Reddit reviews Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus
We found 19 Reddit comments about Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus. Here are the top ones, ranked by their Reddit score.
Used Book in Good Condition
Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.
Amazon search for Dover Books on mathematics
There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm
Pre-Calculus / Problem-Solving
Calculus
Linear Algebra
Differential Equations
Number Theory
Proof-Writing
Analysis
Complex Analysis
Functional Analysis
Partial Differential Equations
Higher-dimensional Calculus and Differential Geometry
Abstract Algebra
Geometry
Topology
Set Theory and Logic
Combinatorics / Discrete Math
Graph Theory
P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
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.
Calculus on Manifolds is an elementary treatment that only assumes basic mathematical maturity along with say a year of calculus. A classic from Michael Spivak, who these days is mostly known for his rigorous calculus book.
If you're looking for other texts, I would suggest Spivak's Calculus and Calculus on Manifolds. At first the text may seem terse, and the exercises difficult, but it will give you a huge advantage for later (intermediate-advanced) undergraduate college math.
It may be a bit obtuse to recommend you start with these texts, so maybe your regular calculus texts, supplemented with linear algebra and differential equations, should be approached first. When you start taking analysis and beyond, though, these books are probably the best way to return to basics.
Yeah, definitely the best book I've read on differential forms was Spivaks Calculus on Manifolds. Its very readable once you have a solid foundational calculus background and is pretty small given what it covers (160pp). If you need to know this stuff then this is definitely the right place to learn it.
This might be of interest, Spivak's Calculus on Manifolds.
There's really no easy way to do it without getting yourself "in the shit", in my opinion. Take a course on multivariate calculus/analysis, or else teach yourself. Work through the proofs in the exercises.
For a somewhat grounded and practical introduction I recommend Multivariable Mathematics: Linear Algebra, Calculus and Manifolds by Theo Shifrin. It's a great reference as well. If you want to dig in to the theoretical beauty, James Munkres' Analysis on Manifolds is a bit of an easier read than the classic Spivak text. Munkres also wrote a book on topology which is full of elegant stuff; topology is one of my favourite subjects in mathematics,
By the way, I also came to mathematics through the study of things like neural networks and probabilistic models. I finally took an advanced calculus course in my last two semesters of undergrad and realized what I'd been missing; I doubt I'd have been intellectually mature enough to tackle it much earlier, though.
I'm a physicist/mathematician, but I think it could be useful for you. Exterior algebra (differential forms) in particular is worth learning because it makes the theory of multivariable calculus much more elegant and simple. With exterior algebra you can see that the fundamental theorem of calculus, Green's theorem, and the divergence theorem are special cases of a generalized Stoke's theorem. Spivak's Calculus on Manifolds book (which is actually not a manifolds book despite the name) teaches calculus at an undergrad level using exterior algebra and differential forms if you're interested in learning this stuff.
Exterior algebra can be considered as part of geometric algebra, so you could continue on to learn geometric algebra if you enjoy exterior algebra.
For me, a "good read" in mathematics should be 1) clear, 2) interestingly written, and 3) unique. I dislike recommending books that have, essentially, the same topics in pretty much the same order as 4-5 other books.
I guess I also just disagree with a lot of people about the
"best" way to learn topology. In my opinion, knowing all the point-set stuff isn't really that important when you're just starting out. Having said that, if you want to read one good book on topology, I'd recommend taking a look at Kinsey's excellent text Topology of Surfaces.
If you're interested in a sequence of books...keep reading.
If you are confident with calculus (I'm assuming through multivariable or vector calculus) and linear algebra, then I'd suggest picking up a copy of Edwards' Advanced Calculus: A Differential Forms Approach. Read that at about the same time as Spivak's Calculus on Manifolds. Next up is Milnor Topology from a Differentiable Viewpoint, Kinsey's book, and then Fulton's Algebraic Topology. At this point, you might have to supplement with some point-set topology nonsense, but there are decent Dover books that you can reference for that. You also might be needing some more algebra, maybe pick up a copy of Axler's already-mentioned-and-excellent Linear Algebra Done Right and, maybe, one of those big, dumb algebra books like Dummit and Foote.
