Linear programming is covered as part of A level maths in the UK, or at least it was when I did it.
P.S. Lancia, just saw your thing about P and S waves. I did a hard science at university but a quarter of my first year was geology (which ruined my grades as I was shit at it), believe me geology is far more fun than maths (even though I was crap at it) and if you are interested in teaching I really recommend that instead just for the laugh!
This is what I was taught as Euler’s substitution. I pulled out my midterm test, so this is copied directly from that:
Solve for x and y:
3x + 2y = -3
5x - 3y = 1
multiply the first equation by 5 and the second by 3 (I assume you see where I got these numbers from). The result is:
15x + 10y = - 15
15x - 9y = 3
Use necessary arithmetic to eliminate one of the variables. In this case we will subtract, eliminating the x. Result is 19y = -18, when solved for y y= - 18/19
Now take the original equation and multiply first by 3 and second by 2, resulting in:
9x + 6y = -9
10x - 6y = 2
do the arithmetic to eliminate a variable, this case y is eliminated. solve for x, x= -7/19.
So, the final answer is (-7/19, -18/19)
If this isn’t Euler’s substitution, which according to wiki it isn’t, what is it?
Well, what you’re doing is solving a system of linear equations:
I’m not sure if the way you’re doing it has any particular name.
It looks close to Gauss-Jordan elimination, although the method you follow by hand is not quite as automated as the algorithm.
Just Gaussian elimination by substitution. It is suitable for solving a system of equations with two or three unknowns, but becomes unwieldy after that unless you have a sparsely populated system, i.e. one in which most of the unknowns only appear in one or two equations. When you get to a system of four or more equations and unknowns, it is much better to go to Gauss-Jordan elimination. For really large problems, however, such as solving a large finite difference or finite element systems in which most of the nodes are not connected to one another, there are a whole host of numerical methods that are used to reduce the computational complexity to a reasonable scale.
Stranger
seems to be a book the OP would be interested in.
That is beyond awesome. Many, many thanks for that link. I’m putting together a list of books to help me get a grip on this stuff, and I will add that one to the list.
Are you familiar with “Lockhart’s Lament”? It’s available online here (PDF).
If you’re looking for recommendations for good mathematics books, we’ve had threads about that; here’s one: Recommend a book on mathematics
I am your same age and going through something very similar, OP, but with Physics instead of Math. It’s been a year now since I realized I just can’t live without it, and I still haven’t really figured out what I will do in the end, but I’m not giving up. I think the hardest thing was to realize that it really does not come easily to me, unlike my “real” career, but in a way, after all these months, the challenge is part of the appeal.
So far I’ve managed to self study and audit college classes up to the junior-year level, and this semester I’m finally taking the plunge and taking a Physics class for credit, along with auditing two more, which is the same load the junior year cohort will be doing. I’m terrified and exhilarated.
Anyway, regarding Math, I had to learn a lot of Math fast as well in order to get the Physics, and it was beautiful Math. While I never thought I was going to understand Calc I, I just finished auditing Calc IV, and I think that I finally got it (Calc I, that is). Linear algebra was a nice respite. Now I’m onto vector calculus, and I’m not any good at it, but it also gives in after a while. Being stubborn has proved immensely useful. Like you, I learn much better from books than from lectures, so I’ve been choosing classes based on the books they cover (and going over the whole book during school breaks before classes start).
In all, I don’t know if I can ever become a physicist, and I’ve spent a lot of time worrying about it, but I know I will continue to study physics every day, because I simply cannot be happy without it. Everything became easier once I realized that.
I disagree that algebra is almost meaningless without calculus. Analysis and Algebra are different mathematical fields, and one can certainly learn, say, linear and abstract algebra with little or no knowledge of calculus.
Physics is dependent on calculus. Look at the exercises in an introductory physics textbook for many examples.
Do you want some physics textbook recommendations?
For a rather difficult and advanced introductory Newtonian mechanics textbook, I recommend this book. It was designed for MIT students, and one of my physics professor said that it was the hardest introductory physics textbook he has ever seen. However, it is written well, and is more concise and has far less fluff than most modern textbooks.
For electrodynamics, I recommend Griffith’s Introduction to Electrodynamics. It is a sophomore/junior level textbook. The solution manual can be found online.
For quantum mechanics, I recommend Shankar’s Principles of Quantum Mechanics over Griffith’s quantum mechanics textbook. Shankar’s book was designed for a self-studying student, and besides assuming basic knowledge of vectors and matrices (and physics), it is a self-sufficient textbook. However, Griffith’s textbook is by far the more popular one for university classes.
Concerning your last paragraph: read this anecdote by Feynman
Thanks! I had never heard of that mechanics book. I just leafed through it on Scribd and it looks really useful for me to patch up my understanding of mechanics.
I’ve been working through Ohanian for electrodynamics, but I can use all the help I can get.
I do have Shankar, and I’m really hoping the class based on it is offered again next semester, because it was last year, but I was nowhere near ready to take it. You’re right, the class I’m about to take goes with Griffith’s instead. I also just audited an intro to Quantum Physics class that used Townsend, and while I didn’t care for the text at all, I liked the contents. I was hoping that auditing that would get QP out if my system, but that backfired and now I can’t wait to learn more.
And thanks for the link. I think that playfulness is probably my biggest missing element. At this point I’m still so in awe of the material, that I don’t feel comfortable playing with it, trying to see what happens in extreme cases, trying to find counter examples, etc. which is something I see the best students in my classes doing. I also know how necessary (and enjoyable) that attitude is in research (at least in linguistics, my actual job), but I’m still petrified when it’s physics.