|If you want Wikipedia’s formulation of Stokes’ Theorem, you get this expression:
which is the same as what we’ve derived once you decipher the notation. Instead of S for a surface, they use the Greek S (a capital sigma); instead of saying that the boundary of the surface is the curve C, they use the notation to indicate that the curve in question is the boundary of without having to say so explicitly. They assume you know that any integral over a surface has to be a surface integral, so they don’t bother to use a double integral sign on the left-hand side of the equation. With that translation, Wikipedia’s version of Stokes’ Theorem is the same as ours.
Except that since it’s written by a math graduate student or advanced math major (probably), it gets into a bit of jargon about “1 forms” and “metrics” in the explanation below that you can safely ignore….
“…which relates the surface integral of the curl of a vector field over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary, is a special case of the general Stokes theorem (with n = 2) once we identify a vector field with a 1 form using the metric on Euclidean three-space. The curve of the line integral ( ∂Σ ) must have positive orientation, meaning that d r points counterclockwise when the surface normal ( d Σ ) points toward the viewer, following the right-hand rule.”
BTW, I love how they provide hyperlinks to stuff that you already know what it is (like “vector field” and “orientation”) but don’t provide links that would be really helpful (such as what a “1 form” is)…
They do, however, in the following part give you a formulation that is component-based rather than vector-based for those of you who don’t like vectors, so maybe that’s worth something.
“It can be rewritten for the student acquainted with forms as
where P, Q and R are the components of F.”