Testing Further topics to satisfy my curiosity.

[BOAS]

Hello,

I am hoping that you can help me with this.

We have just finished the math course that is part of my physics foundation year ( I didn't do A-level maths, so it's like a preparation year at university). We covered lots of topics, calculus up to first order D.E's, Taylor's and Maclaurin's theorem, Vector and Scalar products etc.

A few times in the course my lecturer would say 'but we don't cover that in this course' and it has left me wanting more! Right now I need to focus on revision of the material for the exam, but I've got a long summer ahead of me and i'd love to learn more. Specifically I have a few questions/requests about where to start;

When we finished looking at Maclaurin's theorem, we had done the approximation of cosx and sinx (among others) and my lecturer said that combining these two series gives us e^ix = cosx + isinx. What subject do I need to study more of to understand how to combine these two series and arrive at the formula?

When looking at Vector products we were shown how to do it, but were told 'we don't go into determinants in this course' so we weren't shown where it comes from. Where's a good place to start in linear algebra so that I can arrive at a place where I understand why we do vector products in the way that we do?

Sorry for the rambling post!
BOAS

 

[Stephen Tashi]

For (USA) exam preparation, drill may be more important than deep understanding. Is that true in your case?

Search the math section of this forum for threads about Euler's formula and determinants. Those topics have been explained many times.

Vector and scalar products have also been explained on the forum in various styles. One can take a very specific viewpoint and explain them in terms of Cartesian coordinate geometry. One can take a more general view and explain them in terms of tensors. One can take a less sub-scripty view and explain them in terms of "geometric algebra". If you like your intellectual concepts treated in a unified manner, geometric algebra would be the way to go. However, it would be a big digression from what will be on your exam.

 

[BOAS]

Oh I agree - I intend on doing many many practice problems for my exam preparation. I really like my textbook and it's full of problems. (Bostock and Chandler). And from now on, all my contact time with the lecturer will be revision of material.

I just want to further my understanding over the summer before I start actual year 1 physics, where I assume many of my questions will be answered about the material. I followed through the khan academy stuff on linear algebra and learned how to solve simple vector problems using it, which I really enjoyed. Seeing how to construct planes and describe their intersections was cool aswell.

My house is in the middle of nowhere, so maths is a constructive hobby.

 

[DrewD]

When we finished looking at Maclaurin's theorem, we had done the approximation of cosx and sinx (among others) and my lecturer said that combining these two series gives us e^ix = cosx + isinx. What subject do I need to study more of to understand how to combine these two series and arrive at the formula?

You need nothing more. You know everything that you need to show this is true using a Maclaurin expansion and the added assumption that it is still valid on $\mathbb{C}$.

 

[BOAS]

You need nothing more. You know everything that you need to show this is true using a Maclaurin expansion and the added assumption that it is still valid on $\mathbb{C}$.

Does that C mean the complex plane?

 

[jtbell]

When we finished looking at Maclaurin's theorem, we had done the approximation of cosx and sinx (among others) and my lecturer said that combining these two series gives us e^ix = cosx + isinx. What subject do I need to study more of to understand how to combine these two series and arrive at the formula?

I'd be surprised if you really need any new material to understand that formula, provided you've learned the basic stuff about arithmetic with complex numbers. Have you seen the Maclaurin series for e^x? If so, what do you get when you replace x with ix everywhere?

 

[BOAS]

I'd be surprised if you really need any new material to understand that formula, provided you've learned the basic stuff about arithmetic with complex numbers. Have you seen the Maclaurin series for e^x? If so, what do you get when you replace x with ix everywhere?

You're right, I can probably do this.

A quick look shows that the maclaurin series for e^x contains consecutive powers of x, whilst sinx and cosx contain odd and even ones respectively.

It's probably off topic for this sub-forum, so i'll go away and see what I come up with.