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
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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
Curl Theorem
Cite this as:
Weisstein, Eric W. "Curl Theorem." From
MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CurlTheorem.html
Subject classifications
