dXdX

dxdx is not a standard mathematical object with a single, universal meaning. Depending on the field and the author, it may be used informally to denote different things, or it may simply be a typographical shorthand for other symbols such as (dx)^2 or the wedge product dx ∧ dx. Because of this, encountering dxdx requires inspecting surrounding notation.

In calculus and differential geometry, differentials are typically written as dx, dy, etc., and products of differentials are handled in two main ways. Treating them as ordinary products yields (dx)^2, which is of second order smallness and is usually neglected in first-order Taylor expansions. In the language of differential forms, dx ∧ dx equals zero because the wedge product is antisymmetric; thus dx ∧ dx and any symmetric product dx dx are considered identically zero in that formalism.

A related concept is the second differential d^2x, used in some expansions and introduced as d^2x = x''(t) dt^2 for x = x(t). In this context, d^2x is a second-order quantity and is not identically zero.

In stochastic calculus, products of differentials (for example dX dX) are treated via quadratic variation and are not ordinary products; statements about such products require the Itô calculus framework. The notation dxdx in this area is uncommon and highly context-dependent.

When encountered in text, clarifying how the author defines or uses dxdx is essential. If no definition is provided, treat it as ambiguous and seek additional explanation.