Differential forms Definition and 137 Threads

Hi all, (Thank you for the continuing responses to my other questions...) I am gaining more and more understanding of differential forms and differential geometry. But now I must ask... Why the words? I understand the exterior derivative, but why is it called "exterior?" Ditto for CLOSED and...

In elementary calculus (and often in courses beyond) we are taught that a differential of a function, ##df## quantifies an infinitesimal change in that function. However, the notion of an infinitesimal is not well-defined and is nonsensical (I mean, one cannot define it in terms of a limit, and...

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'm trying to self-study differential geometry and have come across interior products of vectors and differential forms. I will use brackets to show the interior product and I would just like to check I am understanding something correctly. Do I need to manipulate the differential form to...

What do Physics Forums members regard as the best first introduction to differential forms ...?

Can anyone suggest a good text or a good online set of notes from which to make a first approach to the topic of differential forms ... ? Similarly a first approach to to tensors ... ? The thought is to use these notions in order to gain an understanding of differential geometry and ... later...

this starts as a calculus question, but springs into where i can get help with david bachman's A GEOMETRIC APPROACH TO DIFFERENTIAL FORMS second edition. looking at paul's notes cheat sheets http://tutorial.math.lamar.edu/cheat_table.aspx we have## \int \frac{1}{\sqrt{a^{2}-x^{2}}} =...

The usual form for tension as a result of the symmetric Cauchy stress tensor is, $$\mathbf{t} = P \mathbf{\hat{n}}$$ or better $$t_i = {P_i}^j n_j$$ Buoyancy would be $$T = \int_{\partial V}{P_i}^j n_j da$$ integrated over a closed surface. I've assumed that the stress tensor ##P##, is, in...

If we have,$$A=d[(\bar{\alpha}-\alpha)(dt+\lambda)]$$ where $$\alpha$$ is a complex function and $$\lambda$$ is a 1-form. t here represents the time coordinate. If we want to calculate $$d\star A=0$$ where $$\star$$ is hodge star, we get if I did my calculations correctly...

Homework Statement Find the value of the 2-form dxdy+3dxdz on the oriented triangle with (0,0,0) (1,2,3) (1,4,0) in that order. Homework EquationsThe Attempt at a Solution I have tried various subtraction of these coordinates and applying them to the formula but the answer is in the back of...

The flat-space source-free Maxwell equations can be written in terms of differential forms as $$d F = 0; \ \ d \star F = 0.$$ And in the theory of gauge fields, one can introduce a connection one-from A from which one can formulate general Maxwell equations (for Yang-Mills fields) by $$ dF + A...

I have been reading a lot about Differential Forms lately because its so sexy. I have a pretty good grasp of how wedge product, hodge star, and differential operator "d" work, and their application to physics (it took me some time to see how d*F=J). I want to continue reading about it because...

Be a vector field \vec{F}=(f_1,f_2,f_3) and \omega^k_{\vec{F}} the k-form associated with it , i know if i do \int \omega^1_{\vec{F}} is the same of a line integral and \int \omega^2_{\vec{F}} i obtain the same result of \int \int_S \vec{F}\cdot d\vec{S}, which is the flux of a vector field in a...

After read this stretch https://en.wikipedia.org/wiki/Closed_and_exact_forms#Vector_field_analogies, my doubts increased exponentially... 1. A scalar field correspond always to a 0-form? 1.1. The laplacian of 0-form is a 2-form? 1.2. But the laplacian of sclar field is another scalar field...

Hi, All: Just curious if anyone knows of any online or otherwise software to help compute the wedge of forms, or maybe some method to help simplify. Not about laziness; I don't have that much experience, and I want to double check; I have around 30 terms ( many of which may cancel out) , and...

Hey guys, I am wondering whether there is any book out there that approaches EM field using differential form and on the same or more advanced than Jackson, I have a solid knowledge of differential form and algebraic topology, thanks :D

Suppose we have a curve, formed by a function f that maps real numbers to real numbers, such that f is everywhere smooth over a subset D of its domain. Let's suppose that, for all x in D, there is a vector space that contains all vectors tangent to the curve at that point, called the tangent...

