I wanted to try to take what I have learned since September – and kind of summarize it. I wanted also to be able to put some order or a sense of what I learned and what I didn’t understand. I also wonder how everything fits together with all my previous knowledge – and have to ask if anything in my thinking changed because of this class.
Initially (still a little now) I felt technologically challenged 🙂 But, I also had to realize that I am not fresh out of high school – it was the first time that I had a class were we had to blog. So, I worked through learning things – I didn’t learn it all at one time – just last week I got bold eneough to experiment with the background and formatting of my blog. There are a few more things I’d like to learn – but, I am finally okay with not knowing everything right now. I have learned to keep reminding myself – if I knew everything – there would be no need for school – wow – imagine that 🙂
The first day of class I was still learning short-cuts on the key-board. Now I do things so quickly and it seems so easy compared to the beginning. That feels really good. I remember not having a problem with LaTax in the begining – but, kind of struggling in project four. Since we couldn’t seem to get things to transfer – I kind of just pasted and cut equations in … not ideal – but, I learned to do what will work and not get stuck.
I found Matlab really difficult for some reason – as I am reading Matlab DeMystified – it seems more logical to me. I seem to have difficulty just doing something without learning the why’s and how’s – so, as easy as it is to just use a line or two of code – I really tried to learn from my mistakes – from the aspect of – why didn’t it work – what went wrong – what does that error code mean – plus learning the math at the same time – it was very challanging at times – but good.
We also did a bit of Mathematica and TeMath. I liked the graphical abilities of these programs a lot.
Even though part of me felt like my learning was all over the place – or five things at once – it challenged me and pushed me – so, I ended up at a different level – which I totally appreciated. I feel like I really matured mathematically through this class.
As I sit and think – I realize the connections from a lot of previous classes and readings. Euler’s Method – Series – integration and differentiation – all became clearer to me.
At the beginning of the class Dr. Davis talked about learning and thinking about O.D.E.’s analytically and behaviorally – when you cannot get an exact solution. We studied linear and non-linear systems and systems of equations.
It was nice to let the computer do the solutions – and learn the theory – the how’s and why’s of using the computer and various software. Once I got the hang of programming the cells in Excel – it seemed so easy – seems like the first week was a very long time ago.
We also learned some techniques if we could not get exact solutions using the computer. We learned to look at behavior. This allowed us to use quantitative techniques – I really enjoyed how Jay compared ode23 with ode45 – it showed visually margins of error – we got to see it numerically in Euler’s Method – the smaller the increments the more accurate. But we could also see error in the qualitative behavior. I still want and need to look at and research the Dirac function from the ode that was not separable in project four – I wonder why this particular function was chosen – I learned how to create slope fields – visually put together – solution curves and slopes for me – it added to my repertoire of uses for slope. Project three helped synthesize theoretically for me how linear algebra techniques are very useful in D.E. I would have never thought at the beginning – we would end up using matrices and determinants. It was pretty cool.
I still feel I need to sit with LaPlace transforms for a while. I don’t feel I’ve done enough of them – because it is not intuitive enough for me. I am still struggling through them … but, just need to sit and solve a bunch.
As far as analytical techniques – we learned about substitution a and separations of variables. I enjoyed learning especially in this class – to step away when I am stuck and figure out what I do know first. Then approaching the problem from there – I guess I feel I matured mathematically most because of this … I now think before I do any solving – I used to just jump in and write . Now I recognize that – I can ask myself – what does this problem remind me of? Have I ever seen anything like it? How do I want to start? Finish? I guess I kind of find a plan of attack now – a set of blue prints if you will. And if the blue prints aren’t working they can be redesigned and it’s not the end of the world. I will always see Matt in a red convertible saying “friggin” 🙂
There are still some questions about things like the predator prey model – how do we calculate the unpredictable – like a fire etc.? Seems like we would have to take so many variables into consideration – that we would be likely to miss something. The model seems to revolve around just the numbers and animals that exist – not why or how. I imagine it would be a huge problem if we did this – so, I guess I need to research this more. We saw Lorenze attractors and Rossler’s. I am trying to figure out if a predator prey model could ever be 3d – I am assuming this is true by what I know so far – but, if we have four linked d.e. does that put us into 3D? Going by Linear Algebra so far – 4 eq. with 4 unknowns would say yes – is that all there is to it? I wonder how over determined and under-determined equations fit in here? By adding more variables to a predator prey system – do we ever get chaos? hmmmm – is that were overdet. and underdet. fit in? Can computers model past 3D? We are looking at a 2D screen seeing 3D – do we need a 3D screen to see 4D? Lap-tops would then be lap-boxes – I imagine that is what holograms are – probably my fascination behind changing the axis’ and getting different views of the graph –
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and if we set the two equal to each other, we get:
