• This section has been greatly enhanced, and tailored to meet our engineering requirements.
• The section outlined here is not intended to teach the elements of mathematics, but it is designed to be a quick reference guide to support the engineer required to use techniques that may not have been used recently.
• For those planning to write the first ABET Fundamentals of Engineering exam, the following topics are commonly on the exam.
- quadratic equation
- straight line equations - slop and perpendicular
- conics, circles, ellipses, etc.
- matrices, determinants, adjoint, inverse, cofactors, multiplication
- limits, L’Hospital’s rule, small angle approximation
- integration of areas
- complex numbers, polar form, conjugate, addition of polar forms
- maxima, minima and inflection points
- first-order differential equations - guessing and separation
- second-order differential equation - linear, homogeneous, non-homogeneous, second-order
- triangles, sine, cosine, etc.
- integration - by parts and separation
- solving equations using inverse matrices, Cramer’s rule, substitution
- eigenvalues, eigenvectors
- dot and cross products, areas of parallelograms, angles and triple product
- divergence and curl - solenoidal and conservative fields
- integration of volumes
- integration using Laplace transforms
- probability - permutations and combinations
- mean, standard deviation, mode, etc.
- log properties
- taylor series
- partial fractions
- basic coordinate transformations - cartesian, cylindrical, spherical
- trig identities
- derivative - basics, natural log, small angles approx., chain rule, partial fractions
35.1.1 Constants and Other Stuff
• A good place to start a short list of mathematical relationships is with greek letters
Figure 35.1 The greek alphabet
• The constants listed are amount some of the main ones, other values can be derived through calculation using modern calculators or computers. The values are typically given with more than 15 places of accuracy so that they can be used for double precision calculations.
Figure 35.2 Some universal constants
35.1.2 Basic Operations
• These operations are generally universal, and are described in sufficient detail for our use.
• Basic properties include,
Figure 35.3 Basic algebra properties
22.214.171.124 - Factorial
• A compact representation of a series of increasing multiples.
Figure 35.4 The basic factorial operator
35.1.3 Exponents and Logarithms
• The basic properties of exponents are so important they demand some sort of mention
Figure 35.5 Properties of exponents
• Logarithms also have a few basic properties of use,
Figure 35.6 Definitions of logarithms
• All logarithms observe a basic set of rules for their application,
Figure 35.7 Properties of logarithms
35.1.4 Polynomial Expansions
• Binomial expansion for polynomials,
Figure 35.8 A general expansion of a polynomial
35.1.5 Practice Problems
1. Are the following expressions equivalent?
2. Simplify the following expressions.
3. Simplify the following expressions.
4. Simplify the following expressions.
5. Rearrange the following equation so that only ‘y’ is on the left hand side.
6. Find the limits below.