Maths 16 and 17 Aug

16th Aug and 17 Aug

The tour of cambridge enabled us  to look through the cambridge campus which consists of 31 colleges  and we went to at the Kings College chapel, Trinity College where (gasp) Newton used to live and St John’s College. The architecture seemed identical at first glance but as we looked at more of the intricate details on each building the differences in age shows.

John,Marie,Melvin,Mr Lim and Jonathan beside the Math Giant
Viren teaching the RJC students how to solve a question

We also realised that we  could only fully understand a subject or topic such as differentiation only after we understood its proofs . During the time given to solve a particular question or so,we had a lot of peer interaction as well which also increased our insight into the subject as we gained knowledge on how it was plausible for us to solve a particular problem or question in a variety of ways.

In the morning, we learn the theory behind Vector Calculus. In this topic, we learnt about scalar product, vector product, triple scalar product and triple vector product. We also knew that for vectors of more than 3 dimensions, we cannot do vector product but use wedge product. We were introduced to the concepts of curl and divergence which are important to understand differentiation of vectors. Can you imagine that vectors can be differentiated? Cool right? In fact they can be integrated as well.

In the afternoon, we did complex numbers. We were introduced to basic properties of complex numbers, nth roots of unity, Euler’s formula and used complex numbers to help us do differentiation and integration of complicated expressions.

17 Aug

In the morning, we were introduced to Calculus of Variations. Essentially, we realized that to determine the nature of stationary points of functions with more than 1 variable, we need to use Euler Larange Equations. To solve this equations, we need to learn integration and differential equations. We forgot much of our integration stuff so find this part a bit tough going. But the tutor was patient enough to show the steps.

In the afternoon we did some examples which were linked with themes in physics such as the study of mechanics, proving the shortest distance between two points is indeed a straight line and so on.  When we apply scientific method to the physical world, we qualify or define things, then we quantify or measure them. Therefore,it’s inevitable that  measurement of quantities  in physics involves mathematics.

We also further learned about a new concept called functionals. We all know that  functions  map from a domain of numbers to the range which is usually real numbers. However, a functional maps a function to the range of real numbers. In other words, it is a function that takes another function and returns a scalar result. It can be easily thought of as a function of a function. One mistake many people made was that they immediately thought of composite functions. But the problem is that composite functions are functions that takes the scalar results of the first function. It is not an easy concept to accept though.

We did realise that the course was very objective driven as the teachers were very focused on completing a particular topic before the lesson ended. Also,our Vector Calculus tutor Sitikantha Das even gave one to one “lessons” when individuals did not understand a particular concept or so.

Sitikantha teaching students

We did feel, however it was essential that everyone had a rough outline of the courseline as at certain times it was arduous to follow the lesson given that it was a new concept. Having a rough outline would also prompt us to read ahead and during class,it would be easier to follow the lesson.

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