Mathematics Beyond the Calculus

This text is about solving various types of equations using practical mathematical methods. Only the essentials of each topic are discussed.

This is not about proving theorems, taking limits, or other matters important to mathematicians.

"However, the emphasis should be somewhat more on how to do the mathematics quickly and easily, and what formulas are true, rather than the mathematicians' interest in methods of rigorous proof." Richard Feynman

Concepts from Linear Algebra - the determinant, the finite matrix, the eigenvalue - are presented without the distractions of mathematical rigor. You learn solution methods that do not involve guesses. Methods you implement in a straightforward manner.

The operational calculus can be traced back to Oliver Heaviside. Though many scientists preceded Heaviside in introducing operational methods, the systematic use of operational methods in physical problems was stimulated only by Heaviside's work. The methods he created are undoubtedly among the most important ever created.

Heaviside was criticized for his lack of mathematical rigor. Yet his numerous mathematical and physical methods and results proved to be correct when mathematical rigor was incorporated.

The Laplace Transform, a basis for a modern day operational calculus, is a straightforward technique for solving ordinary, partial differential, and, with a few complications, difference equations and a type of integral equation. On the other hand the Z transform solves difference equations without complications. And, Heaviside's differential operator D = d/dt augments the transform methods.

The Laplace Transform transforms equations in one real variable domain, usually the time t domain, to a complex variable domain where the problem at hand is essentially solved. The inverse transform from the complex variable domain to the real variable domain completes the solution. Understanding the inverse transform requires knowledge of the theory of functions of complex variables.

Our main interest in functions of a complex variable is integration, because integration of the complicated integrals of inverse transforms is amazingly simplified.

The methods of the differential and integral calculus are extended to complex numbers and functions of complex variables. The results produce tremendous analytic methods.

We show how ordinary differential equations. systems of ordinary differential equations, partial differential equations, and difference equations are readily solved by transform and/or differential operational methods. We show that each type of equation is solved in essentially the same way.

We just define the Fourier Series, and show how to create Fourier series representing waveforms.

Integral Equations - This is a hugh subject, which we limit to how the Laplace transform solves integral equations that include the convolution integral.

Galois Finite Fields GF(2m) are not used to solve equations per se. They are used to implement functions such as error correcting codes, speech recognition, phase array antennas, and Doppler radar. Functions NOT implemented here.