Pedagogical no relation infinitesimal
Introduction
A non-zero element of an ordered field 
  
    
      
        
          F
        
      
    
    {\displaystyle \mathbb {F} }
   is infinitesimal if and only if its absolute value is smaller than any element of 
  
    
      
        
          F
        
      
    
    {\displaystyle \mathbb {F} }
   of the form 
  
    
      
        
          
            1
            n
          
        
      
    
    {\displaystyle {\frac {1}{n}}}
  , for 
  
    
      
        n
      
    
    {\displaystyle n}
   a standard natural number. Ordered fields that have infinitesimal elements are also called non-Archimedean. More generally, nonstandard analysis is any form of mathematics that relies on nonstandard models and the transfer principle. A field that satisfies the transfer principle for real numbers is called a real closed field, and nonstandard real analysis uses these fields as nonstandard models of the real numbers.
Robinson's original approach was based on these nonstandard models of the field of real numbers. His classic foundational book on the subject Nonstandard Analysis was published in 1966 and is still in print. On page 88, Robinson writes:Pedagogical
H. Jerome Keisler, David Tall, and other educators maintain that the use of infinitesimals is more intuitive and more easily grasped by students than the "epsilon–delta" approach to analytic concepts. This approach can sometimes provide easier proofs of results than the corresponding epsilon–delta formulation of the proof. Much of the simplification comes from applying very easy rules of nonstandard arithmetic, as follows:infinitesimal × finite = infinitesimalinfinitesimal + infinitesimal = infinitesimaltogether with the transfer principle mentioned below.
Another pedagogical application of nonstandard analysis is Edward Nelson's treatment of the theory of stochastic processes.Other applications
Other results were received along the line of reinterpreting or reproving previously known results. Of particular interest is Teturo Kamae's proof of the individual ergodic theorem or L. van den Dries and Alex Wilkie's treatment of Gromov's theorem on groups of polynomial growth. Nonstandard analysis was used by Larry Manevitz and Shmuel Weinberger to prove a result in algebraic topology.The real contributions of nonstandard analysis lie however in the concepts and theorems that utilize the new extended language of nonstandard set theory. Among the list of new applications in mathematics there are new approaches to probability,
hydrodynamics, measure theory, nonsmooth and harmonic analysis, etc.
There are also applications of nonstandard analysis to the theory of stochastic processes, particularly constructions of Brownian motion as random walks. Albeverio et al. have an excellent introduction to this area of research.Applications to calculus
As an application to mathematical education, H. Jerome Keisler wrote Elementary Calculus: An Infinitesimal Approach. Covering nonstandard calculus, it develops differential and integral calculus using the hyperreal numbers, which include infinitesimal elements. These applications of nonstandard analysis depend on the existence of the standard part of a finite hyperreal r. The standard part of r, denoted st(r), is a standard real number infinitely close to r. One of the visualization devices Keisler uses is that of an imaginary infinite-magnification microscope to distinguish points infinitely close together. Keisler's book is now out of print, but is freely available from his website; see references below.
no relation