## tisdag 23 februari 2021

### Engineering Calculus 2100

Engineering mathematics education (at e g Chalmers University of Technology) has not changed much since I was a student in the 1960's, more than 50 years ago. It is still a combination of Classical Calculus of one and several variables + Linear Algebra + some "numerics" (see curriculum).

Classical Calculus is Symbolic Calculus with the objective of expressing solutions to certain basic model problems of mathematical modeling, mainly differential equations such as Laplace equation in a square, in terms of Fourier series, or expressing primitive functions of elementary functions in terms of elementary functions denoted by symbols like exp(x), sin(x) or J_n(z). The essence is symbolic computation as a play with symbols using pen and paper according to certain rules defined in books of Calculus. This formed the expertise of the traditional engineer before the computer, and with the computer has been automated in systems like Maple and Mathematica.

Symbolic solution worked fine for simple model problems but for real problems the engineer had to resort to "numerics" replacing symbol manipulation with "brute" digital computation. With the power of the modern computer "numerics" has developed into a form of Digital Calculus filling the symbols of Symbolic Calculus with concrete meaning by "discretisation" and allowing virtually any system of differential equations in any geometry to solved. The discretisation aspect of Digital Calculus is automated in FEniCS, which upon input of a system of differential equations with coefficients and geometry automatically discretises the differential equations into a systems of algebraic equations which are automatically solved by digital computation. The World can thus be simulated by specifying models using symbols and letting the computer bring out the expression of the symbols, like a cell bringing genetic code into physical expression. This is a revolution.

With FEniCS the expert engineer can focus on the high level task of formulating mathematical models of the World in symbolic form, and can leave the "low-level" task of discretisation and number crunching to the computer.

This brings revolutionary changes to engineering mathematics education, which comes to expression in the DigiMat Program documented at DigiMat Encylopedia:

The scope changes from

• Classical Calculus: simple model problems + "some tricks of numerics",

into

• Digital Calculus: general problems + powerful automated numerics = Engineering Calculus 2100.
Classical Calculus was initiated by Leibniz and Newton in the late 17th century and then perfected by a mathematicians like Euler, Laplace and Cauchy, who reformed engineering mathematics education at Ecole Polytechniques in Paris in the 1820s (Cours d'analyse) into what is still world standard today.

Digital Calculus brings revolutionary changes to engineering mathematics education, the first since Cauchy. Standard books do not bring the new essentials. Stay tuned!

PS Concerning the term "numerics" as opposed to pure mathematics as symbolic mathematics, it is natural to compare with the subdivision of fluid mechanics into theoretical/pure fluid mechanics and "hydraulics", with the distinction well expressed by Nobel Laureate Hinshelwood.