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In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged.〔 section §7.〕 The index subset must generally either be all ''covariant'' or all ''contravariant''. For example, : holds when the tensor is antisymmetric on it first three indices. If a tensor changes sign under exchange of ''any'' pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor of order ''p'' may be referred to as a ''p''-form, and a completely antisymmetric contravariant tensor may be referred to as a ''p''-vector. ==Antisymmetric and symmetric tensors== A tensor A that is antisymmetric on indices ''i'' and ''j'' has the property that the contraction with a tensor B that is symmetric on indices ''i'' and ''j'' is identically 0. For a general tensor U with components and a pair of indices ''i'' and ''j'', U has symmetric and antisymmetric parts defined as: :(U_+U_) || || (symmetric part) |- | || ||(antisymmetric part). |} Similar definitions can be given for other pairs of indices. As the term "part" suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in : 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「antisymmetric tensor」の詳細全文を読む スポンサード リンク
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