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In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold which can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain trivial line bundle, called the density bundle. An element of the density bundle at ''x'' is a function that assigns a volume for the parallelotope spanned by the ''n'' given tangent vectors at ''x''. From the operational point of view, a density is a collection of functions on coordinate charts which become multiplied by the absolute value of the Jacobian determinant in the change of coordinates. Densities can be generalized into ''s''-densities, whose coordinate representations become multiplied by the ''s''-th power of the absolute value of the jacobian determinant. On an oriented manifold 1-densities can be canonically identified with the ''n''-forms on ''M''. On non-orientable manifolds this identification cannot be made, since the density bundle is the tensor product of the orientation bundle of ''M'' and the ''n''-th exterior product bundle of ''T *M'' (see pseudotensor.) == Motivation (Densities in vector spaces) == In general, there does not exist a natural concept of a "volume" for a parallelotope generated by vectors in a ''n''-dimensional vector space ''V''. However, if one wishes to define a function that assigns a volume for any such parallelotope, it should satisfy the following properties: * If any of the vectors ''vk'' is multiplied by , the volume should be multiplied by |''λ''|. * If any linear combination of the vectors ''v''1, ..., ''vj''−1, ''vj''+1, ..., ''vn'' is added to the vector ''vj'', the volume should stay invariant. These conditions are equivalent to the statement that ''μ'' is given by a translation-invariant measure on ''V'', and they can be rephrased as : Any such mapping is called a density on the vector space ''V''. The set Vol(''V'') of all densities on ''V'' forms a one-dimensional vector space, and any ''n''-form ''ω'' on ''V'' defines a density on ''V'' by : 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「density on a manifold」の詳細全文を読む スポンサード リンク
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