Finally, the books I really want to recommend. Spivak's A Comprehensive Introduction to Differential Geometry, Guillemin and Pollack Differential Topology (which is a fucking steal at 30 bucks...the last printing cost at least $80) and Bott & Tu Differential Forms in Algebraic Topology. I like to think of Bott & Tu as "calculus for grown-ups". You will have to supplement these books with others of the cookie-cutter variety in order to really understand them. Oh, and it's going to take years to read and fully understand them, as well :) My advisor once claimed that she learned something new every time she re-read Bott & Tu...and I'm starting to agree with her. It's a deep book. But when you're done reading these three books, you'll have a real education in topology.
For vector calculus, you might enjoy the less formal British text Div, Grad, Curl, and All That by H. M. Schey; for group theory in brief, consider the free textbook Elements of Abstract and Linear Algebra by Edwin H. Connell.
Alternatives to Schey's book include the much more formal Calculus on Manifolds by Michael Spivak, which does have more exercises than Schey but uses most of them to develop the theory, rather than as the mindless drills that fill an ordinary textbook; Michael E. Corral's free textbook Vector Calculus isn't huge but is written closer to an ordinary textbook.
OK, then let's try this again, this time using more calculus and less topology-specific results. I'm going to be using LaTeX markup here; see the sidebar to /r/math for a free browser plugin that'll translate my code into readable mathematics.
The following is from Michael Spivak's Calculus on Manifolds, and it's pretty close to the result you want, but with more restrictions in terms of differentiability and such:
(a) Let
[; f \colon \mathbf{R}^2} \to \mathbf{R} ;]be a continuously differentiable function. Show that[; f ;]is not 1-1. Hint: If, for example,[; D_1 f(x,y) \neq 0 ;]for all[; (x,y) ;]in some open set[; A, ;]consider[; g \colon A \to \mathbf{R}^2 ;]defined by[; g(x,y) = \left( f(x,y), y \right). ;](b) Generalize this result to the case of a continuously differentiable function
[; f \colon \mathbf{R}^n \to \mathbf{R}^m ;]with[; m<n. ;]The basic idea for (a) is that if there were such an continuously differentiable injection
[; f \colon \mathbf{R}^2 \to \mathbf{R}, ;]then (1) we can find some subset[; A \subseteq \mathbf{R}^2 ;]such that (depending on your convention for notation)[; D_1 f(x,y) = \partial_1 f(x,y) = \partial_x f(x,y) = \frac{\partial f}{\partial x} (x,y) \neq 0 ;]for all
[; (x,y) \in A, ;]and (2) the function[; g \colon A \to \mathbf{R}^2 ;]must have a local continuously differentiable inverse. (This is by the Inverse Function Theorem.)The problem, however, arises when you consider the actual form of a local inverse for
[; g, ;]since[; g^{-1} ;]will be independent of the second coordinate. Accordingly,[; g ;]cannot be injective, whence[; f ;]cannot be injective.I imagine the generalization to part (b) is similar. The important thing here is that given a function
[; f \colon \mathbf{R}^m \times \mathbf{R}^n \to \mathbf{R}^m, \text{ where } m<n, ;]one can construct the associated function
[; \begin{align*}<br /> g \colon \mathbf{R}^m \times \mathbf{R}^n &amp;\to \mathbf{R}^m \times \mathbf{R}^n\\<br /> (\mathbf{x}, \mathbf{y}) &amp;\mapsto \left( f(\mathbf{x},\mathbf{y}), \mathbf{y} \right).<br /> \end{align*} ;]In the above example, we're considering the case
[; m=n=1, ;]and we're considering the equivalence[; \mathbf{R}^1 \times \mathbf{R}^1 \simeq \mathbf{R}^2. ;]The advantage is that
[; g ;]now maps between two spaces of the same dimension, so one can often apply the Inverse Function Theorem. (In fact, this is a common way to deduce the Implicit Function Theorem from the Inverse Function Theorem, so you see this technique often enough that it's worth your time to remember it.)These exercises require stronger assumptions—i.e., continuous differentiability rather than mere continuity—but perhaps this'll at least be a bit more accessible because it doesn't invoke quite so much topology. Hope this helps, and good luck!