Hello! I think I got something wrong here, maybe someone can help me out. Lets consider a n-manifold. A differential n-form describing a signed volume element will then transform as: f(x^i) dx^1 \wedge dx^2 \wedge \cdots \wedge dx^n = f(y^i) \;\text{det}\left( \frac{\partial x^i}{\partial...

Author: John Hubbard, Barbara Hubbard Title: Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach Amazon Link: https://www.amazon.com/dp/0971576653/?tag=pfamazon01-20

Homework Statement I have to take the curved space - time homogenous and inhomogeneous maxwell equations, \triangledown ^{a}F_{ab} = -4\pi j_{b} and \triangledown _{[a}F_{bc]} = 0, and show they can be put in terms of differential forms as dF = 0 and d*F = 4\pi *j (here * is the hodge dual...

Hello, I have a somewhat conceptual question about differential forms. I have been studying differential forms off and on for some time now and things are starting to come together for me. However, there is an irritating gap in my understanding. Regarding the geometric significance or...

Homework Statement This is not actually a homework but a personal work. Here it is: Using the differential forms: F=\tfrac{1}{2!}{{F}_{\mu \nu }}d{{x}^{\mu }}\wedge d{{x}^{\nu }} and J=\tfrac{1}{3!}{{J}^{\mu }}{{\varepsilon }_{\mu \alpha \beta \gamma }}d{{x}^{\alpha }}\wedge d{{x}^{\beta...

Author: Raoul Bott, Loring Tu Title: Differential Forms in Algebraic Topology Amazon Link: https://www.amazon.com/dp/1441928154/?tag=pfamazon01-20 Prerequisities: Differential Geometry, Algebraic Topology Level: Grad Table of Contents: Introduction De Rham Theory The de Rham Complex...

So say I have a n-1 form \sum^{n}_{i=1}x^{2}_{i}dx_{1}...\widehat{dx_{i}}...dx_{n} and I want to find the exterior derivative, how do I know where to put which partial derivative for each term, would it simply be?? \sum^{n}_{i=1}...

Hi, I understand the concept of the differential of a (differentiable) function at a point as a linear transformation that "best" approximates the increment of the function there. So for example the differential of a function f : D \subseteq \mathbb{R}^2 \to \mathbb{R} could maybe be df = 8...

I have been working through Spivak's fine book, but the part about differential forms and tangent spaces has left me confused. In particular, Spivak defines the Tangent Space \mathbb R^n_p of \mathbb R^n at the point p as the set of tuples (p,x),x\in\mathbb R^n. Afterwards, Vector fields are...

I'm having problems in my differential geometry class. Does anyone know of a good tutorial or set of notes? The 1-form, 2-form, 3-form stuff is confusing and so is the wedge product. Anyways, here's the problem. After some algebra I arrived at my answer, but I'm unsure of how to incorporate...

Hello everybody, This is my first time on Physics forums. I am a sophomore in high school who LOVES math. I have lots of free time this summer and would like to learn multivariable calculus and/or linear algebra (whichever is a prerequisite for the other, depending on the textbook I choose)...

Let me preface by saying I am a physics major. So I am coming at differential forms from the perspective of physics, i.e. work, flows, em fields, etc. My question is this. My understanding is that a basic 1-form dx, dy, or dz takes a vector v = (v1,v2,v3) and gives back the corresponding...

So about a hundred years ago there was a live (sort of) differential forms thread hosted by someone named Lethe that was really helpful but short-lived. There have been some other diffl forms threads, too, such as the one centered on Bachman's book, but they all seem to peter out without any...

Can someone recommend a good textbook for learning differential forms for someone with an understanding of calculus at the level of Spivak? Thanks.

Hello Everyone, I'm currently working through a differential geometry book that uses Clifford's algebra instead of differential forms. If anybody has knowledge of both, would you please explain what the differences between the approaches are, and what (if any) are the advantages of each...

A sphere has charge density \rho=k\cdot r. Using the integral form of Gauss's Law, one easily finds that the electric field is E=\frac{k\cdot r^2}{4\epsilon} anywhere inside the sphere. However, \nabla\cdot E=\frac{k\cdot r}{2\epsilon}, which is half of what should be expected from the...

