Background Information
Jump to section:
 Calculus with Analytical Geometry
 Differential Equations with Laplace Transforms
 Partial Differential Equations & Boundry Value Problems
 Linear Algebra with Excel Applications
 Applied Statistics with Excel Examples
 Fourier Transforms with Engineering Applications
"Use It or Lose It"
Among engineers, the old saying “use it or lose it” aptly applies to the technical skills we learned in our undergraduate or graduate courses. An absence from the classroom of only a few years can significantly diminish our math and basic engineering skills.
Math as a Tool
As an engineer with over 25 years of university experience teaching both undergraduates and graduates, I have found that students who lack a basic understanding of engineering fundamentals, and equally importantly, of how to use "math as a tool," often encounter difficulties, not only as they pursue their engineering education, but also when they try to handle more technically advanced work assignments.
Goal
As an educator and an engineer, my goal for all the math courses is to instill in participants a working knowledge of how math is used as a tool for problem solving, from initial identification of assumptions through final interpretation of results. By taking these courses in a convenient twoday format, participants can quickly get back up on the "learning curve," without spending weeks or months in often unproductive selfstudy or retaking a formal undergraduate or graduate level math course.
Objective
The objective of these twoday refresher courses is to provide the opportunity for technical professionals to upgrade their basic engineering and math skills quickly and conveniently. The courses are intended for engineers and others who have a need or desire to enhance their technical ability and increase their professional competency. The courses have proven particularly helpful for those taking technical classes, pursuing either graduate or undergraduate studies, or desiring just a review or an introductory overview. Emphasis is placed on how to set up, analyze, and interpret engineering problems.
Review and Overview
Courses designated as "Review" are directed toward participants who have had some exposure to the subject, as would be the case for the Calculus or the Differential Equations courses. Courses designated as "Introductory Overview" are directed toward participants who have had no previous experience with the subject but still have a need to learn more about the topic and to be introduced to the topic in as nonthreatening a way as possible. This might be the case for the Statistics or the Partial Differential Equations courses.
Course Materials
Each participant receives a copy of the instructors course notes and an appropriate review text containing many examples and workedout problems.
Some WorkRelated Aspects of the Math Refresher Courses
 Engineers, particularly those given new assignments, are often asked to review technical reports or professional journal articles to become familiar with a technical project. Typically, this literature contains mathematical equations and derivations that the engineer needs to interpret in order to understand the relevance of the technical material presented.
 Like foreign languages, one’s skill with mathematics can diminish without consistent use. For sometimes years, many engineers are assigned tasks that do not require them to use the math skills they once had. As a result, if their jobs require them to “think mathematically,” they need to get back up on the “knowledge curve.” They could attempt to reacquire these skill on their own, but that could take weeks and still not provide the competency and insight needed. The goal of the math refresher short courses is to review and enhance these basics skills in the context of practical engineering examples and to accomplish that in a convenient twoday format.
 There is a great deal of emphasis today on the use of computers to help design and develop systems and components. However, much of what is being programmed is based on analytical analysis of fundamental laws and principles of engineering. Examples of the old adage “garbage in  garbage out” can easily occur if programmers do not have a good understanding of both the analytic methods and the physical laws they are asked to implement.
 When required to take specific technical training courses, many engineers find themselves overwhelmed by “the math” and do not derive full benefit from these courses as a result.
Calculus with Analytical Geometry
The objective of this course is to provide a review of calculus and analytical geometry. Emphasis is placed on how to use math as a tool to set up and interpret engineering problems.
This course is intended for those with a workrelated need, those taking technical training classes, pursuing either graduate or undergraduate studies, or desiring just a refresher.
Topics include:

 Analytica Geometry
 Differential Calculus
 Derivatives of Algebraic and Trigonometric Functions
 Extremes: Maximum & Minimum
 Related Rates Problems
 Integrals: Indefinite, Definite, and Numerical
 Integration Techniques: Substitution, byparts, partial fractions
 Topics of interest to the course participants
 Engineering Applications Include:
 Energy and Work
 Lengths, Surface Areas, and Volumes
 Moments of Intertia
 Maximum and Minimum
Differential Equations with Laplace Transforms
The objective of this course is to provide a review of differential equations with Laplace transforms. Emphasis is placed on how to use math as a tool to set up and interpret engineering problems.
This course is intended for those with a workrelated need, those taking technical training classes, pursuing either graduate or undergraduate studies, or desiring just a refresher.
