Quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two. For example, In mathematics, a quadratic form is a polynomial with terms all of degree two. For example, 4 x 2 + 2 x y − 3 y 2 {displaystyle 4x^{2}+2xy-3y^{2}} is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K,such as the real or complex numbers, and we speak of a quadratic form over K.Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology (intersection forms of four-manifolds), and Lie theory (the Killing form). Quadratic forms are not to be confused with a quadratic equation which has only one variable and includes terms of degree two or less. A quadratic form is one case of the more general concept of homogeneous polynomials. 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.The notation ⟨ a 1 , … , a n ⟩ {displaystyle langle a_{1},dots ,a_{n} angle } is often used for the quadratic form 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 Zp. 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.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: 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 q : V → K a quadratic form on V.