# What is Calculus on manifolds: Definition and 30 Discussions

Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965) by Michael Spivak is a brief, rigorous, and modern textbook of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates.

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1. ### I Differential k-form vs (0,k) tensor field

Hi, I would like to ask for a clarification about the difference between a differential k-form and a generic (0,k) tensor field. Take for instance a (non simple) differential 2-form defined on a 2D differential manifold with coordinates ##\{x^{\mu}\}##. It can be assigned as linear combination...
2. ### I Motivating definitions in calculus on manifolds

Hi I am a person who always have had a hard time picking up new definitions. Once I do, the rest kinda falls into place. In the case of abstract algebra, Stillwell's Elements of Algebra saved me. However, in the case of Spivak's Calculus on Manifolds, I get demotivated when I get to concepts...
3. ### I Notation in Spivak's Calculus on Manifolds

I have a question regarding the usage of notation on problem 2-11. Find ##f'(x, y)## where ## f(x,y) = \int ^{x + y} _{a} g = [h \circ (\pi _1 + \pi _2 )] (x, y)## where ##h = \int ^t _a g## and ##g : R \rightarrow R## Since no differential is given, what exactly are we integrating with...
4. ### A Integrating the topics of forms, manifolds, and algebra

Hello, As you might discern from previous posts, I have been teaching myself: Calculus on manifolds Differential forms Lie Algebra, Group Push forward, pull back. I am an engineer approaching this late in life and with a deficient background in math. It is all coming together and I almost...
5. ### Geometry Spivak's differential geometry vs calculus on manifolds

Hi, I am just about to finish working through the integration chapter of calculus on manifolds, and I am wondering whether it would be better to get spivaks first volume of differential geometry (or another book, recommendations?) and start on that, or to finish calculus on manifolds first...
6. ### Calculus Theoretical Multivariable Calculus books

Dear Physics Forum advisers, Could you recommend books that treat the multivariable calculus from a theoretical aspect (and applications too, if possible)? I have been reading Rudin's PMA and Apostol's Mathematical Analysis, but their treatment of vector calculus is very confusing and not...
7. ### Spivak's Calculus on Manifolds: Theorem 5-3

I am trying to finish the last chapter of Spivak's Calculus on Manifolds book. I am stuck in trying to understand something that seems like it's supposed to be trivial but I can't figure it out. Suppose M is a manifold and \omega is a p-form on M. If f: W \rightarrow \mathbb{R}^n is a...
8. ### Calculus on Manifolds: Lie Algebra, Lie Groups & Exterior Algebra

So this is beginning to feel like the beginning of the 4th movement of Beethoven's Ninth: it is all coming together. Manifolds,Lie Algebra, Lie Groups and Exterior Algebra. And now I have another simple question that is more linguistic in nature. What does one mean by "Calculus on Manifolds"...
9. ### Should I Study Functional Analysis or Calculus on Manifolds?

I have the opportunity to pursue an independent study in functional analysis (using Kreyszig's book) or calculus on manifolds (using Tu's book) next semester. I think that both of the subjects are interesting and I would like to study them both at some point in my life, but I can only choose one...
10. ### Spivak Calculus on Manifolds

I have one question about Spivak's Calculus on Manifolds book. I have not learned directional derivatives and understand that these are left as exercises in his book, which would make one think these are not that important whereas he focuses on total derivatives or what you may name them...
11. ### Calculus Spivak Calculus on Manifolds and Epsilon delta proofs

I am currently having some issue understanding, what you may find trivial, epsilon-delta proofs. I have worked through Apostol Vol.1 and ran through Spivak and I found Apostol just uses neighborhoods in proofs instead of the epsilon-delta approach, while nesting neighborhoods is 'acceptable' I...
12. ### Calculus on Manifolds: Meaning & Benefits | Mechanical Engineer

Hello, I am a mechanical engineer and I am teaching my self the topic of this subject line. I now have a working understanding of the following: manifolds, exterior algebra, wedge product and some other issues. (I give you this and the next sentence so I can CONTEXTUALIZE my question.) I...
13. ### What is considered Calculus on Manifolds?

One can do calculus on a differentiable manifold, what does that mean? Does it mean you can use differential forms on the manifold, or that you can find tangent vectors, What is certified as "calculus on a manifold".
14. ### Analysis Calculus on Manifolds by Spivak

Author: Michael Spivak Title: Calculus on Manifolds Amazon link: https://www.amazon.com/dp/0805390219/?tag=pfamazon01-20 Prerequisities: Rigorous Calculus Level: Undergrad Table of Contents: Foreword Preface Functions on Euclidean Space Norm and Inner Product Subsets of Euclidean...
15. ### Supplement for spivak's calculus on manifolds

im trying to read calculus on manifolds by michael spivak and am having a tough time with it. if anyone could recommend a more accessible book (perhaps one with solved problems) id really appreciate it.
16. ### Calculus on manifolds - limit of an integral

