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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク volume element ： ウィキペディア英語版 volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :$dV\; =\; \backslash rho(u\_1,u\_2,u\_3)\backslash ,du\_1\backslash ,du\_2\backslash ,du\_3$ where the $u\_i$ are the coordinates, so that the volume of any set $B$ can be computed by :$\backslash operatorname(B)\; =\; \backslash int\_B\; \backslash rho(u\_1,u\_2,u\_3)\backslash ,du\_1\backslash ,du\_2\backslash ,du\_3.$ For example, in spherical coordinates $dV\; =\; u\_1^2\backslash sin\; u\_2\backslash ,du\_1\backslash ,du\_2\backslash ,du\_3$, and so $\backslash rho\; =\; u\_1^2\backslash sin\; u\_2$. The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as the area element, and in this setting it is useful for doing surface integrals. Under changes of coordinates, the volume element changes by the absolute value of the Jacobian determinant of the coordinate transformation (by the change of variables formula). This fact allows volume elements to be defined as a kind of measure on a manifold. On an orientable differentiable manifold, a volume element typically arises from a volume form: a top degree differential form. On a non-orientable manifold, the volume element is typically the absolute value of a (locally defined) volume form: it defines a 1-density. ==Volume element in Euclidean space== In Euclidean space, the volume element is given by the product of the differentials of the Cartesian coordinates :$dV\; =\; dx\backslash ,dy\backslash ,dz.$ In different coordinate systems of the form $x=x(u\_1,u\_2,u\_3),\; y=y(u\_1,u\_2,u\_3),\; z=z(u\_1,u\_2,u\_3)$, the volume element changes by the Jacobian of the coordinate change: :$dV\; =\; \backslash left|\backslash frac\backslash right|\backslash ,du\_1\backslash ,du\_2\backslash ,du\_3.$ For example, in spherical coordinates :$\backslash begin$ x&=\rho\cos\theta\sin\phi\\ y&=\rho\sin\theta\sin\phi\\ z&=\rho\cos\phi \end the Jacobian is :$\backslash left\; |\backslash frac\backslash right|\; =\; \backslash rho^2\backslash sin\backslash phi$ so that :$dV\; =\; \backslash rho^2\backslash sin\backslash phi\backslash ,d\backslash rho\backslash ,d\backslash theta\backslash ,d\backslash phi.$ This can be seen as a special case of the fact that differential forms transform through a pullback $F^$ * as :$F^$ *(u \; dy^1 \wedge \cdots \wedge dy^n) = (u \circ F) \det \left(\frac\right) dx^1 \wedge \cdots \wedge dx^n 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「volume element」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.