differentiaaleja

In calculus, the term differentiaali (plural differentiaaleja) denotes the differential of a function, i.e., its linear approximation to a small change in the input. The concept is central in single-variable and multivariable calculus and also appears in differential geometry as a 1-form. In Finnish mathematical usage, differentiaali refers to this notion, with differentiaaleja as the plural form.

For a differentiable function f: R^n → R, the differential at a point a is a linear map df_a: R^n → R defined by df_a(h) = ∑_i ∂f/∂x_i(a) h_i. In coordinate form, df_a = ∑_i ∂f/∂x_i(a) dx_i. In one variable, dy = f'(x) dx is the familiar differential.

This differential serves as the first-order term in the Taylor expansion Δf ≈ df_a(h). It is linear in the input increment h and describes how small changes in the inputs affect the output. It also encodes gradient information: df_a(h) = ∇f(a) · h.

Examples: If f(x,y) = x^2 + y^3, then df = 2x dx + 3y^2 dy. If f(t) = t^2, then df = 2t dt. The symbols dx_i or dt (different names for differential of a coordinate) indicate an infinitesimal change in that coordinate.

Beyond basic calculus, differentiaaleja are treated as 1-forms in differential geometry, and the concept generalizes to pullbacks and exterior derivatives. They are used in substitutions in integration and in expressing the chain rule in a coordinate-free form.