2xy

2xy denotes twice the product of the real (or complex) variables x and y. As a function of two variables, it is a homogeneous polynomial of total degree 2 and serves as a simple bilinear form in x and y. The expression is symmetric, since 2xy = 2yx, and it appears as the cross term in the expansion of a square: (x + y)² = x² + 2xy + y².

In algebra, 2xy is the off-diagonal cross term when forming general quadratic expressions in x and y. It helps describe quadratic forms and appears in changes of variables and in the study of parabolic or hyperbolic surfaces when set equal to a constant.

Geometrically, the equation 2xy = c (or equivalently xy = c/2) defines a rectangular hyperbola in the plane. For c > 0, the curve lies in the first and third quadrants; for c < 0, it lies in the second and fourth quadrants. The level sets of the function f(x, y) = 2xy have hyperbolic shapes.

In calculus and linear algebra, 2xy has several notable properties. The partial derivatives are ∂(2xy)/∂x = 2y and ∂(2xy)/∂y = 2x. It can be represented as xᵗA y with A = [[0, 1], [1, 0]], since [x y] [[0, 1], [1, 0]] [x; y] = 2xy.

Examples: if x = 1 and y = 3, then 2xy = 6. If x = 2, y = −5, then 2xy = −20.

See also: quadratic form, cross term, binomial expansion, rectangular hyperbola.