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In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus , which originally referred to the " infinity - eth " item in a sequence .
The length of the curve is given by the formula = | ′ | where | ′ | is the Euclidean norm of the tangent vector ′ to the curve. To justify this formula, define the arc length as limit of the sum of linear segment lengths for a regular partition of [ a , b ] {\displaystyle [a,b]} as the number of segments approaches infinity.
Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous rates of change , and the slopes of curves , while the latter concerns accumulation of quantities, and areas under or between curves.
The rigorous counterpart of such a calculation would be that if ε is a non-zero infinitesimal, then 1/ε is infinite. For any finite hyperreal number x , the standard part , st( x ), is defined as the unique closest real number to x ; it necessarily differs from x only infinitesimally.
In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then Δ y = f ′ ( x ) Δ x + ε Δ x {\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x} for some infinitesimal ε , where Δ y = f ( x + Δ x ) − f ( x ...
The formula for the magnitude of the solid angle in steradians is ... the position vector of an infinitesimal area of surface dS with ... "Calculation of solid angle ...
In 1655, John Wallis first used the notation for such a number in his De sectionibus conicis, [19] and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of . [20] But in Arithmetica infinitorum (1656), [21] he indicates infinite series, infinite products and infinite continued fractions by ...
In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus.It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.