1. THE MATHEMATICS ENGINEERS USE EVERYDAY IN INDUSTRY W.G. STEENKEN GE AVIATION OHIO MATHEMATICS AND SCIENCE COALITION NOVEMBER 19, 2015

2. BACKGROUND Bachelors, Masters, and Ph.D. in Mechanical Engineering Mathematics Courses –High School Algebra I and II Geometry Trigonometry Solid Geometry –Undergraduate Calculus, I, II, III Advanced Mathematics for Engineers Introduction to Mathematical Statistics Mathematics of Physics and Modern Engineering –Graduate Advanced Calculus Introduction to Probability Theory and Its Applications Quantum Chemistry and Statistical Mechanics –On Job Training Maximum Value Statistics Measurement and Analysis of Random Data

3. BACKGROUND (Concluded Fifty Year Career in Engineering (Aviation and Propulsion) –General Dynamics, Fort Worth, Texas (6 years) –GE Aviation, Cincinnati, Ohio (44 years) Lived on a Farm for 40 Years

4. MATHEMATICAL TOPICS GEOMETRY –Congruence and Similarity (Dependent on Ratio) –Proofs (Triangles, Parallel and Perpendicular Lines) –Volumes for Cylinders, Pyramids, Cones and Spheres –Intro to Right Triangles Trigonometry Bill X X X X Others % 68 26 63 74

5. MATHEMATICAL TOPICS Algebra II –Expressions, Equations, and Inequalities –Ratios, Proportions, Percents –Functions, Equations, and Graphs –Linear Systems –Quadratic Functions and Equations –Polynomials and Polynomial Functions –Radical Functions and Rational Exponents –Exponential and Logarithmic Functions –Rational functions –Sequences and Series –Quadratic Relations and Conic Sections –Probability and Statistics –Matrices –Periodic Functions and Trigonometry –Trigonometric Identities and Equations Calculus –Differential Calculus –Conic Sections –Maxima, Minima, and Inflections –Infinite Series –Differential Equations –Integral Calculus Bill X X X X X X X X X X Others % 95 100 74 89 84 37 68 21 37 16 79 42 32 53 X X X 0 37 11 37 42

6. MATHEMATICAL TOPICS (Concluded) Other Topics –Vector Algebra –Matrices / Matrix Algebra –Vector Field Theory –Partial Differential Equations –Complex Variables –Elementary Topology –Transformations Statistics –Mean –Standard Deviation –Amplitude Probability Density Function –Correlation Functions –Power Spectral Density Functions Bill Others % 37 26 5 21 11 5 32 X X X X X 84 95 32 53 42

7. MY THESIS OF MAJOR IMPORTANCE –Expressions, Equations, and Inequalities –Ratios, Proportions, Percents –Linear Systems BUT USAGE WAS MUCH BROADER (  74%) –Intro to Right Triangles Trigonometry –Expressions, Equations, and Inequalities –Ratios, Proportions, and Percents –Functions, Equations, and Graphs –Linear Systems –Quadratic Functions and Equations –Polynomials and Polynomial Functions –Probability and Statistics –Statistics (Mean and Standard Deviation) 12/14/2015Engine Operability7

8. SUMMARY Use of broad spectrum of mathematics strands is greater than I thought As a mathematics educator said to me, “To use mathematics well, you need to have a wide knowledge of the mathematical strands” Perhaps this statement is best supported by some recent examples from my consulting work 12/14/2015Engine Operability8

9. EXAMPLES OF ENGINEERING MATHEMATICS

10. (1 + x) -n = 1 + nx + + n(n+1)x 2 2! PROBLEM What is the value of 1/0.98 ~1.01 Remember the series Thus Actual answer 1/0.98 = = 1 + (.02) = 1.01 1 (1-.02) 1/2 1212 ~~ ~ 1/0.98 = 1.015 (% Error of estimate = -0.015)

11. INTERPOLATION P2P2 B P1P1 PxPx X1X1 X2X2 X

12. PROBLEM INVOLVING LOTS OF DATA