Curl Theorem

A special case of Stokes' theorem in which F is a vector field and M is an oriented, compact embedded 2-manifold with boundary in R^3, and a generalization of Green's theorem from the plane into three-dimensional space. The curl theorem states

 int_S(del xF)·da=int_(partialS)F·ds,

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



 int_Sdaxdel F=int_CFds.

and if



 int_S(daxdel )xP=int_CdsxP.

See also

Change of Variables Theorem, Curl, Divergence Theorem, Green's Theorem, Stokes' Theorem

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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.

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Curl Theorem

Cite this as:

Weisstein, Eric W. "Curl Theorem." From MathWorld--A Wolfram Web Resource.

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