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In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (''dα'' = 0), and an exact form is a differential form that is the exterior derivative of another differential form ''β''. Thus, an exact form is in the ''image'' of ''d'', and a closed form is in the ''kernel'' of ''d''. For an exact form ''α'', for some differential form ''β'' of one-lesser degree than ''α''. The form ''β'' is called a "potential form" or "primitive" for ''α''. Since , ''β'' is not unique, but can be modified by the addition of the differential of a two-step-lower-order form. Because , any exact form is automatically closed. The question of whether ''every'' closed form is exact depends on the topology of the domain of interest. On a contractible domain, every closed form is exact by the Poincaré lemma. More general questions of this kind on an arbitrary differentiable manifold are the subject of de Rham cohomology, that allows one to obtain purely topological information using differential methods. == Examples == The simplest example of a form which is closed but not exact is the 1-form "d''θ''" (quotes because it is not the derivative of a globally defined function), defined on the punctured plane which is locally given as the derivative of the argument - note that argument is locally but not globally defined, since a loop around the origin increases (or decreases, depending on direction) the argument by 2''π'', which corresponds to the integral: : and for general paths is known as the winding number. The ''differential'' of the argument is however globally defined (except at the origin), since differentiation only requires local data and different values of the argument differ by a constant, so the derivatives of different local definitions are equal; this line of thought is generalized in the notion of covering spaces. Explicitly, the form is given as: : which is not defined at the origin. This can be computed from a formula for the argument, most simply via arctan(''y''/''x'') (''y''/''x'' is the slope of the line passing through (''x'',''y''), and arctan converts slope to angle), recognizing 1/(''x''2+''y''2) as corresponding to the derivative of arctan, which is 1/(''x''2+1) (these agree on the line ''y''=1). While the differential is correctly computed by symbolically differentiating this expression, this formula is only strictly correct on the halfplane ''x''>0, and properly one must use a correct formula for the argument. This form generates the de Rham cohomology group meaning that any closed form is the sum of an exact form and a multiple of where accounts for a non-trivial contour integral around the origin, which is the only obstruction to a closed form on the punctured plane (locally the derivative of a potential function) being the derivative of a globally defined function. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「closed and exact differential forms」の詳細全文を読む スポンサード リンク
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