Differential of a function

Differential of a function In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable. The differential dy is defined by where is the derivative of f with respect to x, and dx is an additional real variable (so that dy is a function of x and dx). The notation is such that the equation

Differential of a function holds, where the derivative is represented in the Leibniz notation dy/dx, and this is consistent with regarding the derivative as the quotient of the differentials. One also writes The precise meaning of the variables dy and dx depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or analytical significance if the differential is regarded as a linear approximation to the increment of a function. In physical applications, the variables dx and dy are often constrained to be very small ("infinitesimal").

Definition The differential is defined in modern treatments of differential calculus as follows.[7] The differential of a function f(x) of a single real variable x is the function df of two independent real variables x and Δx given by One or both of the arguments may be suppressed, i.e., one may see df(x) or simply df. If y = f(x), the differential may also be written as dy. Since dx(x, Δx) = Δx it is conventional to write dx = Δx, so that the following equality holds:

Definition This notion of differential is broadly applicable when a linear approximation to a function is sought, in which the value of the increment Δx is small enough. More precisely, if f is a differentiable function at x, then the difference in y-values satisfies

Definition where the error ε in the approximation satisfies ε/Δx → 0 as Δx → 0. In other words, one has the approximate identity in which the error can be made as small as desired relative to Δx by constraining Δx to be sufficiently small; that is to say, as Δx → 0. For this reason, the differential of a function is known as the principal (linear) part in the increment of a function: the differential is a linear function of the increment Δx, and although the error ε may be nonlinear, it tends to zero rapidly as Δx tends to zero.

General formulation If there exists an m × n matrix A such that A consistent notion of differential can be developed for a function f Rn → Rm between two Euclidean spaces. Let x,Δx ∈ Rn be a pair of Euclidean vectors. The increment in the function f is If there exists an m × n matrix A such that in which the vector ε → 0 as Δx → 0, then f is by definition differentiable at the point x. The matrix A is sometimes known as the Jacobian matrix, and the linear transformation that associates to the increment Δx ∈ Rn the vector AΔx ∈ Rm is, in this general setting, known as the differential df(x) of f at the point x. This is precisely the Fréchet derivative, and the same construction can be made to work for a function between any Banach spaces.

General formulation Another fruitful point of view is to define the differential directly as a kind of directional derivative: which is the approach already taken for defining higher order differentials (and is most nearly the definition set forth by Cauchy). If t represents time and x position, then h represents a velocity instead of a displacement as we have heretofore regarded it. This yields yet another refinement of the notion of differential: that it should be a linear function of a kinematic velocity. The set of all velocities through a given point of space is known as the tangent space, and so df gives a linear function on the tangent space: a differential form.

General formulation With this interpretation, the differential of f is known as the exterior derivative, and has broad application in differential geometry because the notion of velocities and the tangent space makes sense on any differentiable manifold. If, in addition, the output value of f also represents a position (in a Euclidean space), then a dimensional analysis confirms that the output value of df must be a velocity. If one treats the differential in this manner, then it is known as the pushforward since it "pushes" velocities from a source space into velocities in a target space.

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