Calculus of Finite Differences. Successive Differentiation

Excerpts

• Calculus of Finite Differences. Successive Differentiation

  The symbol $\Delta x$ is called the *difference* of $x$, being the difference between the value of the variable $x$, before and after its increase.

• Calculus of Finite Differences. Successive Differentiation

  And the student must recollect, that in like manner as $\Delta$ is not the symbol of a number, but of an operation, so $\Delta^{2}$ does not denote a number multiplied by itself, but an operation repeated upon its own result; just as the logarithm of the logarithm of $x$ might be written $\log^{2} x$; $(\log x)^{2}$ being reserved to signify the square of the logarithm of $x$.

• Calculus of Finite Differences. Successive Differentiation

  If $y$ be a function which decreases when $x$ is increased, $y_{1} - y$, or $\Delta y$ is negative.

• Calculus of Finite Differences. Successive Differentiation

  And as we have denoted the operation which deduces the second column from the first by $\Delta$, so that which deduces the third from the second may be denoted by $\Delta\Delta$, which is abbreviated into $\Delta^{2}$.

• Calculus of Finite Differences. Successive Differentiation

  Hence we have a succession of ratios $\dfrac{dy}{dx}$, $\dfrac{d^{2} y}{dx^{2}}$, $\dfrac{d^{3} y}{dx^{3}}$, etc., which tend towards finite limits when $dx$ is diminished.

• Calculus of Finite Differences. Successive Differentiation

  Write $dx$ for $\Delta x$, etc., and recollect that $h - dx$, $h - 2\, dx$, etc., continually approximate to $h$.

Equations

• Calculus of Finite Differences. Successive Differentiation

  \Delta^{2} y_{1} - \Delta^{2} y = \Delta^{3} y

  The third difference of y at the first step equals the second difference at the first step minus the second difference at the start, so each order of difference is taken from the one before it.

• Calculus of Finite Differences. Successive Differentiation

  y_{2} = y_{1} + \Delta y_{1}

  Each value of the function is the previous value plus the first difference at the previous value, which generates the table of successive values.

• Calculus of Finite Differences. Successive Differentiation

  y_{1} = y + \Delta y

  The value of the function after one step equals its starting value plus the first difference.

• Calculus of Finite Differences. Successive Differentiation

  \Delta y_{2} = \Delta y + 2\Delta^{2} y + \Delta^{3} y

  The first difference two steps on equals the first difference at the start plus twice the second difference plus the third difference.

• Calculus of Finite Differences. Successive Differentiation

  n\Delta x = h

  Taking n equal steps of size \Delta x to go from x to x + h means n times the step equals h.

• Calculus of Finite Differences. Successive Differentiation

  \phi(x + h) = y + \frac{dy}{dx}\, h + \frac{d^{2} y}{dx^{2}}\, \frac{h^{2}}{2} + \frac{d^{3} y}{dx^{3}}\, \frac{h^{3}}{2·3} + \etc.\Add{,}

  The value of the function at x + h is expanded as the value at x plus successive higher-order derivatives multiplied by powers of h over factorials.

Problems

No exercises in this chapter.