1. Ares I-X Completes a Successful Flight Test NASA's Ares I-X test rocket lifted off Oct. 28, 2009, at 11:30 a.m. EDT from Kennedy Space Centre in Florida for a two-minute powered flight. ©NASA’s Mashall Space Flight Center 2009, sourced from http://www.flickr.com/photos/28634332@N05/4054766770/ Available under creative commons license ©NASA’s Mashall Space Flight Center 2009, sourced from http://www.flickr.com/photos/28634332@N05/4054031697/ Available under creative commons license

2. CIVE2602 - Engineering Mathematics 2.2 Lecturer: Dr Duncan Borman Application of partial derivatives -Total differential -Small Errors Lecture 11 Limits, Sequences and Partial differentiation

3. Total differential Say we have a function Let=small increment in Corresponding increment in will be: We can re write as:

4. Total differential Rearranging gives: From the definition of a partial derivative this gives: This can also be written in terms of a differential as: We can re write as:

5. ©David Arvidsson 2010, sourced from http://www.flickr.com/photos/achoice/4718106723 / Available under creative commons license THE APPROXIMATION OF SMALL ERRORS e.g. 1 Consider a spherical ball being of mass m and radius r. Calculate it’s density N.B Density depends on 2 variables m and r, Volume of a sphere Suppose that the mass m can be measured to an accuracy of 0.5% and the radius r to an accuracy of 1%. To what accuracy can the density be estimated? Small change in density Rate of change density as m changes (r constant) Small change in mass Small change in radius Rate of change density as r changes (m constant)

6. + % error in N.B. +ve sign WORST CASE !! We are interested in the total max error in Thus the density can be estimated to an accuracy of 3.5 % (approx) Remember definition of a percentage error is:

7. % error in Is this realistic? If m = 4kg and r = 1m = 0.9549 If m = 4.02 kg and r = 0.99m (Containing 0.5% and 1% errors described at start of Q) = 0.98908 Error % = = 3.579% (compare with the estimated 3.5%)

8. Consider the volume V of a cone; it depends on the cone's height h and its radius r according to the formulaheightradius The partial derivative of V with respect to r is It describes the rate with which a cone's volume changes if its radius is varied and its height is kept constant. and represents the rate with which the volume changes if its height is varied and its radius is kept constant. The partial derivative with respect to h is Q) Suppose that the radius r can be measured to an accuracy of 1% and the height h to an accuracy of 2%. To what accuracy can the Volume, V be estimated?

9. + % error in Thus the Volume, V can be estimated to an accuracy of 4 % (approx) Radius r accuracy of 1%. Height h accuracy of 2%.

10. + % error in Thus the Volume, V can be estimated to an accuracy of 4 % (approx) Radius r accuracy of 1%. Height h accuracy of 2%.

11. + % error in Thus the Volume, V can be estimated to an accuracy of 4 % (approx) Radius r accuracy of 1%. Height h accuracy of 2%.

12. Multiple choice Choose A,B,C or D for each of these: If A B C D 1)

13. Multiple choice Choose A,B,C or D for each of these: If A B C D 2)

14. Multiple choice Choose A,B,C or D for each of these: If A B C D 3)

15. Multiple choice Choose A,B,C or D for each of these: If A B C D 4)

16. Multiple choice Choose A,B,C or D for each of these: If A B C D 5)

17. Multiple choice Choose A,B,C or D for each of these: If and are two independent variables. Which is true? A B C D 6) All are true

18. Week 5 – Task Results Best time 100% in 20 mins Most people over 30mins Remember making an attempt on ALL the tasks is a compulsory part of the module – if you are one of the few who’ve not done it please do it as soon as possible.

19. A Cylinder has an elliptical cross-section a b It is subjected to an external torque T at the ends If a may be measured to an accuracy of 0.5% b may be measured to an accuracy of 1% T may be measured to an accuracy of 0.5% To what accuracy can be measured? The maximum shear stress is given by

20. a: 1%. b: 0.5% c:1% a b L For WORST CASE We find size of errors For errors we require the worst possible case, so: = ½ % + 2 (1)% + 2% =

21. Q: A cube has sides of length a and mass m. If a and m can be measured within an accuracy of 2% and 3%, respectively, then to what accuracy can the density of the cube be estimated? Work in pairs or 3’s - work this out (you can use your notes) Put your hand up when you think you’ve an answer – I’ll want to see some working.

22. Q: A cube has sides of length a and mass m. If a and m can be measured within an accuracy of 2% and 3%, respectively, then to what accuracy can the density of the cube be estimated? N.B Density depends on 2 variables m and a, Clickers?/chocolate Work in pairs or 3’s - work this out (you can use your notes) Put your hand up when you think you’ve an answer – I’ll want to see some working.

23. Q: A cube has sides of length a and mass m. If a and m can be measured within an accuracy of 2% and 3%, respectively, then to what accuracy can the density of the cube be estimated? N.B Density depends on 2 variables m and a,

24. Differential dm, da, df, dt etc Use to Approximate small errors Week 6 task available REMEMBER ALL LECTURE NOTES AND LOTS MORE ARE ON THE VLE Dr Stewart will do module feedback in two weeks to cover his and these lectures a b R r L

25. Why look at partial derivatives? Partial differential equations are equations involving unknown functions and their partial derivatives. They appear in lots of real world phenomena and engineering applications. Some examples are: 1)The wave equation is an important second-order linear partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves Both use 2 nd order partial derivatives The heat equation is an important partial differential equation which describes the distribution of heat (or variation in temperature) in a given region over time.