I second Michael Spivak's Calculus if you haven't done a proper analysis course before. It's a 100% rigorous treatment of calculus from first principles and is probably better thought of as "analysis in one dimension." I post on a subreddit of folks working through the book pretty frequently: /r/calculusstudygroup
After that, I like Kolmogorov and Fomin's Introductory Real Analysis and Walter Rudin's Principles of Mathematical Analysis.
There's also Michael Spivak's Calculus on Manifolds, which focuses purely on multi-variable calculus on manifolds. Torus calculus!
Strictly speaking it's "Analysis in Several Variables" and it uses the Spivak "Calculus on Manifolds" book.
http://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219/ref=sr_1_1?ie=UTF8&amp;qid=1314643509&amp;sr=8-1
I don't know of any video lectures that covers these topics, but I do know of a couple of good books that should be good resources to reference if you find Rudin a bit too terse in some places:
I would recommend a combination of 2 and 3 for the differential forms and stuff from Rudin, and 1 for single variable real analysis.
This is not the first time this situation has occurred. Prim's Algorithm was orignally discovered by a fellow named Jarnik in the 1930s. The Divergence Theorem was discovered and rediscovered by LaGrange, Gauss, Ostrogradsky and Green. Stokes theorem was first discovered by Lord Kelvin - at least according to the cover art of Calculus on Manifolds.
For single variable calculus, like everyone else I would recommend Calculus - Spivak. If you have already seen mechanical caluculus, mechanical meaning plug and chug type problems, this is a great book. It will teach you some analysis on the real line and get your proof writing chops up to speed.
For multivariable calculus, I have three books that I like. Despite the bad reviews on amazon, I think Vector Calculus - Marsden & Tromba is a good text. Lots of it is plug and chug, but the problems are nice.
One book which is proofed based, but still full of examples is Advanced Calculus of Several Variables - Edwards Jr.. This is a nice book and is very cheap.
Lastly, I would like to give a bump to Calculus on Manifolds - Spivak. This book is very proofed based, so if you are not comfortable with this, I would sit back and learn from of the others first.
You should check out Spivaks Calculus on Manifolds.
http://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219
Read the first chapter or 2 and see how you like it, if you feel overwhelmed check some of the other recommendations out.
It is however a good book, and you should read it sooner or later.
Rudins principles of mathematical analysis is also excellent, however it
is not strictly multi-dimensional analysis.
Read at least chapter 2 and 3, they lay a very important groundwork.
I wouldn't bother with Apostol's Calculus. For analysis, you should really look at the first two volumes of Stein and Shakarchi's Princeton Lectures in Analysis.
Vol I: Fourier Analysis
Vol II: Complex Analysis
Then, you should pick up:
Munkres, Analysis on Manifolds or something similar, you could try Spivak's book but it's a bit terse. (on a personal note, I tried doing Spivak's book when I was a freshman. It was a big mistake).
In truth, most introductory undergrad analysis texts are actually more invested in trying to teach you the rigorous language of modern analysis than in expositing on ideas and theorems of analysis. For example, Rudin's Principles is basically to acquaint you with the language of modern analysis -- it has no substantial mathematical result. This is where the Stein Shakarchi books really shines. The first book really goes into some actual mathematics (fourier analysis even on finite abelian groups and it even builds enough math to prove Dirichlet's famous theorem in Number Theory), assuming only Riemann Integration (the integration theory taught in Spivak).
For Algebra, I'd suggest you look into Artin's Algebra. This is truly a fantastic textbook by one of the great modern algebraic geometers (Artin was Grothendieck's student and he set up the foundations of etale cohomology).
This should hold you up till you become a sophomore. At that point, talk to someone in the math department.
Would probably have to say Calculus on Manifolds by Spivak.
I think you were looking for things that weren't necessarily textbooks, but I think this book is still popular...amongst analysis courses.