A special case of Stokes' theorem in which vector
field and 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. Kaplan, W. "Stokes's Theorem." §5.12 in Advanced
Calculus, 4th ed. Morse,
P. M. and Feshbach, H. "Stokes' Theorem." In Methods
of Theoretical Physics, Part I. Referenced
on Wolfram|Alpha Curl Theorem
Cite this as:
Weisstein, Eric W. "Curl Theorem." From
MathWorld https://mathworld.wolfram.com/CurlTheorem.html
Subject classifications
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
