Fundamentals of Analysis in Physics

Fundamentals of Mathematical Treatments

Author(s): Masatoshi Kajita

Pp: 1-20 (20)

DOI: 10.2174/9789815049107122010002

* (Excluding Mailing and Handling)

Abstract

Physical phenomena can be understood by solving equations that lead to
physical laws. The first objective of this chapter is to solve certain equations that are
required for physical analysis. First, an iterative solution of the equation is introduced.
Using this approach, the numerical solution of an equation f(x) = 0 can be obtained
also when the function f(x) is too complicated for the solution to be obtained as an
explicit formula.
Many physical equations can be expressed using differential and integral
mathematical representations, which might not be familiar to all college students. The
fundamental concepts of the differential and integral were introduced. Several
fundamental mathematical formulae are reviewed.
The second objective is to solve the differential equations that are required for the
physical analysis. First, some solutions of simple differential equations given by
explicit formulas are introduced, which are important for their physical interpretation.
However, the equations for technical use are generally too complicated. Several
methods for obtaining numerical solutions are introduced, which are useful for
analyzing motion orbits. There is also a phenomenon that cannot be predicted by
solving equations, which is called “chaos.”
Finally, the fundamentals of the eigenvalues of matrices are introduced, which are
important for understanding quantum mechanics. This chapter was prepared for
undergraduate students who are not familiar with differential and integral calculus and
matrices.


Keywords: Iterative solution, Differential, Partial derivative, Integral, Taylor expansion, Euler method, Middle point method, Runge-Kutta method, Chaos, Lyapunov exponent, matrix, Determinant, Eigen value, Eigen vector.

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