A differential form is nothing other than a generalization of infinitesimal volumes such as dxdydz that appear in the usual Riemann integral onto a differentiable manifold. Conversely, to define a differential form, it is necessary to first define an infinitesimal volume of general dimensions in Euclidean space. This is the so-called exterior algebra, and since it is a formulation of volume elements, conventions such as 0 if the same edge appears more than once, or the sign being reversed when two adjacent edges are swapped should naturally be introduced. Source: Heibonsha World Encyclopedia, 2nd Edition Information |
通常のリーマン積分で現れる,微小体積dxdydzなどを微分可能多様体上に一般化するのが,微分形式にほかならない。逆にいえば,微分形式を定めるには,一般次元の微小体積をユークリッド空間でまず定めておくことが必要である。これがいわゆる外積代数であって,したがって体積要素の定式化である以上は,同じ〈辺〉が2度以上現れると0とか,相隣る二つの〈辺〉が入れかえられると符号が逆になるなどの規約は自然に導入されるべきものである。
出典 株式会社平凡社世界大百科事典 第2版について 情報 |
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