Calculus on manifolds Definition and 34 Threads

In this lecture, the lecturer claims that the ##C^{\infty}##-module of smooth vector fields defined on a smooth manifold can lack to admit a basis (not even infinite dimensional). Indeed the set of smooth vector fields can be given an (infinite dimensional) vector space structure over the field...

As follow up of this thread in Special and General Relativity subforum, I'd like to better investigate the following topic. Consider the 4d euclidean space in which there are 10 ##\mathbb R##-linear independent KVFs. Their span at each point is 4 dimensional (i.e. at any point they span the...

A spherically symmetric manifold has, by definition, a set of 3 independent Killing Vector Fields (KVFs) satisfying: $$\begin{align}[R,S] &=T \nonumber \\ [S,T] &=R \nonumber \\ [T,R] &=S \nonumber \end{align}$$ These 3 KVFs define a linear subspace of the (infinite dimensional) vector space...

Consider the expression $$T(\omega, X, Y) := \omega \nabla_X Y$$ where ##\omega,X,Y## are a covector field and two vector fields respectively. Is ##T## a (1,2) tensor field ? From my understanding the answer is negative. The point is that ##T(\,. , \, . , \, .)## is not ##C^{\infty}##-linear...

Consider the following 1-form ##\omega## defined on ##U = \mathbb R^2 \, \backslash \{ 0 \}##: $$\omega = \frac {y} {x^2 + y^2} dx + \frac {-x} {x^2 + y^2} dy$$ It is closed on ##U## since ##\partial \left (\frac {y} {x^2 + y^2} \right) / \partial y = \partial \left (\frac {-x} {x^2 + y^2}...

TL;DR Summary: In search of a suitable topic for an interesting undergraduate dissertation. I am a final year Mathematics and Computing undergraduate. I am expected to submit an extensive B.Sc. thesis in four months. I have previously studied multivariable calculus, differential fields and...

I am a final year Mathematics and Computing undergraduate. I am expected to submit an extensive B.Sc. thesis in four months. I have previously studied multivariable calculus, differential forms, chains, and a little bit of Theory of manifolds (Calculus on Manifolds, Michael Spivak). I am...

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...

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...

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...

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...

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...

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...

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...

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"...

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...

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...

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...

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...

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".

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...

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.

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

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...

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...

What are some preliminary texts/knowledge before approaching: 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...

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...

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...

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...

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...

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...

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...

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...