Non-standard calculus


\(\newcommand\fLim{\textsf{lim}}\)For a real function \(f : \mathbb{R} \to \mathbb{R}\), \(\fLim_-~a~f\) (the limit of \(f\) as its input approaches \(a\) from left) and \(\fLim_+~a~f\) (the limit of \(f\) as its input approaches \(a\) from right) are: \[ \newcommand\fSt{\textsf{st}} \newcommand\tVec{\textsf{Vec}} \newcommand\mbR{\mathbb{R}} \begin{align*} \fLim_-~a~f &= \fSt~(f~(a-\delta)) \\ \fLim_+~a~f &= \fSt~(f~(a+\delta)) \end{align*} \] where \(\delta\) is the unit hyperreal infinitesimal. The limit of of \(f\) as its input approaches \(a\) exists and is equal to both limits iff both limits exist and are equal: \[ \frac{\fLim_-~a~f = b \qquad \fLim_+~a~f = b}{\fLim~a~f = b}. \]

The limit as input grows unbounded positively is: \[ \newcommand\fLimgub{\textsf{limgub}} \fLimgub = \fLim_-~\omega \] where \(\omega\) is the unit hypernatural infinite: \[ \delta \cdot \omega = 1. \]


Find the limit of the following \(f\) as \(x\) approaches \(1\) from the right: \[ f~x = \frac{x^2-1}{x-1}. \] First, we compute \(f~(x+\delta)\) with \(x=1\): \[ \begin{align*} f~(1+\delta) &= \frac{(1+\delta)^2-1}{(1+\delta)-1} \\ &= \frac{1+2 \delta + \delta^2-1}{1+\delta-1} \\ &= \frac{2 \delta + \delta^2}{\delta} \\ &= 2+\delta. \end{align*} \] Then, we take the standard part: \[ \begin{align*} \fLim_+~1~f &= \fSt~(f~(1+\delta)) \\ &= \fSt~(2+\delta) \\ &= 2. \end{align*} \]


The derivative of \(f\) at \(x\): \[ \newcommand\fDer{\textsf{der}} \fDer~f~x = \fSt \left( \frac{f~(x+\delta)-f~x}{\delta} \right). \]

Partial derivative

The partial derivative of vector-to-real function \(f\) at \(x\) with respect to the input component at index \(k\) is: \[ \newcommand\fPd{\textsf{pd}} \\ \fPd~k~f~x = \fSt\left(\frac{f~(x+\delta \cdot e_k)-f~x}{\delta}\right) \] where \(e_k\) is the vector whose element at index \(k\) is one and others zero. The index begins at zero.

Partial derivation commutes in this way: \[ \fPd~m~(\fPd~n~f) = \fPd~n~(\fPd~m~f) \] which can also be written: \[ \fPd~m \circ \fPd~n = \fPd~n \circ \fPd~m. \]

Directional derivative

The directional derivative of a function \(f\) along a vector \(d\) generalizes the notion of partial derivative: \[ \newcommand\fDd{\textsf{dd}} \begin{align*} \fDd~d~f~x &= \fSt\left(\frac{f~(x+\delta \cdot d) - f~x}{\delta}\right) \\ \fPd~k~f~x &= \fDd~e_k~f~x. \end{align*} \]

Riemann integral

Let \(s = [a,b]\) be a real line. Let \(\newcommand\fPart{\textsf{part}} \newcommand\fAint{\textsf{aint}} \newcommand\fInt{\textsf{int}} \newcommand\fSum{\textsf{sum}} \newcommand\fMap{\textsf{map}} \fPart~n~s\) be the partitioning of \(s\) into \(n\) contiguous parts. Let \(\fAint~n~s~f\) be the \(n\)-approximation of the integral of \(f\) over \(s\): \[ \fAint~n~s~f = \fSum~(\fMap~(\lambda~u.~f~(p~u)\cdot m~u)~(\fPart~n~s)) \] where \(p~u\) is any point in \(u\) and \(m~u\) is the measure of \(u\). For simplicity, we can use the left point of \(u\) as \(p~u\), and we divide the parts into the same size. The Riemann integral is defined as: \[ \fInt = \fAint~\omega. \]

Other resources

Wikipedia has an article about non-standard calculus.