Fluxion

Template:Short description Template:About {{#invoke:sidebar|collapsible | class = plainlist | titlestyle = padding-bottom:0.25em; | pretitle = Part of a series of articles about | title = Calculus | image = $\underset{a}{\overset{b}{\int }}{f}^{\prime }(t)dt=f(b)-f(a)$ | listtitlestyle = text-align:center; | liststyle = border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa; | expanded = | abovestyle = padding:0.15em 0.25em 0.3em;font-weight:normal; | above =

• Fundamental theorem

• Limits
• Continuity

• Rolle's theorem
• Mean value theorem
• Inverse function theorem

| list2name = differential | list2titlestyle = display:block;margin-top:0.65em; | list2title = Template:Bigger | list2 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | heading1 = Definitions
 | content1 =

• Derivative (generalizations)
• Differential
  • infinitesimal
  • of a function
  • total

 | heading2 = Concepts
 | content2 =

• Differentiation notation
• Second derivative
• Implicit differentiation
• Logarithmic differentiation
• Related rates
• Taylor's theorem

 | heading3 = Rules and identities
 | content3 =

• Sum
• Product
• Chain
• Power
• Quotient
• L'Hôpital's rule
• Inverse
• General Leibniz
• Faà di Bruno's formula
• Reynolds

}}

| list3name = integral | list3title = Template:Bigger | list3 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

• Lists of integrals
• Integral transform
• Leibniz integral rule

| heading2 = Definitions

 | content2 =

• Antiderivative
• Integral (improper)
• Riemann integral
• Lebesgue integration
• Contour integration
• Integral of inverse functions

 | heading3 = Integration by
 | content3 =

• Parts
• Discs
• Cylindrical shells
• Substitution (trigonometric, tangent half-angle, Euler)
• Euler's formula
• Partial fractions (Heaviside's method)
• Changing order
• Reduction formulae
• Differentiating under the integral sign
• Risch algorithm

}}

| list4name = series | list4title = Template:Bigger | list4 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

• Geometric (arithmetico-geometric)
• Harmonic
• Alternating
• Power
• Binomial
• Taylor

 | heading2 = Convergence tests
 | content2 =

• Summand limit (term test)
• Ratio
• Root
• Integral
• Direct comparison
• 
  Limit comparison
• Alternating series
• Cauchy condensation
• Dirichlet
• Abel

}}

| list5name = vector | list5title = Template:Bigger | list5 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

• Gradient
• Divergence
• Curl
• Laplacian
• Directional derivative
• Identities

 | heading2 = Theorems
 | content2 =

• Gradient
• Green's
• Stokes'
• Divergence
• generalized Stokes
• Helmholtz decomposition

}}

| list6name = multivariable | list6title = Template:Bigger | list6 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | heading1 = Formalisms
 | content1 =

• Matrix
• Tensor
• Exterior
• Geometric

 | heading2 = Definitions
 | content2 =

• Partial derivative
• Multiple integral
• Line integral
• Surface integral
• Volume integral
• Jacobian
• Hessian

}}

| list7name = advanced | list7title = Template:Bigger | list7 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

• Calculus on Euclidean space
• Generalized functions
• Limit of distributions

}}

| list8name = specialized | list8title = Template:Bigger | list8 =

• Fractional
• Malliavin
• Stochastic
• Variations

| list9name = miscellanea | list9title = Template:Bigger | list9 =

• Precalculus
• History
• Glossary
• List of topics
• Integration Bee
• Mathematical analysis
• Nonstandard analysis

}}

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

A fluxion is the instantaneous rate of change, or gradient, of a fluent (a time-varying quantity, or function) at a given point. 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. Fluxions and fluents made up Newton's early calculus.

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: Template:Quote The gibberish string was in fact a hash code (by denoting the frequency of each letter) of the Latin phrase Data æqvatione qvotcvnqve flventes qvantitates involvente, flvxiones invenire: et vice versa, meaning: "Given an equation that consists of any number of flowing quantities, to find the fluxions: and vice versa".

Example

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

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

Here Template:Tmath is an infinitely small amount of time. So, the term Template:Tmath is second order infinite small term and according to Newton, we can now ignore Template:Tmath because of its second order infinite smallness comparing to first order infinite smallness of Template:Tmath. So, the final equation gets the form:

$\dot{y}=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }t}=\frac{4o}{o}=4$

He justified the use of Template:Tmath 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, denounced Newton's fluxions in his essay The Analyst, published in 1734. Berkeley refused to believe that they were accurate because of the use of the infinitesimal Template:Tmath. 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 Template:Tmath as infinitely small, preferring to define it as approaching zero, using a similar definition to the concept of limit. 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 remains in use today.

See also

• History of calculus
• Newton's notation
• Hyperreal number: A modern formalization of the reals that includes infinity and infinitesimals
• Nonstandard analysis