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In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, : is a quadratic form in the variables ''x'' and ''y''. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric), differential topology (intersection forms of four-manifolds), and Lie theory (the Killing form). == Introduction == Quadratic forms are homogeneous quadratic polynomials in ''n'' variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form: : : : where ''a'', ..., ''f'' are the coefficients.〔A tradition going back to Gauss dictates the use of manifestly even coefficients for the products of distinct variables, i.e. 2''b'' in place of ''b'' in binary forms and 2''d'', 2''e'', 2''f'' in place of ''d'', ''e'', ''f'' in ternary forms. Both conventions occur in the literature〕 Note that quadratic functions, such as in the one variable case, are not quadratic forms, as they are typically not homogeneous (unless ''b'' and ''c'' are both 0). The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain field. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring, frequently the integers Z or the ''p''-adic integers Z''p''.〔away from 2, i. e. if 2 is invertible in the ring, quadratic forms are equivalent to symmetric bilinear forms (by the polarization identities), but at 2 they are different concepts; this distinction is particularly important for quadratic forms over the integers.〕 Binary quadratic forms have been extensively studied in number theory, in particular, in the theory of quadratic fields, continued fractions, and modular forms. The theory of integral quadratic forms in ''n'' variables has important applications to algebraic topology. Using homogeneous coordinates, a non-zero quadratic form in ''n'' variables defines an (''n''−2)-dimensional quadric in the (''n''−1)-dimensional projective space. This is a basic construction in projective geometry. In this way one may visualize 3-dimensional real quadratic forms as conic sections. A closely related notion with geometric overtones is a quadratic space, which is a pair (''V'',''q''), with ''V'' a vector space over a field ''K'', and a quadratic form on ''V''. An example is given by the three-dimensional Euclidean space and the square of the Euclidean norm expressing the distance between a point with coordinates (''x'',''y'',''z'') and the origin: : 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「quadratic form」の詳細全文を読む スポンサード リンク
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