For your first case, I don't see a boundary line to the surface. For the second case, the surface is a cylinder, and I think they might be asking you to compute ## \int \vec{F} \cdot \, dS ##. If that is the case, you could also use Gauss law and compute ## \int \nabla \cdot \vec{F} \, d^3x ##, but certainly not Stokes theorem. (the Gauss's law version would also include in its result the integration over the endfaces of the cylinder). ## \\ ## Additional item: For vector curl use " \nabla \times " in Latex. To get Latex, put " ## " on both sides of your statement or expression. (The vector gradient is " \nabla" in Latex. The divergence is " \nabla \cdot ".)fonseh said:
fonseh said:
Stoke's theorem is a mathematical theorem that relates the surface integral of a vector field to the line integral of its curl along the boundary of the surface. Its purpose is to provide a way to evaluate a surface integral by converting it into a line integral, which is often easier to calculate.
In order for Stoke's theorem to be applicable, the surface must be smooth and closed, meaning it has no boundary. Additionally, the vector field must be continuously differentiable in the region enclosed by the surface.
Stoke's theorem is used in physics and engineering to calculate flux, circulation, and other important quantities in vector fields. It is often used in applications such as fluid mechanics, electromagnetism, and thermodynamics.
Yes, Stoke's theorem can be applied in three dimensions. In fact, it can be generalized to higher dimensions as well.
While both Stoke's theorem and Green's theorem relate surface integrals to line integrals, they differ in the dimensionality of their domains. Stoke's theorem applies to three-dimensional surfaces, while Green's theorem applies to two-dimensional regions in the plane. Additionally, Stoke's theorem involves the curl of a vector field, while Green's theorem involves the divergence of a vector field.
