A special case of Stokes' theorem in which is a vector
field and
is an oriented, compact embedded 2manifold
with boundary in ,
and a generalization of Green's theorem from the
plane into threedimensional 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. 6164,
1985.Kaplan, W. "Stokes's Theorem." §5.12 in Advanced
Calculus, 4th ed. Reading, MA: AddisonWesley, pp. 326330, 1991.Morse,
P. M. and Feshbach, H. "Stokes' Theorem." In Methods
of Theoretical Physics, Part I. New York: McGrawHill, p. 43, 1953.Referenced
on WolframAlpha
Curl Theorem
Cite this as:
Weisstein, Eric W. "Curl Theorem." From
MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/CurlTheorem.html
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