Homework Statement Let \phi \in C^{\infty}_{0}(\mathbb{R}^2) and f: \mathbb{R}^2 \to \mathbb{R} a smooth, non-negative function. For c > 0, let < F_c, \phi > := \int_{\{f(x,y) \le c\}} \phi(x,y)\mbox{dx dy} . Supposing the gradient of \frac{\partial f}{\partial x} is nonzero everywhere on M...
17. ### Spivak calculus on manifolds

I hope this is not the wrong place to ask this... Can anybody tell me if it is possible to find "Spivak calculus on manifolds" on line (a PDF copy for example) Thanks
18. ### Spivak (Calculus on Manifolds) proof of stolkes theorem

http://planetmath.org/?op=getobj&from=objects&id=4370 that's pretty much the proof of Stolkes Theorem given in Spivak but I'm having a lot of difficulty understanding the details specifically...when the piecewise function is defined for j>1 the integral is 0 and for j=1 the integral is...
19. ### Differentiation on Euclidean Space (Calculus on Manifolds)

Homework Statement This is from Spivak's Calculus on Manifolds, problem 2-12(a). Prove that if f:Rn \times Rm \rightarrow Rp is bilinear, then lim(h, k) --> 0 \frac{|f(h, k)|}{|(h, k)|} = 0 Homework Equations The definition of bilinear function in this case: If for x, x1, x2...
20. ### Spivak Calculus on Manifolds

Homework Statement Given a Jordan-measurable set in the yz-plane, use Fubini's Thm to derive an expression for the volume of the set in R3 obtained by revolving the set about the z-axisHomework Equations The Attempt at a Solution I solved this problem very easily using change of variable...
21. ### Spivak's calculus on manifolds

I am currently working through spivak's calculus on manifolds (which i love by the way) in one of my class. my question is about his notation for partial derivatives. i completely understand why he uses it and how the classical notation has some ambiguity to it. however, i can't help but...
22. ### Spivak's Calculus on Manifolds?

What are some preliminary texts/knowledge before approaching: Spivak's Calculus on Manifolds?
23. ### Integration on chains in Spivak's calculus on manifolds

I would like to discuss this chapter with someone who has read the book. From looking at other books, I realize that Spivak does things a little differently. He seems to be putting less structure on his chains (for instance, no mention of orientation, no 1-1 requirement and so on), and as a...
24. ### Check my work (Spivak problem in Calculus on Manifolds)

Problem: given compact set C and open set U with C \subsetU, show there is a compact set D \subset U with C \subset interior of D. My thinking: Since C is compact it is closed, and U-C is open. Since U is an open cover of C there is a finite collection D of finite open subsets of U that...
25. ### Understanding Spivak's "Calculus on Manifolds" - Ken Cohen's Confusion

Working through Spivak "Calculus on Manifolds." On p. 7, he explains that "the interior of any set A is open, and the same is true for the exterior of A, which is, in fact, the interior of R\overline{}n-A." Later, he says "Clearly no finite number of the open sets in O wil cover R or, for...
26. ### Typo in spivak's calculus on manifolds?

In the first problem set of chapter 1, problem 1-8(b) deals with angle preserving transformations. In the newest edition of the book the problem is stated If there is a basis x_1, x_2, ..., x_n and numbers a_1, a_2, ..., a_n such that Tx_i = a_i x_i, then the transformation T is angle...
27. ### Spivak's Calculus on Manifolds problem (I). Integration.

Homework Statement If A\subset\mathbb{R}^{n} is a rectangle show tath C\subset A is Jordan-measurable iff \forall\epsilon>0,\, \exists P (with P a partition of A) such that \sum_{S\in S_{1}}v(S)-\sum_{S\in S_{2}}v(S)<\epsilon for S_{1} the collection of all subrectangles S induced by P such...
28. ### Taking a course in calculus on manifolds.

im thinking of taking in 2008 the second semester a course in analysis of manifolds. now some of the preliminaries although not obligatory, are differnetial geometry and topology, i will not have them at that time, so i think to learn it by my own, will baby rudin and adult rudin books will...
29. ### Definition of cross product in Spivak's 'Calculus on Manifolds'

Homework Statement In Calculus on Manifold pp.83-84, Spivak writes that "if v_1,...,v_{n-1} are vectors in R^n and f:R^n-->R is defined by f(w)=det(v_1,...,v_{n-1},w), then f is an alternating 1-tensor on R^n; therefore there is a unique z in R^n such that <w,z>=f(w) (and this z is denoted v_1...
30. ### Spivak calculus on manifolds solutions? (someone asked this b4 and got ignored)

Does anyone know if there's worked out solution to the problems in spivak's calculus on manifolds? It's awfully easy to get stuck in the problems and for some of them I don't even know where to start... Also, if there isn't any, any good problem and 'SOLUTION' source for analysis on manifolds...