A special case of Stokes' theorem in which 
is a vector
field and 
is an oriented, compact embedded 2-manifold with boundary in 
, and a generalization of Green's
theorem from the plane into three-dimensional space. The curl theorem states
 |
(1) |
where the left side is a surface integral and the right side is a line integral.
There are also alternate forms of the theorem. If
 |
(2) |
then
 |
(3) |
and if
 |
(4) |
then
 |
(5) |
See also
Change of Variables Theorem, Curl, Divergence Theorem, Green's Theorem, Stokes' Theorem
Explore with Wolfram|Alpha
References
Arfken, G. "Stokes's Theorem." §1.12 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 61-64, 1985.Kaplan, W. "Stokes's Theorem." §5.12 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 326-330, 1991.Morse, P. M. and Feshbach, H. "Stokes' Theorem." In Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 43, 1953.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Curl Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CurlTheorem.html