Appendix D: Mathematical Toolbox

Logarithms

Base 10 (Common logaritm)

If y = 10x, x is called the logarithm of y to the base 10. This is written log y = x.

Properties of logs:

(D-1): log ab = log a + log b
(D-2): log a/b = log a - log b
(D-3): log an = n log a, where n can be any number, positive, negative, or zero, or fractional.

NOTE: log (a+b) is not equal to log a + log b

Base e (Natural, or Naperian, logarithms)

If y = ex, x is called the natural log of y. This is written x = ln y.

e has the approximate value 2.71828...

(D-4): ln y = 2.303 log y

Exponents

Properties of exponents

(D-5): xn+m = xnxm
(D-6): xn-m = xn/xm
(D-7): (xn)m = xnm

NOTE: (a+b)n is not equal to an + bn

Trigonometric Functions

These are defined in terms of the right triangle in Figure D-1.

(D-8): Sin q = y/r = y/(x2+y2)1/2
(D-9): Cos q = x/r = x/(x2+y2)1/2
(D-10): Tan q = x/y

Some trigonometric identities and other useful facts:

(D-11): sin2x + cos2x = 1
(D-12): sin 2x = 2 (sin x)(cos x)
(D-13): cos 2x = cos2x - sin2x
(D-14): e2pi = cos 2p + i sin 2p
(D-15): sin x » x when x is very small
(D-16): For a triangle of sides a,b,c and angles a (between b,c), ß between a,c), and g (between a,b), a/sin a = b/sin ß = c/sin g

Quadratic Equations

General form: ax2 + bx + c = 0

This is called a second-order polynomial. There are 2 values of x (roots) that will satisfy the equation, given by the quadratic formula:

(D-17): x = [-b ± (b2-4ac)1/2]/2a

An equation of form axn + bxn-1 + ... + z = 0 is called an nth order polynomial. There are n values of x (roots) that will satisfy such an equation. For n > 3, such equations are usually solved graphically, or by an iterative numerical procedure called the Newton-Raphson method.

Basic Calculus

Suppose the value of the dependent variable, y, is determined by the value of the independent variable, x, according to some expression. Then we say that "y is a function of x." This is written y = f(x). See Figure D-2.

Derivative

Construct a Cartesian plot of the function, y = f(x). The derivative of y with respect to x, written dy/dx, at some particular value of x = xo, is the slope of the plot at xo.

Integral

The integral of f(x) over the range x1 to x2, written òf(x)dx, is the area under the curve between x1 and x2.

Some Useful Relationships

(D-18): Circumference of a circle = pd = 2pr
(D-19): Area of a circle = pr2
(D-20): Area of a triangle = (base)(height)/2
(D-21): Surface area of a sphere = 4pr2
(D-22): Volume of a cylinder = pr2h
(D-23): Volume of a sphere = 4pr3/3

Physical Laws

(D-24): Newton's Second Law: F = ma (force = mass * acceleration)
(D-25): Work: w = F·d (work = force * distance)
(D-26): Kinetic Energy: KE = mv2/2
(D-27): Momentum: p = mv (momentum = mass * velocity)
(D-28): Universal Law of Gravitation: F = Gm1m2/r2 (two bodies are attracted by a force proportional to the product of their masses and to the inverse square of the distance between them).

(D-29): Coulomb's Law: F = q1q2/4peor2 (two charges experience an electrical force between them proportional to the product of their charges and to the inverse square of the distance between them).

For two electrons, Felectrical = 1042 Fgrav (i.e., the Coulomb force is much stronger than the gravitational force, all other things being equal).

Important Units:

mass: g or kg
distance: cm or m
velocity: cm/s or m/s
acceleration: cm/s2 or m/s2
 force: dyne (1 dyne = 1 g-cm/s2)
newton (1 newton = 1 kg-m/s2)
energy (work): erg (1 erg = 1 g-cm2/s2)
joule (1 joule = 1 kg-m2/s2)