fredag 22 november 2013

BodyandSoul vs Standard Calculus 2

Einstein presenting his equation of general relativity $R_{ik}=0$ supposedly(?) describing the world. Simple and general. 

A Standard Calculus text book, like Calculus: A Complete Course by Adams and Essex, is filled with symbolic formulas covering more than 1000 small print pages and is difficult for the student to digest and heavy to carry along.

The objective of a Standard Calculus text book appears to be to convince the student of the usefulness of Calculus through mass demonstration by presenting so many specific problems, which can be solved with pen and paper using Symbolic Calculus, that there can be only a few left which cannot be solved this way. In short, the objective is to show that Symbolic Calculus works by presenting very many specific examples. But the massiveness is misleading since in fact very few problems can be solved symbolically with pen and paper.

The essence of the BodyandSoul approach as Constructive Calculus is the opposite: Instead of many specific problems solved by symbolic mathematics with pen and paper, one general problem containing all the specific problems of Standard Calculus and many more, is considered. The essence of the theory is then to show how and why any given instance of the general problem can be solved by the computer, as expressed in a Fundamental Theorem of Calculus.

The one general problem of Constructive Calculus, in one variable to start with, is the Initial Value Problem (IVP): Construct a function $u(t)$ of time $t$ such that
  • $\frac{du}{dt} = f$ for $t > 0$ with $u(0)=u_0$,         (*) 
where $f=f(u,t)$ is a given function of $u$ and $t$ and $u_0$ a given initial value, by successive time stepping according to
  • $u(t+\Delta t)$ = $u(t) + f(u(t),t)\Delta t$ with $u(0)=u_0$, 
with $\Delta t >0$ small. This is formally a finite time step version of $du = fdt$ or $\frac{du}{dt}=f$ with vanishingly small time step $dt$, and $du \approx u(t+\Delta t) - u(t) =f\Delta t\approx fdt$.

If $f$ depends only on $t$, the solution $u(t)$ is the integral
  • $\int_0^t f(s)\, ds + x_0$.
If $f=u$ and $u_0=1$, the solution is $u(t)=\exp(t)$. 

More generally, with simple dependencies of $u$ and $t$ all elementary functions (exponential, trigonometric, Bessel, ...) are constructed this way and their properties follow from the specifics of the IPV they solve.Calculus in one variable can thus be reduced to a study of the IVP (*). 

Similarly, Calculus in several variables can essentially be reduced to an IVP of a generalization of (*) with $u=u(t,x)$ and $x$ a multi-dimensional space coordinate and $f$ depending on partial derivatives of $u$ with respect to space coordinates, which is solved by time stepping after finite element discretization in space. 

Constructive Calculus can thus be summarized as $\frac{du}{dt} = f$ solved by time stepping $du=fdt$. Constructive Calculus combines simplicity with generality, which is a prime goal of (computer) science and mathematics, to be compared with the difficulty of all the specific cases of Symbolic Calculus.

PS We may summarize as follows:
  • Constructive Caculus is simple and general.
  • Symbolic Calculus is difficult and special.

                                     Mass demonstration for (or against?) Symbolic Calculus

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