XdY

XdY is a term that can have different meanings depending on context, and there is no single universally fixed definition. In mathematics, it is most often read as the product of a function or variable X with the differential of another function Y, written as X dY.

In calculus and differential geometry, X dY is typically treated as a 1-form: a linear object that can be integrated along a curve. The expression appears in contexts such as integration by parts for differential forms, where ∫ X dY can be rewritten as X Y evaluated at endpoints minus ∫ Y dX. In differential geometry, the exterior derivative of this 1-form satisfies d(X dY) = dX ∧ dY, illustrating how X dY interacts with the differential structure of the space.

XdY can also arise in stochastic calculus, where dY denotes an infinitesimal change in a stochastic process. In Itô calculus, products like X dY feature in the expansion of d(XY), which includes additional terms such as d⟨X, Y⟩, the quadratic covariation. These interpretations depend crucially on the nature of X and Y (deterministic functions, stochastic processes, etc.) and on the surrounding mathematical framework.

Beyond formal mathematics, XdY is often used as a stylized label or brand name in branding and product design, chosen for its distinctive typography rather than a specific technical meaning.

See also: differential, differential form, integration by parts, exterior derivative, Itô calculus.