differentialförskjutningar

Differential is a term used in several disciplines, most often referring to a small change in a quantity or to an operator that maps a function to its rate of change. In single-variable calculus, the differential dy of a function y = f(x) is defined as dy = f'(x) dx, expressing how a small change in x produces a corresponding change in y. This concept underpins linear approximation and is a building block for integration and differentiation.

For functions of multiple variables, the total differential generalizes to df ≈ ∑ ∂f/∂x_i dx_i, describing how infinitesimal changes in each coordinate contribute to a change in the function. Differentials extend further in differential geometry as differential forms, which are algebraic objects that can be integrated over curves, surfaces, and more generally on manifolds. The exterior derivative d acts on differential forms and satisfies d^2 = 0, leading to central results in topology and geometry.

In physics and engineering, differentials appear in descriptions of physical quantities changing over time or space, and in the formulation and solution of differential equations that relate rates of change to quantities of interest. In mechanical contexts, the term differential also denotes a mechanical differential gear, which distributes torque between wheels to allow different rotational speeds.

Overall, the concept of a differential links infinitesimal changes, linear approximations, and integral formulations across mathematics and applied sciences. If a specific field or term was intended, further clarification can tailor the article appropriately.