Relationship To The Stokes Theorem
Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the xy-plane:
We can augment the two-dimensional field into a three-dimensional field with a z component that is always 0. Write F for the vector-valued function . Start with the left side of Green's theorem:
Kelvin–Stokes Theorem:
The surface is just the region in the plane, with the unit normals pointing up (in the positive z direction) to match the "positive orientation" definitions for both theorems.
The expression inside the integral becomes
Thus we get the right side of Green's theorem
Read more about this topic: Green's Theorem
Famous quotes containing the words relationship to, relationship, stokes and/or theorem:
“Film music should have the same relationship to the film drama that somebody’s piano playing in my living room has to the book I am reading.”
—Igor Stravinsky (1882–1971)
“Harvey: Oh, you kids these days, I’m telling you. You think the only relationship a man and a woman can have is a romantic one.
Gil: That sure is what we think. You got something better?
Harvey: Oh, romance is very nice. A good thing for youngsters like you, but Helene and I have found something we think is more appropriate to our stage of life—companionship.
Gil: Companionship? I’ve got a flea-bitten old hound at home who’ll give me that.”
—Tom Waldman (d. 1985)
“Some pray to marry the man they love,
My prayer will somewhat vary;
I humbly pray to Heaven above
That I love the man I marry.”
—Rose Pastor Stokes (1879–1933)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)