Math Background
I. Taylor Series Expansion
If a function, f(x), is analytic in some region (i.e. it has a derivative and is continuous) then the function can be expanded about a point x = a inside the region as
Where the expansion coefficients, bn, are found by the formula
The Taylor expansion is used in many areas of physics and engineering especially when one is trying to model a difficult non-linear problem (transistor operating curve, thermal expansion, etc) as a linear system in some limited range of operation.
II. Binomial Approximation
The Binomial Expansion is a powerful method for approximating small effects
in physics and engineering problems. It is extremely useful in both special
relativity and electromagnetism problems even when you have a calculator.
If you look in the Schaum's Math Handbook (Eq. 3.3 on page 5), you will see that the expansion of the nth power of (1+x) is given by
The Binomial approximation states that when x < 1
III. Differential Equations
Most physics and engineering problems involve solving equations with derivatives in them. These equations are called differential equations. Because there is no general solution trick for solving the wide range of different equations which physicists and engineers run into, students take several courses learning different tricks for solving these equations. Although you will not know all of these tricks off your head, you should know how to quickly solve some of the more important, simpler cases which form the basis for many of our models.
1. Linear Differential Equation – The power of any derivative is either first or zero.
Ex: “Linear DE”
Ex: “Non-Linear DE”
2. Constant Coefficient – The coefficients in front of each derivative is constant
Ex: “Constant Coefficient DE”
Ex: “Non-Constant Coefficient DE”
III. Special Constant Coefficient Linear Differential Equation
One of the most important differential equations in physics and engineering is the equation:
where is a constant.
Case I: “Linear Case”
You have previously seen this equation in freshman physics where the left-hand side of the equation is acceleration and the right-hand side is the net external force. This is Newton’s 1st Law.
“The solution is the straight line from College Algebra”
Case II: “Parabolic Case”
You have previously seen this equation in freshman physics for the simple un-damped harmonic oscillator. The general solution to this equation can be written in three different but equivalent ways:
a) “Cosine and Sine”
b) “Trig. Function & Phase Angle”
c) “Complex Exponential”
This equation models a bound system which oscillates at an angular frequency about some equilibrium point.
Case III: “Hyperbolic Case”
You should have previously dealt with this equation in freshman physics in capacitor and inductor circuits as well as in chemistry and biology. The general solution to this equation can be written in the following different ways:
a) “Hyperbolic Trig. Functions”
b) “Exponential Form”
where .
This equation models exponential decay or growth systems. These include problems involving compound interest in Engineering Economy and quantum tunneling in Modern Physics.
IV. Constant Coefficient Linear Differential Equations
The results in Section III are just special cases of a more general class of differential equations called constant coefficient differential equations. These equations can be solved by replacing the function with an exponential solution.
This is best demonstrated by example.
Example: Find the general solution to the following differential equation
We start by letting and calculating the required time derivatives
and .
We now substitute our derivative results into our differential equation.
Since this equation must be true for any value of x and not just for x = 0, we know that the term in the parenthesis must be zero.
Thus, we have replaced solving a differential equation for solving a polynomial equation in k. For this example, we can use the quadratic formula to solve for k.
Thus, the general solution to the problem is
You should have seen this equation in freshman physics for the damped harmonic oscillator (under-damped, critically damped, and over damped case).
V. Complex Numbers
A complex number can be represented by an ordered pair of numbers (a,b) where a represents the distance along the real axis and b represents the distance along the imaginary axis. It can also be seen as a vector of magnitude and an angle q.
The Cartesian representation of a complex number is best for adding and subtracting complex numbers.
Using our knowledge of trigonometry and vectors, we can rewrite the Cartesian form in terms of the magnitude of the vector and the angle.
The term in the parenthesis must represent the angle and have a length of one since the magnitude of z is . Using the Taylor series expansions for sine and cosine, you can show that this is the complex exponential function.
This is called the Euler representation of a complex number and is superior when multiplying or dividing complex numbers.