Fxdx

fxdx is a string that commonly appears in calculus as shorthand for the product of a function f evaluated at x and the differential dx, written as f(x) dx. It is not a standalone function or constant, but a notational form used in integrals and differential forms to indicate the variable of integration and the infinitesimal element with respect to which accumulation is performed.

In integral notation, ∫ f(x) dx uses dx to denote the differential element with respect to the variable x. The dx indicates the variable of integration and serves to specify how the accumulation is performed. In the limit sense, Riemann sums approximate f(x_i) Δx, where Δx plays the role of the differential dx as the partition becomes finer.

In differential geometry and multivariable calculus, f(x) dx can be interpreted as a 1-form on a one-dimensional manifold like the real line. In higher dimensions, one writes components such as f_i(x) dx_i, forming a collection of 1-forms that can be integrated along curves or surfaces.

Notationally, variants such as f(x) dx or simply f dx are common, especially in informal writing. The dx symbol is not a stand-alone number; it is a differential that indicates the variable of integration and the corresponding infinitesimal change, and its precise interpretation depends on the mathematical context.

Example: for f(x) = x^2, the integral ∫ x^2 dx equals x^3/3 + C, illustrating how f(x) dx serves as the integrand. The concept underpins the link between differentiation and integration via the fundamental theorem of calculus.