- •1. TABLE OF CONTENTS
- •2. MATHEMATICAL TOOLS
- •2.1 INTRODUCTION
- •2.1.1 Constants and Other Stuff
- •2.1.2 Basic Operations
- •2.1.2.1 - Factorial
- •2.1.3 Exponents and Logarithms
- •2.1.4 Polynomial Expansions
- •2.2 FUNCTIONS
- •2.2.1 Discrete and Continuous Probability Distributions
- •2.2.2 Basic Polynomials
- •2.2.3 Partial Fractions
- •2.2.4 Summation and Series
- •2.3 SPATIAL RELATIONSHIPS
- •2.3.1 Trigonometry
- •2.3.2 Hyperbolic Functions
- •2.3.2.1 - Practice Problems
- •2.3.3 Geometry
- •2.3.4 Planes, Lines, etc.
- •2.4 COORDINATE SYSTEMS
- •2.4.1 Complex Numbers
- •2.4.2 Cylindrical Coordinates
- •2.4.3 Spherical Coordinates
- •2.5 MATRICES AND VECTORS
- •2.5.1 Vectors
- •2.5.2 Dot (Scalar) Product
- •2.5.3 Cross Product
- •2.5.4 Triple Product
- •2.5.5 Matrices
- •2.5.6 Solving Linear Equations with Matrices
- •2.5.7 Practice Problems
- •2.6 CALCULUS
- •2.6.1 Single Variable Functions
- •2.6.1.1 - Differentiation
- •2.6.1.2 - Integration
- •2.6.2 Vector Calculus
- •2.6.3 Differential Equations
- •2.6.3.1 - First Order Differential Equations
- •2.6.3.1.1 - Guessing
- •2.6.3.1.2 - Separable Equations
- •2.6.3.1.3 - Homogeneous Equations and Substitution
- •2.6.3.2 - Second Order Differential Equations
- •2.6.3.2.1 - Linear Homogeneous
- •2.6.3.2.2 - Nonhomogeneous Linear Equations
- •2.6.3.3 - Higher Order Differential Equations
- •2.6.3.4 - Partial Differential Equations
- •2.6.4 Other Calculus Stuff
- •2.7 NUMERICAL METHODS
- •2.7.1 Approximation of Integrals and Derivatives from Sampled Data
- •2.7.2 Euler First Order Integration
- •2.7.3 Taylor Series Integration
- •2.7.4 Runge-Kutta Integration
- •2.7.5 Newton-Raphson to Find Roots
- •2.8 LAPLACE TRANSFORMS
- •2.8.1 Laplace Transform Tables
- •2.9 z-TRANSFORMS
- •2.10 FOURIER SERIES
- •2.11 TOPICS NOT COVERED (YET)
- •2.12 REFERENCES/BIBLIOGRAPHY
- •3. WRITING REPORTS
- •3.1 WHY WRITE REPORTS?
- •3.2 THE TECHNICAL DEPTH OF THE REPORT
- •3.3 TYPES OF REPORTS
- •3.3.1 Laboratory
- •3.3.1.1 - An Example First Draft of a Report
- •3.3.1.2 - An Example Final Draft of a Report
- •3.3.2 Research
- •3.3.3 Project
- •3.3.4 Executive
- •3.3.5 Consulting
- •3.3.6 Interim
- •3.4 ELEMENTS
- •3.4.1 Figures
- •3.4.2 Tables
- •3.4.3 Equations
- •3.4.4 Experimental Data
- •3.4.5 References
- •3.4.6 Acknowledgments
- •3.4.7 Appendices
- •3.5 GENERAL FORMATTING
- •Title: High Tech Presentations The Easy Way
- •1.0 PRESENTATIONS IN GENERAL
- •2.0 GOOD PRESENTATION TECHNIQUES
- •2.1 VISUALS
- •2.2 SPEAKING TIPS
- •3.0 PRESENTATION TECHNOLOGY
- •3.1 COMMON HARDWARE/SOFTWARE
- •3.2 PRESENTING WITH TECHNOLOGY
- •X.0 EXAMPLES OF PRESENTATIONS
- •4.0 OTHER TECHNOLOGY ISSUES
- •4.1 NETWORKS
- •4.1.1 Computer Addresses
- •4.1.2 NETWORK TYPES
- •4.1.2.1 Permanent Wires
- •4.1.2.2 Phone Lines
- •4.1.3 NETWORK PROTOCOLS
- •4.1.3.1 FTP - File Transfer Protocol
- •4.1.3.2 HTTP - Hypertext Transfer Protocol
- •4.1.3.3 Novell
- •4.1.4 DATA FORMATS
- •4.1.4.1 HTML - Hyper Text Markup Language
- •4.1.4.1.1 Publishing Web Pages
- •4.1.4.2 URLs
- •4.1.4.3 Hints
- •4.1.4.4 Specialized Editors
- •4.1.4.6 Compression
- •4.1.4.7 Java
- •4.1.4.8 Javascript
- •4.1.4.9 ActiveX
- •4.1.4.10 Graphics
- •4.1.4.11 Animation
- •4.1.4.12 Video
- •4.1.4.13 Sounds
- •4.1.4.14 Other Program Files
- •4.2 PULLING ALL THE PROTOCOLS AND FORMATS TOGETHER WITH BROWSWERS
- •REFERENCES
- •AA:1. ENGINEERING JOKES
- •AA:1.1 AN ENGINEER, A LAWYER AND A.....
