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this is from the book Calculus made easy
Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrentev.
Personally read only the first chapter, but the book is praised by lots of people. I bet Mr. Nikolaevich has read it.
You can find it on Amazon https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163
For real analysis I really enjoyed Understanding Analysis for how clear the material was presented for a first course. For abstract algebra I found A book of abstract algebra to be very concise and easy to read for a first course. Those two textbooks were a lifesaver for me since I had a hard time with those two courses using the notes and textbook for the class. We were taught out of rudin and dummit and foote as mainly a reference book and had to rely on notes primarily but those two texts were incredibly helpful to understand the material.
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If any undergrads are struggling with those two courses I would highly recommend you check out those two textbooks. They are by far the easiest introduction to those two fields I have found. I also like that you can find solutions to all the exercises so it makes them very valuable for self study also. Both books also have a reasonable amount of excises so that you can in theory do nearly every problem in the book which is also nice compared to standard texts with way too many exercises to realistically go through.
The texts by Stewart and Larson are the two most common introductions to calculus, and both are fantastic. Calculus is such a powerful tool that you don't need to worry about tailoring it to one field. Just as addition and multiplication will be used by economists and physicists alike, so will integration and differentiation.
I don't think that is a very compelling argument, unless we believe mathematicians can do no notational wrong :-) The imprecise, ambiguous, sometimes obfuscatory notation that arises in multivariable calculus and the calculus of variations is a well known and frequently discussed issue. I think we underestimate the difficulty it causes to students, especially to students coming from other disciplines who aren't steeped in the mathematical vernacular.
It's been problematic enough that there are some high profile and semi-accepted attempts to refine the notation, such as the functional notation used in Spivak's Calculus on Manifolds, which is based in an earlier attempt from the 50s if I remember correctly. Another presentation of physics motivated in large part by fixing the notation is Sussman & Wisdom's Structure and Interpretation of Classical Mechanics which adopts Spivak's notation, and also uses computer programs to describe algorithms more precisely.
Try picking up a book. I recommend this one. You can also use Rudin but it will be more difficult.
If you are using notes and online research, it may be that the exercises you've been working on are coming from many different areas and aren't really focused on one topic in particular. This may be the reason that every problem seems to require a new trick.
While it's certainly not the best or broadest advice, I've always found that, whenever a problem starts to get excessively complicated, the mean value theorem always seems to be the why-didn't-I-think-of-that trick that solves it.
Ordinary Differential Equations from the Dover Books on Mathematics series. I Just took my final for Diff Eq a few days ago and the book was miles better than the one my school suggested and is the best written math textbook I have encountered during my math minor. My Diff Eq course only covered about the first 40% of the book so there's still a TON of info that you can learn or reference later. It is currently $14 USD on amazon and my copy is almost 3" thick so it really is a great deal. A lot of the reviewers are engineering and science students that said the book helped them learn the subject and pass their classes no problem. Highly Highly recommend. ISBN-10: 9780486649405
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https://www.amazon.com/gp/product/0486649407/ref=ppx_yo_dt_b_asin_title_o08_s00?ie=UTF8&psc=1
Damn, son. That's way bigger than my guesstimate.
The amazon prices I checked out pinned the collection closer to $400, which granted is still really, really impressive.
In case you're curious this was my textbook. It's come down by a lot in price over a couple years. Brand new it was $365 in the shrink wrap from my school's store!
Eh, either way I'm wrong, just by a different amount.
Definitely read How to Think About Analysis by Lara Alcock. It's exactly the book you want, before jumping into regular textbooks.
Cheers, thanks for the extra info.
Just like another commenter has mentioned, she really ought to try Michael Spivak's Calculus. It will be hard -- perhaps even too much so, in which case maybe she wants to try a proof-writing book like Hammack's Book of Proof first -- but it is the correct introduction to "real" calculus/math if she is really interested in getting serious about math.
Best of luck to her and to you. Math is a trying but beautiful endeavour to pursue, whether it's as a hobby (for me) or as work (for many others whose math skills far exceed mine).
Read this book if you haven't already. This is in my opinion the gold standard, and should close all the holes you feel you have in your game.
He's referring to "Calculus on Manifolds" also by Spivak that is sometimes used as the course text for 257.
Pour les div, grad, curl, laplacien, ... et le calcul vectoriel. Je ne peux que te conseiller ce livre. Il est classé dans mon top 5 des meilleurs livres de math ever. Succinct et intuitif.
I see, makes sense.
And apologies: I said Taylor, I meant Stewart.
So my recommendation would be to start with pretty much any edition of Stewart, like for instance https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552/ref=sr_1_4?crid=27TLBWSQDEBRF&keywords=stewart+calculus&qid=1666117266&qu=eyJxc2MiOiIzLjAyIiwicXNhIjoiMi45NyIsInFzcCI6IjIuOTYifQ%3D%3D&sprefix=stewar...
