diferencials

Differentials, sometimes called differential forms, are infinitesimal changes used to express the linear approximation of changes in a function. In one-variable calculus, if y = f(x) is differentiable, the differential dy is defined by dy = f′(x) dx, where dx represents an independent infinitesimal change in x. The differential captures how much y changes to first order when x changes by a small amount, and Δy ≈ dy for small Δx.

Example: f(x) = x^2. Then dy = 2x dx. At x = 3 and dx = 0.1, dy ≈ 0.6, so Δy ≈ 0.63.

For functions of several variables, the differential is extended to the total differential: df = ∑ ∂f/∂x_i dx_i. It gives Δf ≈ df. Example: f(x, y) = x^2 + y^2, df = 2x dx + 2y dy.

In differential geometry, differentials are viewed as 1-forms, such as dx, dy, and df, and they can be integrated along curves. A differential that can be written as df for some function f is called an exact differential. The concept generalizes to higher-order forms and to manifolds, forming a core part of the calculus on differentiable spaces.

Applications of differentials include linearization and error propagation, where small changes in inputs are translated into approximated changes in outputs. They also underpin many techniques in physics and engineering, such as expressing infinitesimal work, fluxes, and the chain rule in differential form.