Fluxion

A fluxion is the instantaneous rate of change, or gradient, of a fluent (a time-varying quantity, or function) at a given point.[1] Fluxions were introduced by Isaac Newton to describe his form of a time derivative (a derivative with respect to time). Newton introduced the concept in 1665 and detailed them in his mathematical treatise, Method of Fluxions.[2] Fluxions and fluents made up Newton's early calculus.[3]

Differential

Definitions
Derivative (generalizations) Differential infinitesimal of a function total
Concepts
Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem
Rules and identities
Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula

Integral

Definitions
Lists of integrals Integral transform
Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions
Integration by
Parts Discs Cylindrical shells Substitution (trigonometric, Weierstrass, Euler) Euler's formula Partial fractions Changing order Reduction formulae Differentiating under the integral sign

Series

Convergence tests
Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor
Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel

Newton's introduction of the notions "fluent" and "fluxion" in his 1736 book

History

Fluxions were central to the Leibniz–Newton calculus controversy, when Newton sent a letter to Gottfried Wilhelm Leibniz explaining them, but concealing his words in code due to his suspicion. He wrote:[4]

I cannot proceed with the explanations of the fluxions now, I have preferred to conceal it thus: 6accdæ13eff7i319n4o4qrr4s8t12vz

The gibberish string was in fact an enciphered Latin phrase, meaning: "Given an equation that consists of any number of flowing quantities, to find the fluxions: and vice versa".[5]

Example

If the fluent $y$ is defined as $y=t^{2}$ (where $t$ is time) the fluxion (derivative) at $t=2$ is:

{\dot {y}}={\frac {\Delta y}{\Delta t}}={\frac {(2+o)^{2}-2^{2}}{(2+o)-2}}={\frac {4+4o+o^{2}-4}{2+o-2}}=4+o

Here $o$ is an infinitely small amount of time[6] and according to Newton, we can now ignore it because of its infinite smallness.[7] He justified the use of $o$ as a non-zero quantity by stating that fluxions were a consequence of movement by an object.

Criticism

Bishop George Berkeley, a prominent philosopher of the time, slammed Newton's fluxions in his essay The Analyst, published in 1734.[8] Berkeley refused to believe that they were accurate because of the use of the infinitesimal $o$ . He did not believe it could be ignored and pointed out that if it was zero, the consequence would be division by zero. Berkeley referred to them as "ghosts of departed quantities", a statement which unnerved mathematicians of the time and led to the eventual disuse of infinitesimals in calculus.

Towards the end of his life Newton revised his interpretation of $o$ as infinitely small, preferring to define it as approaching zero, using a similar definition to the concept of limit.[9] He believed this put fluxions back on safe ground. By this time, Leibniz's derivative (and his notation) had largely replaced Newton's fluxions and fluents, and remain in use today.

References

Newton, Sir Isaac (1736). The Method of Fluxions and Infinite Series: With Its Application to the Geometry of Curve-lines. Henry Woodfall; and sold by John Nourse. Retrieved 6 March 2017.
Weisstein, Eric W. "Fluxion". MathWorld.
Fluxion at the Encyclopædia Britannica
Turnbull, Isaac Newton. Ed. by H.W. (2008). The correspondence of Isaac Newton (Digitally printed version, pbk. re-issue. ed.). Cambridge [u.a.]: Univ. Press. ISBN 9780521737821.
Clegg, Brian (2003). A brief history of infinity: the quest to think the unthinkable. London: Constable. ISBN 9781841196503.
Buckmire, Ron. "History of Mathematics" (PDF). Retrieved 28 January 2017.
"Isaac Newton (1642-1727)". www.mhhe.com. Retrieved 6 March 2017.
Berkeley, George (1734). The Analyst: a Discourse addressed to an Infidel Mathematician . London. p. 25 – via Wikisource.
Kitcher, Philip (March 1973). "Fluxions, Limits, and Infinite Littlenesse. A Study of Newton's Presentation of the Calculus". Isis. 64 (1): 33–49. doi:10.1086/351042.

