## torsdag 26 maj 2016

### Fatal Attraction of Fundamental Theorem of Calculus?

Calculus books proudly present the Fundamental Theorem of Calculus as the trick of computing an integral
• I=$\int_a^b f(x)dx$,
not by tedious summation of little pieces as a Riemann sum
• $\sum_i f(x_i)h_i$
on a partition $\{x_i\}$ of the interval $(a,b)$ with step size $h_i = x_{i+1} - x_i$, but by the formula
• $I = F(b) - F(a)$,
where $F(x)$ is a primitive function to $f(x)$ satisfying $\frac{dF}{dx} = f$,

The trick is thus to compute an integral, which by construction is a sum of very many terms, not by doing the summation following the construction, but instead taking just one big leap using a primitive function.

On the other hand, to compute a derivative no trick is needed according to the book; you just compute the derivative using simple rules and a catalog of already computed derivatives.

In a world of analytical mathematics, computing integrals is thus valued higher than computing derivatives, and this is therefore what fills Calculus books.

In a world of computational mathematics, the roles are switched. To compute an integral as a sum can be viewed to be computationally trivial, while computing a derivative $\frac{dF}{dx}$ is a bit more tricky because it involves dividing increments $dF$ by small increments $dx$.

This connects to Poisson's equation $\Delta\phi =\rho$ of Newton's theory of gravitation discussed in recent posts. What is here to be viewed as given and what is derived? The standard view is that the mass density $\rho$ is given and the gravitational potential $\phi$ is derived from $\rho$ as an integral
• $\phi (x) = \frac{1}{4\pi}\int\frac{\rho (y)}{\vert x-y\vert}dy$,
seemingly by instant action at distance.

In alternative Newtonian gravitation, as discussed in recent posts, we view instead $\phi$ as primordial and $\rho =\Delta\phi$ as being derived by differentiation, with the advantage of requiring only local action.

We thus have two opposing views:
• putting together = integration requiring (instant) action at distance with dull tool.
• splitting apart = differentiation involving local action with sharp tool.
It is not clear what to prefer?