What maths is needed for Engineering and Science courses at Uni?

1. What maths is needed for Engineering and Science courses at Uni? • where the High School topics lead to in first yearMary Coupland, University of Technology SydneyMary.Coupland@uts.edu.au Linking Engineers and Scientists with Teachers, UTS 29-30 Mary 2014

2. Importance of mathematics and science to engineers • Tools for problem solving • Means of professional communication • Means of defining/explaining/ understanding engineering concepts • Used to model a system before building it • Required for professional accreditation

3. More immediately… Mathematics is used in Engineering and Science subjects to teach engineering and scientific concepts

4. Mathematics beyond high school Algebra and calculus are developed into • Linear algebra, including 3D geometry • Analysis • Differential equations • Multivariable calculus and vector calculus • Numerical analysis • Complex analysis

5. Choosing one of these:Differential equations • From high school:

6. Many situations in engineering are modelled by second order D.E.s • Simple one: y’’ = -w2y leading to SHM (then add friction). Here, y might be displacement or an electric current. • We get equations like this: • We solve them by assuming a solution of the form . Then = and . • We get , reduces to , called the characteristic equation.

7. Second order DE’s The theory of quadratic equations is then useful. The different kinds of roots of the characteristic equation determine the kind of solution, y as a function of x, of the differential equation.

8. Repeated real roots of the characteristic equation correspond to critical damping which stops any oscillations at all, in the shortest time. Distinct real roots correspond to an overdamped system, which takes longer to come to equilibrium.

9. Comparison of overdamping (purple) with critical damping (blue).

10. Overdamping (purple) Critical damping (blue)

11. Complex roots of the characteristic equation correspond to weakly damped oscillations. This is .

12. Note the mathematical complexity… • To solve equations like this: • We assume a solution of the form . • We get , reduces to , called the characteristic equation. The nature of the roots of this quadratic equation determine the kinds of solution of the original DE.

13. The same second order DE applies to mechanical vibrations and RLC electric circuits.http://en.wikipedia.org/wiki/File:RLC_series_circuit.png

14. First order D.E.s • Applications occur in electrical systems • Other applications occur in mixing problems used to model the environment, e.g. pollution in a lake/ river system, mixing of air in an air conditioning system,spreading of heat through insulation.

15. Linear algebra • Initial idea: how to solve systems of linear equations • Simple example: • Once written in matrix form, AX=B, the same method can be applied to larger systems

16. A system of equations can be written in matrix form:

17. A system of equations can be written in matrix form, AX=B:

18. In matrix notation we solve by using the inverse matrix: AX=B A-1AX=A-1B X=A-1B

19. Example from structural engineering We need the idea of resolving forces into horizontal and vertical components F 

20. Example from structural engineering Resolving forces into components |F|sin F |F|cos 

21. Diagram of a roof trusswww.onlinetips.org/images/roof-truss.gif

22. Examples from www.diydoctor.org.uk/.../Trusses%201.jpg

23. Example from the University of British Columbia Website:http://commons.bcit.ca/math/examples/

25. The equations for each joint:

26. Writing all the equations in matrix form:

27. Linear algebra • Theoretical questions arise – • Can we solve any system? • How many solutions are there? • What happens when we have more equations than unknowns? • What happens when we have more unknowns than equations ?

28. Analysis A first year uni topic is Taylor Series, the representation of functions by infinite series. For example, the geometric series

29. Analysis A first year uni topic is Taylor Series, the representation of functions by infinite series. For example, the geometric series when

30. Analysis • We can turn this around and say that the function when

31. Analysis (continued) We can show that infinite series can be found for other functions.

32. Analysis (continued) • We need to know when these series converge. (They are not geometric series) • We use power series like these to solve harder differential equations (equations involving derivatives) • We also use power series as a tool for situations where it is useful to replace a function by its series in order to do further calculations

33. Analysis (continued) Power series are used to derive the error bounds for various methods in numerical integration. Error estimate for Simpson’s Rule The error is less than or equal to where M is a constant. M is chosen such that the fourth derivative of f (x) satisfies the inequality for all x in the interval [a,b].

34. Multi-variable calculus • At school: functions where y depends on x. • We extend this to functions where z depends on x and y. • Then extend further to functions of many variables. • In these cases we use partial differentiation to describe rates of change.

35. Numerical analysis • Numerical methods are used to solve equations (for example Newton’s method). • Further questions: How accurate is it? Will it always work? • Other numerical methods are used to estimate derivatives and integrals.

36. Applications of integration Considering the deflection of a beam: What does this depend on?

37. Applications of integration Considering the deflection of a beam: What does this depend on? length load supporting conditions material how the material is distributed

38. Applications of integration Considering the deflection ( d ) of a beam: What does this depend on? length l load w supporting conditions K material E how the material is distributedI, the second moment of area

39. Deflection of a beam Note that todecrease the deflection, we need to increase the terms on the denominator.

40. The second moment of area The second moment of area of a tiny shape of area A, about an axis, is the product of A with the square of the distance of that shape from that axis. If the axis is the x axis, the distance is y. For a larger shape we add up all the second moments of all the tiny pieces of area:

41. y Finding the second moment of area of a rectangle about the x axis. The breadth is B units and the height is D units.

42. To calculate this…

43. Consequences for shapes of beams

44. Other Mathematical Sciences used by Engineers • Discrete mathematics • Probability and statistics • Optimisation and graph theory • Management Science / operations research

45. What’s the take away message? High School maths leads on to uni maths in interesting ways that allow us to model complicated systems in the real world. You can assure your students that if they intend to go on to Science or Engineering courses, they will need as much maths as possible!

46. Thank you. Questions?