The later editions just exist to be more expensive but don't really offer any additional value.
Once you've mastered everything in that up to Taylor series, I would maybe switch to a book on Linear Algebra. There are many good ones, and you might just "try out" the books by Strang, Lay, Friedberg, ... if you're really ballsy you can attempt Axler's. Axler's is not recommended as a first book, but you might get something out of the attempt at least.
Anyway, it's hard to imagine you'll even get to the Linear Algebra book before class starts. But if somehow you manage to get through Calculus and Linear Algebra, a very profitable thing to do is look into logic and discrete math. (This would cover, for instance, combinatorics among other things.) A good book for that is by Rosen.
https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552/ref=mp_s_a_1_4?keywords=james+stewart+calculus+8th+edition&qid=1661608473&sr=8-4 this is book essentially, I’ve tutored this for years know and I’m actually surprised you need to know the techniques since the application is more in depth, good luck!
This is the textbook I used when taking AP calc AB and BC.
Stewart’s Calculus is also a pretty common book at the university level.
Yes, you should read through
this book. It is basically what's going to be in Calc 2. If you can't finish this book in a month you should just give up the accelerated course.
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Thanks!
Wish you luck!
>To know the answer to that question of whether allowing proofs in mathematics curriculums is too much for kids, we would have to look at psychological studied and papers in mathematics education.
Well, to me kids can handle some simple proofs. Generalising shouldn't be that difficult.
https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 is a very concise book about real analysis.
You can use a few of the proofs there as reference.
As for Susskind's book, you can find it on Amazon, at least in the US. I'm not sure why it's not titled Classical Mechanics like the other two in the series, but there you are. Whether you choose that or Thorne and Blandford, or another reference, I hope your search for the meaning of tensors is fruitful!
I have a physics bg, but become increasingly interested in mathematics as years passed by. So I got into the habit of self-studying, mostly from translated Soviet Dover books.
I found that the following provides a nice overview of the basic math sub-fields, and it should be helpful for you too:
https://www.amazon.es/Mathematics-Content-Methods-Meaning-Dover/dp/0486409163/
You might like my goodreads list too.
Keep up the good work, believe in yourself, look out for friendly math communities either online (eg. this one) and offline, and never again let others define what you like and what you don't like :).
It is fairly standard for EE students to be unprepared for the required math when they take their first fields course. This is 9/10 times a failing on the part of the university and not the student.
Unfortunately, your professor is correct, though hopefully he said it in a nice way. I got exactly the same speech from my fields professor 30 years ago; fortunately he cared about us, and spent a week going over vector calculus concepts, and recommended to us this book, which though not mathematically rigorous, does the job.
Also, one other bit of advice: try not to despair. I was where you are, as have been lots of other people. It is a difficult, but not insurmountable obstacle.
If you (or anyone else) has electromagnetics questions, feel free to PM me.
If you'd like a particular reference, Michael Spivak wrote a classic "calculus-on-Euclidian-space" called Calculus, but has what I understand to be a pretty solid introduction to what you're looking for called Calculus on Manifolds. (Links to the Amazon pages)
I would suggest "The Theoretical Minimum"
https://www.amazon.com/Theoretical-Minimum-Start-Doing-Physics/dp/0465075681
It's not a pop sci book that give handwavy woo-woo explanations, it delves into the math but explains what the equations mean. A college educated person would have no problem with it.
Oh alright, in that case I would strongly suggest that you take a look at Thomas' Calculus (the editions does not matter) I teached an advanced placement class with this, and I left my students with strong computational and proof skills, in the case that you want to emphasis more analysis Spivak's Calculus
Also these books are really easy to get :)
There is a really good book that goes into this
And a youtube series taught by the author
https://www.youtube.com/playlist?list=PL-rICyRc1Qz144U91HTd6zY9pDVVPwskg
Ordinary Differential Equations by Tenanbaum and Pollard. The text I used at school just concentrated on some seemingly wild-ass method to solve them whereas this book concentrates on the problems themselves and how to solve them. If THAT makes any sense.
I've never actually watched this, but Leonard Susskind's Theoretical Minimum set of courses might be worthwhile for you. There's also a book with the same title by him, which sounds like what you're looking for. The book and the courses are both stand-alone, the first isn't a textbook you need for the second.
I'm pretty sure the courses are free to watch, so I'd suggest starting with them. The eBook or paperback versions aren't that expensive, though.
I always suggest this as an overview of the most important math fields up to the 70s, written by famous Soviet mathematicians.
I consider it pretty accessible to anyone with an interest in higher math, with a strong background of high school math and physics and a deep desire to learn more.
Well you can understand/get-the-frelong-of some stuff/concepts by reading high level books such as this:
https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163
But imho, if you want to understand a mathematical concept deeply, you'll have to go through the hard road of solving actual problems.