måndag 11 november 2013

Standard Calculus as Ill-Posed Unstable Backward Magic

        Jacques Hadamard (1865-1963) was a gentle man with strong opinions on mathematics.

Previous posts on the Fundamental Theorem of Calculus have exposed two approaches to the connection between primitive function/integral $x(t)$, derivative $Dx = \frac{dx}{dt}$ and integrand $v(t)$ connected by the equations:
  1. $Dx(t) = v(t)$
  2. $x(t)=\int_0^t v(s)\, ds$.
In the standard approach as presented in e.g. the standard text book Calculus: A Complete Course by Adams and Essex, the integral $x(t)$ as an area under the graph $t\rightarrow v(t)$ is the primary given object and the proof of the Fundamental Theorem consists of showing that $x(t)$ satisfies the differential equation $Dx = v$. 

In the BodyandSoul approach the primary given object is the differential equation $Dx=v$ with $v$ given data and $x$ unknown to determine, and the proof of the Fundamental theorem consists of showing that this equation can be solved by time stepping producing the integral $x(t)$ as the solution. The process from input data $v(t)$ to output solution $x(t)$ by solving $Dx=v$ by time stepping, is well-posed or stable in the sense that small perturbations of data or solution process results in small perturbations of the solution $x(t)$.  

The mathematician Hadamard identified well-posedness and stability to be a necessary requirement in order for a mathematical problem to be meaningful, in the sense that a meaningful solution can be found. The process of integration from integrand $v(t)$ to integral $x(t)$ is well-posed and meaningful.

On the other hand, the process from integral/primitive function $x(t)$ to derivative $Dx(t)$, is ill-posed and unstable, in the sense that small perturbations in $x(t)$ may give rise to large perturbations in the derivative, because
  • $Dx(t)=\lim_{\Delta t\rightarrow 0}\frac{x(t+\Delta t)-x(t)}{\Delta t}$
and a small perturbation in $x(t+\Delta t)$ or $x(t)$ gets divided by the quantity $\Delta t$ tending to zero and thus gets amplified by the large factor $1/\Delta t$. The standard approach to the Fundamental Theorem puts the emphasis on the ill-posed or unstable process of differentiation. 

We sum up as follows: 
  1. The standard approach to the Fundamental Theorem is ill-posed, unstable and of questionable meaning. As illposed problem it rests on symbolic mathematics of infinite precision, which appears as magics.
  2. The approach in BodyandSoul is well-posed, stable and clearly meaningful. As well-posed problem it can be solved by numerical mathematics in finite precision, which is reasonable and not magics.
These aspects would be possible to discuss constructively with the man on the street, but may be very difficult to present to a teacher of standard Calculus for which Adams' book is the bible.

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