![]() Defining the differential as a kind of differential form, specifically the exterior derivative of a function.Main article: Differential (infinitesimal)Īlthough the notion of having an infinitesimal increment dx is not well-defined in modern mathematical analysis, a variety of techniques exist for defining the infinitesimal differential so that the differential of a function can be handled in a manner that does not clash with the Leibniz notation. 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. 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. 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. 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. This yields yet another refinement of the notion of differential: that it should be a linear function of a kinematic velocity. If t represents time and x position, then h represents a velocity instead of a displacement as we have heretofore regarded it. Which is the approach already taken for defining higher order differentials (and is most nearly the definition set forth by Cauchy). Of a point in rectilinear motion according to the law $ s = f ( t) $.D f ( x ) = f ′ ( x ) d x. ![]() 10 Derivatives and differentials of composite functions.īe defined in some neighbourhood of a point $ x _ ( t) $.9 Differential calculus of functions in several variables.8 Principal theorems and applications of differential calculus.3 Mechanical interpretation of the derivative.2 Geometric interpretation of the derivative.The central concepts of differential calculus - the derivative and the differential - and the apparatus developed in this connection furnish tools for the study of functions which locally look like linear functions or polynomials, and it is in fact such functions which are of interest, more than other functions, in applications. Differential calculus is based on the concepts of real number function limit and continuity - highly important mathematical concepts, which were formulated and assigned their modern content during the development of mathematical analysis and during studies of its foundations. Differential calculus is usually understood to mean classical differential calculus, which deals with real-valued functions of one or more real variables, but its modern definition may also include differential calculus in abstract spaces. The creation of differential and integral calculus initiated a period of rapid development in mathematics and in related applied disciplines. Leibniz towards the end of the 17th century, but their justification by the concept of limit was only developed in the work of A.L. Differential and integral calculus were created, in general terms, by I. Descartes was the principal factor in the creation of differential calculus. The introduction of variable magnitudes into mathematics by R. Together they form the base of mathematical analysis, which is extremely important in the natural sciences and in technology. The development of differential calculus is closely connected with that of integral calculus. A branch of mathematics dealing with the concepts of derivative and differential and the manner of using them in the study of functions.
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