) are traversed in both clockwise and counterclockwise directions, so the corresponding line integrals cancel automatically from the sum of the line integrals giving all of the boundary of the boundary. No contradiction there either! Oct 8, 2015 at 15:51
• Title of the question looks very interesting....! Oct 8, 2015 at 15:54
• Oct 8, 2015 at 15:56