Calculus
Precalculus	Binomial theorem Concave function Continuous function Factorial Finite difference Free variables and bound variables Graph of a function Linear function Radian Rolle's theorem Secant Slope Tangent
Limits	Indeterminate form Limit of a function One-sided limit Limit of a sequence Order of approximation (ε, δ)-definition of limit
Differential calculus	Derivative Differential Differential equation Differential operator Mean value theorem Notation Leibniz's notation Newton's notation Rules of differentiation linearity Power Sum Chain L'Hôpital's Product General Leibniz's rule Quotient Other techniques Implicit differentiation Inverse functions and differentiation Logarithmic derivative Related rates Stationary points First derivative test Second derivative test Extreme value theorem Maxima and minima Further applications Newton's method Taylor's theorem
Integral calculus	Antiderivative Arc length Basic properties Constant of integration Differentiation under the integral sign Fundamental theorem of calculus Differentiating under the integral sign Integration by parts Integration by substitution trigonometric Euler Weierstrass Partial fractions in integration Quadratic integral Trapezoidal rule Volumes Washer method Shell method
Vector calculus	Derivatives Curl Directional derivative Divergence Gradient Laplacian Basic theorems Line integrals Green's Stokes' Gauss'
Multivariable calculus	Divergence theorem Geometric Hessian matrix Jacobian matrix and determinant Lagrange multiplier Line integral Matrix Multiple integral Partial derivative Surface integral Volume integral Advanced topics Differential forms Exterior derivative Generalized Stokes' Theorem Tensor calculus
Sequences and series	Arithmetico–geometric sequence Types of series Alternating Binomial Fourier Geometric Harmonic Infinite Power Maclaurin Taylor Telescoping Tests of convergence Abel's Alternating series Cauchy condensation Direct comparison Dirichlet's Integral Limit comparison Ratio Root Term
Special functions and numbers	Bernoulli numbers e (mathematical constant) Exponential function Natural logarithm Stirling's approximation
History of calculus	Adequality Brook Taylor Colin Maclaurin Generality of algebra Gottfried Wilhelm Leibniz Infinitesimal Infinitesimal calculus Isaac Newton Fluxion Law of Continuity Leonhard Euler Method of Fluxions The Method of Mechanical Theorems
Lists	Differentiation rules List of integrals of exponential functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions List of integrals of inverse trigonometric functions List of integrals of irrational functions List of integrals of logarithmic functions List of integrals of rational functions List of integrals of trigonometric functions Secant Secant cubed List of limits Lists of integrals
Miscellaneous topics	Differential geometry curvature of curves of surfaces Euler–Maclaurin formula Gabriel's Horn Proof that 22/7 exceeds π Regiomontanus' angle maximization problem Steinmetz solid

Sir Isaac Newton
Publications	Fluxions (1671) De Motu (1684) Principia (1687; writing) Opticks (1704) Queries (1704) Arithmetica (1707) De Analysi (1711)
Other writings	Quaestiones (1661–1665) "standing on the shoulders of giants" (1675) Notes on the Jewish Temple (c. 1680) "General Scholium" (1713; "hypotheses non fingo" ) Ancient Kingdoms Amended (1728) Corruptions of Scripture (1754)
Contributions	Calculus fluxion Impact depth Inertia Newton disc Newton polygon Newton–Okounkov body Newton's reflector Newtonian telescope Newton scale Newton's metal Newton's cradle Spectrum Structural coloration
Newtonianism	Bucket argument Newton's inequalities Newton's law of cooling Newton's law of universal gravitation post-Newtonian expansion parameterized gravitational constant Newton–Cartan theory Schrödinger–Newton equation Newton's laws of motion Kepler's laws Newtonian dynamics Newton's method in optimization Apollonius's problem truncated Newton method Gauss–Newton algorithm Newton's rings Newton's theorem about ovals Newton–Pepys problem Newtonian potential Newtonian fluid Classical mechanics Corpuscular theory of light Leibniz–Newton calculus controversy Newton's notation Rotating spheres Newton's cannonball Newton–Cotes formulas Newton's method generalized Gauss–Newton method Newton fractal Newton's identities Newton polynomial Newton's theorem of revolving orbits Newton–Euler equations Newton number kissing number problem Newton's quotient Parallelogram of force Newton–Puiseux theorem Absolute space and time Luminiferous aether Newtonian series table
Personal life	Woolsthorpe Manor (birthplace) Cranbury Park (home) Early life Later life Religious views Occult studies Scientific Revolution Copernican Revolution
Relations	Catherine Barton (niece) John Conduitt (nephew-in-law) Isaac Barrow (professor) William Clarke (mentor) Benjamin Pulleyn (tutor) John Keill (disciple) William Stukeley (friend) William Jones (friend) Abraham de Moivre (friend)
Depictions	Newton by Blake (monotype) Newton by Paolozzi (sculpture)
Namesake	Isaac Newton Institute Isaac Newton Medal Isaac Newton Telescope Isaac Newton Group of Telescopes Newton (unit)
Categories	► Isaac Newton

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Fluxion

History

Example

Criticism

See also

References