Proper sequence for re-learning mathematics

[s0laris]

Hi everyone,

After completing an undergraduate engineering degree, I walked away with a feeling that all I was taught was to crunch numbers, lacking an intuitive understanding of solution mechanisms.
Now, with spare time, I got the desire to re-learn my upper mathematics curriculum. One of the things I found frustrating while in college was that many times topics would be approached that required background that was not previously taught. For example, I think that complex analysis should precede differential equations - my curriculum lacked the course all together. So my question is: what is the best sequence of topics to obtain a truly comprehensive sense for the material (including anything that might otherwise not be included in main coursework)?

Thanks in advance

 

[homeomorphic]

I think a lot of your problem might be solved by just a little bit of extra math, plus thinking really deeply about the material, rather than adding a lot more math. I was in a similar situation to you, but I took it to an extreme and got a PhD in math, but it ended up being a bit of a wild goose chase for me, since I am really more like an engineer or physicist at heart who is more concerned with nature/technology/applications than mathematical abstraction. Some of the math I learned was helpful, and I did get to the point where I understood a lot of things better about engineering math, but a lot of that had more to do with thinking about it more deeply on my own initiative than with studying all these big math theories, which I was required to do in my coursework. One big thing that did happen is that I became really good at linear algebra, and if I ever get back into engineering, I think that might be the number one thing that came out of the math PhD. Another benefit was the challenge, which makes a lot of things just seem easy by comparison.

For example, I think that complex analysis should precede differential equations - my curriculum lacked the course all together.

A whole course in complex analysis seems like overkill to me for basic ordinary diff eq. It's true that a lot of people will come out of high school with too poor of an understanding of complex arithmetic for diff eq, but that's chapter 1 of complex analysis, not the whole course. If it's PDE, then you might want to know complex analysis for some things, but I think it's not crucial for a first course.

Looking back at my experience as an EE major for a while, I found that a lot of the weaknesses in my understanding were corrected by more advanced courses and just reading chapter 1 of Visual Complex Analysis. For example, I took some circuits classes and initially, I didn't really understand phasor analysis, beyond being able to plug and chug with it because I didn't understand complex numbers well enough. Reading just that 1st chapter of VCA seemed to fix that, with a little extra thought on my part, applying it to circuits. That would also help with diff eq.

A lot of people probably don't understand the number e and its role in diff eq, despite being able to plug and chug with it because they take it on faith that the derivative of e^x is itself. I took care of that by figuring out on my own how to construct a function whose slope equals its height the first time I took calculus. It took me quite a bit longer to realize why it was actually the number e raised to the power x, rather than just a function, which you could call exp(x), and e = exp(1), by definition.

Another thing that bothered me was the way Fourier series were sort of pulled out of a hat. Historically, the motivation for that comes from the wave equation and vibrating strings. Any nice enough function pushed along the string will satisfy the wave equation. If you look at it from another point of view, though, suggested by studying oscillations, you arrive at the viewpoint that the motion of the string is a superposition of sine waves at frequencies that are multiples of the lowest, fundamental frequency. Put the two viewpoints together and voila, it seems as though any nice enough function can be represented as a sum of sine waves. This is the sort of thing you might read about in a book about the history of math or a physics book about waves and oscillations or classical mechanics.

Those are a few examples of conceptual dissatisfactions I had as an engineering student. Probably, I've covered a significant portion of it, actually.

 

[homeomorphic]

I suppose you are referring to the inverse Laplace transform with the complex analysis and diff eq because in standard introductory diff eq, that's the only place where it comes up (tangentially) beyond complex arithmetic. Cauchy's formula and contour integration yield the inversion for the Laplace transform. But that's a lot of work to do to just to prove that the Laplace transform is one to one (and onto). Technically, you do use that if you do Laplace transform tables because a priori, maybe two different functions would have the same Laplace transform. You'd be stuck. But even for a real Nazi of understanding like me, that didn't bother me so much when I took diff eq and earlier EE classes. What would have made me happier is not having the Laplace transform itself pulled out of a hat. Part of that was addressed by a later course in signal processing, but the other part could have just been explained right off the bat.

http://mathoverflow.net/questions/383/motivating-the-laplace-transform-definition

There is some method to the madness of the curriculum, even if I don't agree with it 100%. It's hard to organize all the content in a logical way and there is always the problem of pleasing the students. Of course, there are a lot of things that could be very easily corrected if people had just a little bit more appreciation for some good intuition and motivation, which is the part that really gets to me sometimes, rather than the choice of material to be covered.

So, anyway, if you are interested in doing all that work, mainly to see that the Laplace transform is one-to-one (I suppose you can also use the transform directly to find the inverses, but it's more typical to use it indirectly along with the tables), you can read about complex contour integrals and Cauchy's integral formula in Visual Complex Analysis, too, but that would be chapter 9. It's a beautiful journey, though, beyond its utility (plus the general math practice you gain could be useful). He doesn't discuss the Laplace transform, but you can probably Google it and find a proof, once you understand Cauchy's formula.

Also helps to understand Fourier series/transform well, and Fourier inversion in that context. So, another book you might take a look at is Discourse on Fourier Series by Lanczos.

 

[jbunniii]

http://mathoverflow.net/questions/383/motivating-the-laplace-transform-definition

The video linked in the first response is very nice!

http://ocw.mit.edu/courses/mathemat...ure-19-introduction-to-the-laplace-transform/

I had always thought of the Laplace transform as a sort of strange generalization of the Fourier transform, but the video makes it clear that it's a perfectly natural way to define a continuous version of a power series.

 

[Matrix0]

for any advice or suggestions!

I can understand your frustration with feeling like you were only taught how to crunch numbers without a deeper understanding of the underlying concepts. It is great to hear that you have the desire to re-learn your upper mathematics curriculum and gain a more comprehensive understanding of the material.

In terms of the best sequence for re-learning mathematics, it can vary depending on the individual and their learning style. However, a general recommendation would be to start with the basics and build a strong foundation before moving on to more advanced topics. This could include topics such as algebra, geometry, and trigonometry.

Once you have a solid understanding of these fundamental concepts, you can then move on to more complex topics such as calculus, differential equations, and linear algebra. It may also be helpful to incorporate topics from other fields, such as physics or computer science, to gain a broader perspective and see how mathematics is applied in different contexts.

Additionally, as you mentioned, it can be beneficial to fill in any gaps in your knowledge by revisiting topics that were not covered in your previous curriculum. This could include topics like complex analysis, which you feel should have been taught before differential equations. It is important to have a well-rounded understanding of mathematics in order to fully grasp more advanced concepts.

Overall, the best sequence for re-learning mathematics would involve building a strong foundation, incorporating topics from different fields, and addressing any gaps in knowledge. It may also be helpful to seek out resources such as textbooks, online courses, or tutoring to supplement your learning and provide additional support. Best of luck in your re-learning journey!