Topics include:
 Linear First & Second Order Differential Equations
 Homogeneous and NonHomogeneous Equations
 Initial and Boundary Value Problems
 Systems of Ordinary Linear Differential Equations
 Laplace Transforms with Method of Residue
 State Variables Formulation of Simple Systems
 Engineering Applications Include:
 Electric Circuits,
 Mechanical Vibrations,
 Strength of Materials,
 Control Systems,
 Fluids Dynamics,
 Heat Transfer
Partial Differential Equations & Boundary Value Problems
The objective of this course is to provide an introductory overview of linear partial differential equations (PDE) as they apply in engineering. Basic physical laws are reviewed and applied to the derivation and interpretation of initial and boundaryvalue problems.
This course is intended for those with a workrelated need, those taking technical training classes, and those pursuing either graduate or undergraduate studies.
No previous experience with Partial Differential Equations is required.
Topics include:
 Review of Ordinary Differential Equations
 Boundary and Initial Value Problems
 Fourier Series and other Orthogonal Series
 Separation of Variables,
 Eigenfunction Expansions,
 Vector Analysis and Green’s Function,
 Differential Equations of Special Functions (Bessel, Legendra, etc.),
 Integral Transform Techniques: (Fourier, Bessel )
 Engineering Applications Include:
 Structural Vibrations,
 Modal Analysis,
 Heat Transfer,
 Fluids Dynamics,
 Sound Waves,
 Wave Guides
Topics of interest to the course participants.
Linear Algebra with Excel Applications
The objective of this course is to provide a basic introductory overview of linear algebra and matrix analysis techniques used in engineering. Basic concepts are developed and explored through examples and geometrical interpretations.
This course is intended for those with a workrelated need, those taking technical training classes, and those pursuing either graduate or undergraduate studies.
No previous experience with linear algebra nor matrices is required.
Topics include:
 Review of basic vector analysis and matrix algebra
 Systems of linear equations
 Square matrices and determines
 Inverse matrices techniques and features
 Orthogonal relations and inner product
 Diagonalization techniques
 Eeigenvalues and eigenvector problems
 Singular value decomposition (SVD)
 Principal component analysis (PCA)
 Engineering applications including Excel examples from:
 Engineering mechanics,
 Mechanical vibrations,
 Electric circuits,
 Data analysis ,
 Sensor calibration
Applied Statistics with Excel Examples
The objective of this course is to provide a basic introductory overview of statistics and probability as they apply in engineering. Basic concepts are developed and explored through simple examples and graphical illustrations. The course also acquaints participants with numerical techniques of statistical analysis using Excel examples.
This course is intended for those with a workrelated need, those taking technical training classes, and those pursuing either graduate or undergraduate studies.
No previous experience with probability or statistics is required.
Topics include:
 Frequency and Probability Distributions
 Means, Variances and Standard Deviations
 Combinational and Conditional Probabilities
 Discrete and Continuous Distribution Models:
e.g. Binomial, Normal, Triangular, Exponential, Weibull, et al  Sampling Techniques and Sample Size Considerations
 Confidence Levels and Intervals
 Probability Plots and Percentiles
 Curve Fitting of Data; GoodnessofFit
 Linear Regression and Correlation Analysis
 Excel Examples of Statistical Analysis
 Engineering Applications:
 Manufacturing Tolerances,
 System Reliability,
 Structural Failure and Fatigue,
 Signal Processing,
 Design of Experiments
Topics of interest to the course participants.
Fourier Transforms with Engineering Applications
The objective of this course is to provide an introductory overview of Fourier Transforms and their use in various engineering applications Basic concepts are developed and explored through simple analytical and graphical examples.
This course is intended for those with a workrelated need, those taking technical training classes, and those pursuing either graduate or undergraduate studies.
No previous experience with Fourier analysis is required.
Topics include:
 Fourier Series of Period Functions:
 Orthogonal functions and their selective analysis properties
 Evaluation of discrete Fourier coefficients and phase
 Time and spatial applications
 Fourier Transform of NonPeriodic Functions:
 Convolution Theorem
 Applications to frequency analysis
 Discrete Fourier Transforms:
 Digital sampling considerations
 Fast Fourier Transforms (FFT)
 Window functions: Flattop, Hanning
 Aliasing and leakage
 Engineering applications include:
 Heat transfer,
 Vibrational modes,
 Signal processing,
 Systems analysis
Topics of interest to the course participants.
Learning Options:
 Onsite tailored instruction: Twoday onsite classes
Course Materials: Each student receives:
 Each student receives a copy of the instructor's course notes.
 Review textbook with workedout examples.