26. Essential for modelling Earthquakes Modelling dam performance/failures Stress analysis of railway bridge ©Alton.art 2008, sourced from http://en.wikipedia.org/wiki/File:101.portrait.altonthompson.jpg Available under creative commons license Outdoor Airflows

27. Find

28. SECOND-ORDER PARTIAL DERIVATIVES Q1

29. Consider the volume V of a cone; it depends on the cone's height h and its radius r according to the formula Now consider by contrast the total derivative of V with respect to r and h. They are, respectively and The difference between the total and partial derivative is the elimination of indirect dependencies between variables.

30. Chain Rule III f is a function of x and y AND x and y are both functions of two other variables If and The chain rule allows us to find the partial derivatives wrt to both s and t

31. Cartesian and polar coordinates (2D) Cartesian (x,y) Polar (r,θ)

32. Cartezian and polar coordinates (2D) http://www.edumedia-sciences.com/a367_l2-polar-coordinates-2d.html

33. Chain Rule III If f(x,y) is the temperature of surface of a circular break disc at the point (x,y) The temperature at any point x,y is f(x,y)=x 2 +y 2 + 6 In Polar coordinates x=r cos(θ) y=r sin(θ) Find what the rate of change of temperature is in the disc is in the radial direction. This would be given by

34. Cartesian coordinates (3D)

35. Polar coordinates (3D)

37. CIVE2602 - Engineering Mathematics 2.2 Feedback forms Q1: I think having lectures and other material available in one place on the VLE will be useful. Q2: Having material hand written using the Tablet computer was difficult to read. Q3: I would have preferred to have PowerPoint style slides that the lecturer was not able to write on. Q4: I found the lectures useful. Q5-Q15 from sheet A – Strongly Agree B – Agree C – Neither agree nor disagree D – Disagree E – Strongly Agree Any other comments can be written on the back of the pink sheet- all feedback welcome.

38. CIVE2602 - Engineering Mathematics 2.2 Lecture- Summary Chain Rule – 2 nd derivative Approximation of small errors Next week – Prof Ingham (2 lectures) Following week – Dr Stewart – Vector Algebra

39. Very important in CFD modelling approaches

40. Summary of 2 nd derivative (when x and y are functions of two other variables s & t) Stage 2 We can differentiate each of these partial derivatives wrt to both s and t And also Stage 3 Stage 1 Find partial derivatives of f wrt x and y Find partial derivatives of x and y wrt s and t

41. Multiple choice Choose A,B,C or D for each of these: Find the partial derivative wrt x If A B C D 1)

42. Multiple choice Choose A,B,C or D for each of these: Find the partial derivative wrt x If A B C D 2)

43. Multiple choice Choose A,B,C or D for each of these: Find the partial derivative wrt x If A B C D 3)

44. Multiple choice Choose A,B,C or D for each of these: Find the partial derivative wrt t If A B C D 4)

45. Multiple choice Choose A,B,C or D for each of these: Find the partial derivative wrt z If A B C D 5)

46. Multiple choice Choose A,B,C or D for each of these: Find the partial derivative wrt m If A B C D 6)

47. What does differentiation mean in 2-D? Gradient ? (if so in which direction x or y )

48. Essential for modelling Arsenal stadium Earthquakes Wind farms ©Vattenfall 2010, sourced from http://www.flickr.com/photos/vattenfall/5019768387/ Available under creative commons license ©Alton.arts 2008, sourced from http://en.wikipedia.org/wiki/File:101.portrait.altonthompson.jpg Available under creative commons license

49. Chinese stadiums- FEA analysis ©Curt Smith 2008, sourced from http://www.flickr.com/photos/curtsm/3005592094/ Available under creative commons license ©A. Aruninta 2008, sourced from http://commons.wikimedia.org/wiki/File:Olympic2008_watercube02_night.jpg?uselang=en-gb Available under creative commons license

50. Arup Consulting engineers, designers, planners and project managers Eden Project – Arup's CFD flow analysis contributed to the design. They also helped develop software to model moisture concentrations in the Humid Tropics - to create a precise atmosphere for maximising plant growth. CFD modeling of air temperature distributions London Coliseum Air flow around building ©Kenneth Allen 2007, sourced from http://www.geograph.org.uk/photo/462742http://www.geograph.org.uk/photo/462742 Available under creative commons license

51. Chain Rule III- example Some times both x and y are functions of 2 other variables. So you might have: x as the number of blue fish in a river - dependent on t and s (t= river height and s= river temperature, e.g. x=2t+s) y as the number of orange fish in a river - dependent on t and s (t= river height and s= river temperature, e.g. y=t+2s) x y t The total number of fish which would be f(x,y)=x+y s Now you might want to find what the rate of increase/decrease of the number of fish in the river as the river height changes or as temp changes. This would be given by and (partial derivatives wrt to t and s) ©Stacie Lynn Baum 2009, sourced from http://www.flickr.com/photos/stacylynn/3202712417 /http://www.flickr.com/photos/stacylynn/3202712417 /, Available under creative commons license ©[Brian] 2008, sourced from http://www.flickr.com/photos/bmelancon/3014042769 / Available under creative commons license ©Gisela Giardino 2006, sourced from http://www.flickr.com/photos/gi/271023578/ Available under creative commons license ©MelRick 2007, sourced from http://www.flickr.com/photos/melrick/510099461/ Available under creative commons license

52. Chain Rule III x y t s Now you might want to find what the rate of increase/decrease of the number of fish in the river is as the river height changes or as temp changes. This would be given by and (partial derivatives wrt to t and s) x=2t+sy=t+2s f(x,y)=x+y Find and Find df/ds

53. Example 1 = 0

54. =

56. = + +

57. Application of Partial differential equations (CFD)