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**.” |