ENG2092: Advanced Engineering Mathematics B

-5x 1% fortnightly tests (held during each second tute)
-5x 5% fortnightly assignments (handed in each second tute)
-1x 3 hour 70% exam
*Note that the weeks in which tests are held and assignments are due alternate

Overall I enjoyed this unit. It was an engineering unit so there is an emphasis on application, however there still remains a decent amount of rigorous proof (specifically in the complex analysis section). I found this unit fairly easy in comparison to the other units I took this semester (PHS2022, ECE2072 and MTH2032) and I feel that is in no small part due to the amount of tute/textbook/exam-like questions I completed.

The first 4 weeks consist of John Head's Complex analysis section. By far, this was the most well taught of the 3 sub-units. John goes into great detail about the proofs behind each statement he makes and goes through very helpful example questions. His written notes are also very useful and come with corresponding typed notes.

Week 1:

You first go through a basic introduction into complex numbers and functions, then learn about the mathematical definition of analyticity, continuity and limits (and you thought you were done with epsilon-delta proofs after MTH2010). You then build on this knowledge with methods of determining the analyticity of complex functions through the use of the Cauchy-Reinmann equations.

Week 2:

This is where the real maths begins. Firstly you learn about elementary complex functions. although it may seem easy, questions relating to this section are often approached in unintuitive ways (i.e. finding the value of i^i by using polar form). While these questions aren't difficult in their complexity, their approach may be difficult to see during an exam situation so it is best to familiarise yourself with them in the off chance they show up on an exam. You then move onto Contour (Line) integrals and their properties. If you've done MTH2010, then you know that these bastards can be annoying to calculate (especially when you have shitty boundaries or an ugly complex field to integrate over. Either way, practice makes perfect with these types of questions. Do all the tute questions relating to these and you'll be fine. You then learn about Cauchy's Integral Theorem (If you are integrating over a closed loop and there is no singularity within the region bounded by the loop then the result is 0) and Green's theorem (although this was taught, I don't believe it is assessed).

Weeks 3 & 4:

Weeks 3 and 4 consist of the vast majority of material assessed in the final exam (Complex analysis section). There is usually about 40-50 marks or so in the complex analysis section and about 30-40 or so of it consists of this material. You first learn of Cauchy's integral formula (used when there is 1 or more singularity within your closed loop (only simple poles are applicable here)). You then build on the first year maths you've done with some work on series (notably Power, Taylor and Laurent Series) that will be used for week 4. In week 4 you continue your work on series as you build towards the Residue theorem. This is used when you have non-simple singularities within your bounded loop (i.e. integrating e^(1/z) around the circular loop of radius 1 about the origin. e^(1/z) has what is known as an essential pole at the origin and you need both the Laurent series and Residue theorem to find this integral).

With the assignments relating to this material, I must stress the need for meticulous working and explanations. The tutors have been known to take off up to half of the marks based solely on explanations alone (even with correct answers). You need to show every step, no matter how little it contributes to the final answer. I managed to average 95% on these assignments and I can tell you that even though I felt as though the assignments could be completed in less than 3-4 pages, I would often write in excess of 8 pages just to make sure there were as few places as possible for marks to be taken off (i.e. rigorous explanations and detailed diagrams). There is a detailed document on the Moodle page named "Guidelines for writing in mathematics" and I implore you to check it out.

I won't go into as much detail in the next two sub-units as I didn't pay as nearly as much attention to them as I did with complex analysis.

Weeks 5-8 consist of Jennifer Flegg's Integral Transforms section. Other than the application to ODE's, I wasn't a big fan of this sub-unit. The lecture notes are pre-typed which I didn't like very much (I tend to take notes during the lecture rather than at home, so it helps when a lecturer writes at the same pace as me) and I felt as though there wasn't a great deal of proof relating to the material (other than proving the transforms themselves). A lot of it we were forced to take for granted.

You start off with Laplace transforms, in which you are transforming between a t domain to an s domain via a relatively simple integral (Worst case scenario you'll have to use integration by parts in order to prove these transforms). The most interesting part of this section is the application of Laplace transforms to ODE's. Given initial conditions, you can transform a linear ODE into the s domain, then after some simple algebraic manipulation you can use then inverse Laplace transform to get a function that solves the ODE (Hope you like Partial Fractions). You then move onto more abstract applications such as convolution in which a relatively annoying integral turns into the multiplication of functions.

You then move onto Complex Fourier series and transforms. Questions relating to this section are relatively straightforward, but silly mistakes can cost you dearly. I recommend that you practice the hell out of questions relating to Integral transforms as many of the questions rely on your ability to recognise a transform or property. Although you are given a complete list of transforms and properties, recognising them is an entirely different issue that is best honed by doing practice questions (the tute sheets are a good source for these).

- John Head (Complex analysis)
- Jennifer Flegg (Integral transforms)
- Jonathan Keith (Statistics)

2 full exams. 1 with answers.

4 out of 5

Yes, with screen capture.

Weeks 9-12 consist of Jonathan Keith's Statistics sub-unit. I felt that by far, this was both the easiest and worst taught section. The lectures were pretty bad and were essentially just the STA1010 lecture notes covered in bad hand writing (sorry Jonathan but it's true). Apart from the inference section, the rest of the material is built upon Maths Methods and Further Maths. You learn about box plots, averages and calculating Standard Deviations and Variances. Then work with different probability distributions (Normal, Binomial and Poisson), confidence intervals and finally inference (which is essentially just a contextualisation of confidence intervals). I urge you to look at a review for STA1010 for more info (just exclude the experimental design, conditional probability calculation and the stuff at the end like ANOVA/Chi-squared test).

Overall this is an easy unit if you do your homework. A couple of hours a week doing questions and looking through the notes should be fine for revision. The exam is relatively easy too. They tend to throw you a couple of curve balls here and there, but 80% of the exam is just applying your knowledge in a familiar way and 20% is just extended versions of it.

-E. Kreysig, Advanced Engineering Mathematics (9th edition).
-G. James, Advanced Modern Engineering Mathematics (4th edition).

Neither of these textbooks are needed to get a good mark as the tute sheets are more than sufficient, but definitely worth looking into if you feel like you need a greater understanding of the material/need some more practice (I would recommend Kreysig over James). I would also recommend buying the STA1010 lecture notes as the stats sub-unit is completely covered in it.

3x 1 hour lectures, 1x 2 hour tute per week

2015 semester 2.

90 HD