differentiaaliset

Differentiaaliset is Finnish plural form of the adjective differentiaalinen, commonly used in mathematics to describe things related to differentials. As a standalone term, it is not a separate mathematical object, but a grammatical form that appears in phrases referring to multiple differentials or to concepts built from them.

In calculus, a differential represents an infinitesimal change in a function due to a small change in its input. For a single-variable function y = f(x), the differential is dy, and dy ≈ f'(x) dx. The differential of a multivariable function f(x1,...,xn) is the total differential df = sum_i (∂f/∂xi) dxi.

Beyond differentials, the notion leads to differential forms, a central object in differential geometry. A differential form of degree 1 assigns a linear differential to each tangent vector, and higher-degree forms combine via the wedge product. The exterior derivative d maps k-forms to (k+1)-forms, providing a coordinate-free way to express concepts such as gradients, curls, and fluxes.

Historically, the differential concept was introduced by Gottfried Wilhelm Leibniz and is foundational in analysis, physics, and geometry. In Finnish mathematical writing, differentiaaliset will typically appear as adjectives describing all these objects (for example, differentiaaliset käsitteet or differentiaaliset muodot), rather than as a distinct concept itself.

Further reading includes differential, differential form, exterior derivative, and manifolds.