- •AA:1.2 GEEKY REFERENCES
- •AA:1.3 QUIPS
- •AA:1.4 ACADEMIA
- •AA:1.4.1 Other Disciplines
- •AA:1.4.2 Faculty
- •AA:1.4.3 Students
- •AA:1.5 COMPUTERS
- •AA:1.5.1 Bill
- •AA:1.5.2 Internet
- •AA:1.6 OTHER STUFF
- •2. PUZZLES
- •2.1 MATH
- •2.2 STRATEGY
- •2.3 GEOMETRY
- •2.4 PLANNING/DESIGN
- •2.5 REFERENCES
- •3. ATOMIC MATERIAL DATA
- •4. MECHANICAL MATERIAL PROPERTIES
- •4.1 FORMULA SHEET
- •5. UNITS AND CONVERSIONS
- •5.1 HOW TO USE UNITS
- •5.2 HOW TO USE SI UNITS
- •5.3 THE TABLE
- •5.4 ASCII, HEX, BINARY CONVERSION
- •5.5 G-CODES
- •6. COMBINED GLOSSARY OF TERMS
page 10
2.1.4 Polynomial Expansions
• Binomial expansion for polynomials,
(a + x ) = a |
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+ na |
n – 1 |
x + n-------------------(n – 1 )a |
n – 2 |
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+ … + x |
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2.2 FUNCTIONS
2.2.1 Discrete and Continuous Probability Distributions
• The Binomial distribution is,
P(m ) = ∑ |
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q,p [0,1 ] |
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• The Poisson distribution is, |
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t ≤ m |
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• The Hypergeometric distribution is, |
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• The Normal distribution is, |
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page 11
2.2.2 Basic Polynomials
• The quadratic equation appears in almost every engineering discipline, therefore is of great importance.
ax |
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+ bx + c = 0 = a(x – r1 )(x – r2 ) |
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– 4ac |
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• Cubic equations also appear on a regular basic, and as a result should also be considered.
x3 + ax2 + bx + c = 0 = (x – r1 )(x – r2 )(x – r3 )
First, calculate, |
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Q = |
3-----------------b – a2 |
R = |
9--------------------------------------ab – 27c – 2a3 |
S = 3 R + Q3 + R2 T = 3 R – Q3 + R2 |
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Then the roots,
a r1 = S + T – -- 3
r2 = |
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– a-- |
+ j--------3(S – T ) |
r3 = |
S------------+ T |
– a-- |
– j--------3(S – T ) |
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• On a few occasions a quartic equation will also have to be solved. This can be done by first reducing the equation to a quadratic,
x4 + ax3 + bx2 + cx + d = 0 = (x – r1 )(x – r2 )(x – r3 )(x – r4 )
First, solve the equation below to get a real root (call it ‘y’),
y3 – by2 + (ac – 4d )y + (4bd – c2 – a2d ) = 0
Next, find the roots of the 2 equations below,
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a2 – 4b + 4y |
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y2 – 4d |
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z + ------------------------------ |
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r3,r4 = z |
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page 12
2.2.3 Partial Fractions
• The next is a flowchart for partial fraction expansions.
start with a function that has a polynomial numerator and denominator
is the order of the
numerator >= yes denominator?
no
Find roots of the denominator and break the equation into partial fraction form with unknown values
use long division to reduce the order of the numerator
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Done
• The partial fraction expansion for,
page 13
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• Consider the example below where the order of the numerator is larger than the denominator.
( ) 5s3 + 3s2 + 8s + 6 x s = -------------------------------------------
s2 + 4
This cannot be solved using partial fractions because the numerator is 3rd order and the denominator is only 2nd order. Therefore long division can be used to reduce the order of the equation.
5s + 3
s2 + 4 5s3 + 3s2 + 8s + 6 5s3 + 20s
3s2 – 12s + 6
3s2 + 12
– 12s – 6
This can now be used to write a new function that has a reduced portion that can be solved with partial fractions.
x(s ) = 5s + 3 + |
–---------------------12s – 6 |
solve |
–---------------------12s – 6 |
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+ |
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• When the order of the denominator terms is greater than 1 it requires an expanded partial fraction form, as shown below.