Non-standard calculus

In mathematics non-standard calculus is the application of non-standard analysis techniques to differential and integral calculus. It provides a rigorous justification of purely formal calculations using infinitesimals to derive facts about derivatives, integrals, and series. Such formal calculations with infinitesimals were widely used before alternative and rigorously justified methods, without infinitesimals were introduced in the 19th century. See history of calculus.

We illustrate this technique with the following naive calculation of the derivative of the function f(t)=t² at the value x:

<math> \frac{(x + \Delta x)^2 - x^2}{\Delta x} = 2 x + \Delta x \cong 2 x<math>

whenever the increment in x is infinitesimal. Thus the derivative of f at x is 2x. This formal calculation can be completely justified by non-standard analysis.

Basic theorems

If f is a real valued function defined on an interval [a, b], then *f is an internal, hyperreal-valued function defined on the hyperreal interval [*a, *b].

Theorem. Let f be a real-valued function defined on an interval [a, b]. The f is differentiable at a < x < b iff for every non-zero infinitesimal h, the value

<math> \Delta_h f := \operatorname{st} \frac{[*f](x+h)-[*f](x)}{h} <math>

is independent of h. In that case, the common value is the derivative of f at x.

This fact follows from the transfer principle of non-standard analysis and overspill.

Note that a similar result holds for differentiability at the endpoints a, b provided the sign of the infinitesimal h is suitably restricted.

For the second theorem, we consider the Cauchy integral. This integral is defined as the limit, if it exists, of a directed family of Cauchy sums; these are sums of the form

<math> \sum_{k=0}^{n-1} f(\xi_k) (x_{k+1} - x_k) <math>

where

<math>a = x_0 \leq \xi_0 \leq x_1 \leq \ldots x_{n-1} \leq \xi_{n-1} \leq x_n = b.<math>

We will call such a sequence of values a Cauchy integral mesh and

<math> \sup_k (x_{k+1} - x_k) <math>

the width of the mesh. In the definition of the Cauchy integral, the limit of the Cauchy sums is taken as the width of the mesh goes to 0.

Theorem. Let f be a real-valued function defined on an interval [a, b]. The f is Cauchy-integrable on [a, b] iff for every internal Cauchy integral mesh of infinitesimal width

<math> S_M = \operatorname{st} \sum_{k=0}^{n-1} [*f](\xi_k) (x_{k+1} - x_k) <math>

is independent of the mesh. In this case, the common value is the Cauchy integral of f over [a, b].

Applications

One immediate application is an extension of the standard definitions of differentiation and integration to internal functions on intervals of hyperreal numbers.

An internal hyperreal-valued function f on [a, b] is S-differentiable at x, provided

<math> \Delta_h f = \operatorname{st} \frac{f(x+h)-f(x)}{h} <math>

exists and is independent of the infinitesimal h. The value is the S derivative at x.

Theorem. Suppose f is S-differentiable at every point of [a, b] where b − a is a bounded hyperreal. Suppose furthermore that

<math> |f'(x)| \leq M \quad a \leq x \leq b. <math>

Then for some infinitesimal ε

<math> |f(b) - f(a)| \leq M (b-a) + \epsilon.<math>

To prove this, let N be a non-standard natural number. Divide the interval [a, b] into N subintervals by placing N − 1 equally spaced intermediate points:

<math>a = x_0 < x_1< \cdots < x_{N-1} < x_N = b<math>

Then

<math> |f(b) - f(a)| \leq \sum_{k=1}^{N-1} |f(x_{k+1}) - f(x_{k})| \leq \sum_{k=1}^{N-1} \left\{|f'(x_k)| + \epsilon_k\right\}|x_{k+1} - x_{k}|.<math>

Now the maximum of any internal set of infinitesimals is infinitesimal. Thus all the ε_k's are dominated by an infinitesimal ε. Therefore,

<math> |f(b) - f(a)| \leq \sum_{k=1}^{N-1} (M + \epsilon)(x_{k+1} - x_{k}) = M(b-a) + \epsilon (b-a)<math>

from which the result follows.

See also

How Archimedes used infinitesimals