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Let be a function in the Lebesgue space ([,]).We say that in ([,]) is a weak derivative of if ′ = ()for all infinitely differentiable functions with () = =.. Generalizing to dimensions, if and are in the space () of locally integrable functions for some open set, and if is a multi-index, we say that is the -weak derivative of if
Let be a Banach space, let ′ be the dual space of , let : ′ be a linear map, and let ′.A vector is a solution of the equation = if and only if for all , () = ().A particular choice of is called a test vector (in general) or a test function (if is a function space).
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.
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
In complex analysis, it is natural to define differentiation via holomorphic functions, which have a number of useful properties, such as repeated differentiability, expressibility as power series, and satisfying the Cauchy integral formula. In real analysis, it is usually more natural to consider differentiable, smooth, or harmonic functions ...
In complex analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing complex numbers . So, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is said to be differentiable at x = a {\textstyle x=a} when
The following notation will be used throughout this article: is a fixed positive integer and is a fixed non-empty open subset of Euclidean space. = {,,, …} denotes the natural numbers.
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. [1] It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.