What is Differential forms: Definition and 135 Discussions
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.
For instance, the expression f(x) dx from one-variable calculus is an example of a 1-form, and can be integrated over an oriented interval [a, b] in the domain of f:
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{\displaystyle \int _{a}^{b}f(x)\,dx.}
Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz is a 2-form that has a surface integral over an oriented surface S:
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{\displaystyle \int _{S}(f(x,y,z)\,dx\wedge dy+g(x,y,z)\,dz\wedge dx+h(x,y,z)\,dy\wedge dz).}
The symbol ∧ denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a 3-form f(x, y, z) dx ∧ dy ∧ dz represents a volume element that can be integrated over an oriented region of space. In general, a k-form is an object that may be integrated over a k-dimensional oriented manifold, and is homogeneous of degree k in the coordinate differentials.
The algebra of differential forms is organized in a way that naturally reflects the orientation of the domain of integration. There is an operation d on differential forms known as the exterior derivative that, when given a k-form as input, produces a (k + 1)-form as output. This operation extends the differential of a function, and is directly related to the divergence and the curl of a vector field in a manner that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the generalized Stokes theorem. In a deeper way, this theorem relates the topology of the domain of integration to the structure of the differential forms themselves; the precise connection is known as de Rham's theorem.
The general setting for the study of differential forms is on a differentiable manifold. Differential 1-forms are naturally dual to vector fields on a manifold, and the pairing between vector fields and 1-forms is extended to arbitrary differential forms by the interior product. The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds. This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided that the information is expressed in terms of differential forms. As an example, the change of variables formula for integration becomes a simple statement that an integral is preserved under pullback.
I'm completely new to differential forms so I am having trouble following the arguments from the following post: https://physics.stackexchange.com/questions/555668/integrating-over-non-trivial-fiber-bundles-chern-simons-theory
Specifically:
1.) In equation (12) of the accepted answer, where did...
Hi, I'm studying the concept of contact manifold -- Contact geometry
A related theorem is Darboux's theorem for one-forms -- Darboux theorem
In the particular case of one-form ##\theta \neq 0## such that ##d\theta## has constant rank 0 then if ##\theta \wedge (d\theta)^0 \neq 0## there exists a...
Hi, I'd like to ask for clarification about the definition of differential of a smooth scalar function ##f: M \rightarrow \mathbb R## between smooth manifolds ##M## and ##\mathbb R##.
As far as I know, the differential of a scalar function ##f## can be understood as:
a linear map ##df()##...
We know that a metric tensor raises or lowers the indices of a tensor, for e.g. a Levi-Civita tensor. If we are in ##4D## spacetime, then
\begin{align}
g_{mn}\epsilon^{npqr}=\epsilon_{m}{}^{pqr}
\end{align}
where ##g_{mn}## is the metric and ##\epsilon^{npqr}## is the Levi-Civita tensor.
The...
The equality is implied in the move from equation 15.43 line 2 to line 3.
I do find Dray's book is admirably clear and absolutely says something I wish to understand, but my 78 year old brain has difficulty. However, in this case I can be precise about where I fail to follow.
Oh! I find...
Why are n-forms called differential forms? What is differential about them? And why was the dx notation adopted for them? It must have something to do with the differential dx in calculus. But dx in calculus is an infinitesimal quantity. I don't see what n-forms have to do with infinitesimal...
Hi,
I am missing the point about the application of Darboux theorem to symplectic manifold case as explained here Darboux Theorem.
We start from a symplectic manifold of even dimension ##n=2m## with a symplectic differential 2-form ##w## defined on it. Since by definition the symplectic 2-form...
First time looking at differential forms. What is the difference of the forms over R^n and on manifolds? Does the exterior product and derivative have different properties? (Is it possible to exaplain this difference without using the tangent space?)
Good Morning
To cut the chase, what is the dx in an integral?
I understand that d/dx is an "operator" on a function; and that one should never split, say, df, from dx in df/dx
That said, I have seen it in an integral, specifically for calculating work.
I do understand the idea of...
I am confused as to how exactly we integrate differential forms. I know how to integrate them in the sense that I can perform the computations and I can prove statements, but I don't understand how it makes sense. Let's integrate a 1-form over a curve for example:
Let ##M## be a smooth...
I am test my knowledge of differential forms and obviously I am missing something because I can't figure out where I am going wrong here:
Let ##C## denote the positively oriented half-circle of radius ##r## parametrized by ##(x,y) = (r \cos t, r \sin t)## for ##t \in (0, \pi)##. The value of...
I read in the book Gravitation by Wheeler that "Any tensor can be completely symmetrized or antisymmetrized with an appropriate linear combination of itself and it's transpose (see page 83; also this is an exercise on page 86 Exercise 3.12).
And in Topology, Geometry and Physics by Michio...
There are a few different textbooks out there on differential geometry geared towards physics applications and also theoretical physics books which use a geometric approach. Yet they use different approaches sometimes. For example kip thrones book “modern classical physics” uses a tensor...
In college I learned Maxwell's equations in the integral form, and I've never been perfectly clear on where the differential forms came from. For example, using \int _{S} and \int _{V} as surface and volume integrals respectively and \Sigma q as the total charge enclosed in the given...
Hello!
I was not quite sure about posting in this category, but I think my question fits here.
I am wondering about Maxwell equations in vacuum written with differential forms, namely:
\begin{equation} \label{pippo}
dF = 0 \qquad d \star F = 0
\end{equation}
I know ##F## is a 2-form, and It...
