Fundamentals of Mathematical Treatments

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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