Homework Statement Show that exterior differentiation of a 0-form f on R3 is essentially the same as calculating the gradient of f. The Attempt at a SolutionLet U be a differentiable 0-form on R3. I think dU = \sum _{j=1} ^n \frac{δF_I}{δx_j}dx_j dx_IHowever, since U is a 0-form, I can...

I'm reading Flanders' Differential Forms with Applications to the Physical Sciences and I have some issues with problems 2 and 3 in chapter 3, which appear to ask the reader to compute the pullback a mapping from X to Y applied to a form over X, and I'm not sure how to interpret such a thing...

Lee 2003: Introduction to Smooth Manifolds ( http://books.google.co.uk/books?id=eqfgZtjQceYC&printsec=frontcover#v=onepage&q&f=false ) (search eg. for "computational"), Lemma 12.10 (b), p. 304: where I is an increasing multi-index: (i_1,...i_k) with each value less than or equal to all those...

Homework Statement Here is my problem http://i51.tinypic.com/34dihx5.jpg However my teacher had some suggestions of solving this problem in a nice mathematical way. Here is the plan of solving which I would like to follow 1) find the Hodge dual to f 2) compute df 3) is f exact...

*Bit of reading involved here, worth it if you have any interest in, or knowledge of, differential forms*. It took me quite a while to find a good explanation of differential forms & I finally found something that made sense, in a sense. Most of what I've written below is just asking you...

Homework Statement Show that d\omega_{ij}+\sum_{k=1}^{n} \omega_{ik}\wedge\omega_{kj} =0 Homework Equations Let G be the group of invertible nxn matrices. This is an open set in the vector space M=Mat(n\times n, R) and our formalism of differential forms applies there with the...

In my reference books differential forms are integrated by means of pullbacks. Actually, integrals of differential forms are DEFINED by means of pullbacks. In other words, the integration domain must have a parametrization. Since differential forms and their integrals are under regularity...

I made a post titled the same thing but it didnt seem to show up for some reason so if i am just reposting this over again i apologize. I recently got the book A geometrical approach to differential forms by David Bachman. At the moment the biggest issue i am having is just visualizing what...

Hi everyone, Homework Statement I've been studying a paper in which there is a connection given by, A = f(r)\sigma_1 dx+g(r)\sigma_2 dy, where \sigma's are half the Pauli matrices. I need to calculate the field strength, F = dA +[A,A]. Homework Equations A = f(r)\sigma_1...

Hi everyone, I've been studying a paper in which there is a connection given by, A = f(r)\sigma_1 dx+g(r)\sigma_2 dy, where \sigma's are half the Pauli matrices. I need to calculate the field strength, F = dA +[A,A]. I have computed it, but a factor is given me problems. I would say, dA =...

I was reading about dual spaces and dual bases in the book Linear Algebra by Friedberg, Spence and Insel (FSI) and they give an example of a linear functional, f_i (x) = a_i where [x]_β = [a_1 a_2 ... a_n] denotes the matrix representation of x in terms of the basis β = {x_1, x_2, ..., x_n} of...

As in the title , I recently somehow want to learn differential form , but ,actually , I do not really know where I should start , or what books I should read .. So,can anyone recommend some useful books ?

Can general relativity be constructed with differential forms?

Hey, A quick question. In the definition of a differential form, we normally require that they be sections of the k-th exterior power of the cotangent bundle. However, on page 14 of Jurdjevic's book on...

Hi, I had a silly idea that probably doesn't work, but I thought I'd ask about it anyway. I understand that vectors can be thought of as derivative operators, e.g. \frac{d}{d\lambda} = \frac{dx^\mu}{d\lambda} \partial_\mu, where lambda parametrizes some curve. I also gather that one-forms...

In math, differential forms are alternating: dx^dy=-dy^dx. But in physics, we seem to exchange the order freely: dxdy=dydx. What's going on? I am comfortable with an answer that involves tensors, differential geometry, physics, volume forms, etc. In fact, this is really something I should...

Hi all, I posted this awhile back in the homework sections of the forums and received only one reply, which suggested that I post it here instead, though I understand that it belongs in the homework section. The fundamental problem is not this particular exercise but about integration of...