The Method of Fluxions

Excerpts

• The Method of Fluxions

  If we suppose $y$ to be any function of $x$, and that $x$ increases with a given velocity, $y$ will also increase or decrease with a velocity depending: (1) upon the velocity of $x$; (2) upon the function which $y$ is of $x$.

• The Method of Fluxions

  If we diminish $dt$, the term $\dfrac{dx}{dt}\, dx$ will diminish without limit, since one factor continually approaches to a given quantity, viz., the velocity of $x$, and the other diminishes without limit.

• The Method of Fluxions

  The processes are the same in both methods, since the ratio of the velocities is the limiting ratio of the corresponding increments, or, according to Leibnitz, the ratio of the infinitely small increments.

Equations

• This equation is in Algebraical Geometry (Algebraical Geometry)
• This equation is in The Notation of the Differential Calculus (The Notation of the Differential Calculus)

• The Method of Fluxions

  \dfrac{dy}{dt} = 2x\, \dfrac{dx}{dt} + \dfrac{dx}{dt}\, dx

  Dividing the increment relation by the small interval of time dt gives the rate of change of y as 2x times the velocity of x plus a term that vanishes as dt shrinks.

• The Method of Fluxions

  \dot{y} = 2x\, \dot{x}

  The fluxion (velocity) of y = x^2 equals 2x times the fluxion (velocity) of x, the Newtonian form of the derivative result.

• The Method of Fluxions

  dy = 2x\, dx

  The differential form of the result: dy equals 2x dx, the form used in the differential method in place of fluxions.

Problems

No exercises in this chapter.