So I heard a k-form is an object (function of k vectors) integrated over a k-dimensional region to yield a number. Well what about integrals like pressure (0-form?)over a surface to yield a vector? Or the integral of gradient (1-form) over a volume to yield a vector?
In particular I’m...
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am reading Chapter 13: Differential Forms ... ... and am currently focused on Section 13.1 Tensor Fields ...
I need some help in order to fully understand some statements by Browder in Section 13.1 ... ...
Andrew Browder in his book: "Mathematical Analysis: An Introduction" ... ... defines a differential form in Section 13.1 which reads as follows:
In the above text from Browder we read the following:
" ... ... A differential form of degree ##r## (or briefly an ##r##-form) in ##U## is a...
I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ...
I need some help in order to fully understand some statements of Shifrin at the start of Chapter 8, Section 2 on the dual space ...
The relevant text from...
Homework Statement
This problem is from V.I Arnold's book Mathematics of Classical Mechanics.
Q) Show that every differential 1-form on line is differential of some function
Homework Equations
The differential of any function is
$$df_{x}(\psi): TM_{x} \rightarrow R$$
The Attempt at a Solution...
Homework Statement
Suppose that a smooth differential ##n-1##-form ##\omega## on ##\mathbb{R}^n## is ##0## outside of a ball of radius ##R##. Show that $$
\int_{\mathbb{R}^n} d\omega = 0.
$$
Homework Equations
[/B]
$$\oint_{\partial K} \omega = \int_K d\omega$$
The Attempt at a Solution...
In the exercises on differential forms I often find expressions such as $$
\omega = 3xz\;dx - 7y^2z\;dy + 2x^2y\;dz
$$ but this is only correct if we're in "flat" space, right?
In general, a differential ##1##-form associates a covector with each point of ##M##. If we use some coordinates...
Here's exercise 1 of chapter 2 in Flanders' book.
Let ##L## be an ##n##-dimensional space. For each ##p##-vector ##\alpha\neq0## we let ##M_\alpha## be the subspace of ##L## consisting of all vectors ##\sigma## satisfying ##\alpha\wedge\sigma=0##. Prove that ##\dim(M_\alpha)\leq p##. Prove also...
I'm reading section 2.7 of Flanders' book about differential forms, but I have some doubts.
Let ##\lambda## be a ##p##-vector in ##\bigwedge^p V## and let ##\sigma^1,\ldots,\sigma^n## be a basis of ##V##. There's a unique ##*\lambda## such that, for all ##\mu\in \bigwedge^{n-p}##,$$
\lambda...
The question provides the vector field (xy, 2yz, 3zx) and asks me to confirm Stokes' theorem (the vector calc version) but I am trying to use the generalized differential forms version. So, I am trying to integrate \omega = xy\,dx + 2yz\,dy + 3zx\,dz along the following triangular boundary...
Hi.
Can anyone recommend a text introducing differential forms along with all the necessary pre-requisites for understanding them? For example, I'm not really familiar with tensor calculus but would like to shortcut studying it completely separately to learning differential forms. If that's too...
I've been trying to get a meaningful understanding of the benefits of using differential forms. I've seen examples of physics formulas that are reduced to a very simple declarative form relative to their tensor counterparts. However to me it just seems like a notation change to implied tensor...
Hello there,
I had some questions regarding k-forms. I was looking in the wiki page of differential forms(https://en.wikipedia.org/wiki/Differential_form) and noticed that it was was introduced to perform integration independent of the co-ordinates. I am not clear how? Is this because given a...
In undergraduate dynamics, they do things like this:
--------------------
v = ds/dt
a = dv/dt
Then, from this, they construct: a ds = v dv
And they use that to solve some problems.
--------------------
Now I have read that it is NOT wise to treat the derivative like a fraction: it obliterates...
In a text I am reading (that I unfortunately can't find online) it says:
"[...] differential forms should be thought of as the basis of the vector space of totally antisymmetric covariant tensors. Changing the usual basis dx^{\mu_1} \otimes ... \otimes dx^{\mu_n} with dx^{\mu_1} \wedge ...
Hi was reading about differential forms, when I tried to solve the example given in this pdf https://www.rose-hulman.edu/~bryan/lottamath/difform.pdf. According to it, the answer is that on the image above. But when I tried to solve this same example by following the expression for ##w## given...
Hi everyone. In reading some popular textbooks I noticed that in (maybe) most of GR and SR we don't encounter situations where we can use wedge-product and differential forms. However, these things are presented to us in most of the textbooks. But... if most of the books present them, it means...
I have seen and gone through this thread over and over again but still it is not clear.
https://www.physicsforums.com/threads/vectors-one-forms-and-gradients.82943/
The gradient in different coordinate systems is dependent on a metric
But the 1-form is not dependent on a metric. It is a...
(I am a mechanical engineer, trying to make up for a poor math education)'
I understand that:
A CLOSED form is a differential form whose exterior derivative is 0.0.
An EXACT form is the exterior derivative of another form.
And it stops right there. I am teaching myself differential forms...
So I understand that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω.
And I understand that one can pull back the integral of a 1-form over a line to the line integral between the...
Hello, does anyone have reference to(or care to write out) fully rigorous proof of Stokes theorem which does not reference Differential Forms? I'm reviewing some physics stuff and I want to relearn it.
I honestly will never use the higher dimensional version but I still want to see a full proof...
Some time ago I was looking around the web for the use of differential equations in General Relativity. Then I found a definition (below) of differential forms, but I noted that the definition on my book is different from this one. Could someone tell me if it is right?
The differential form of a function is
\partial{f(x^1,...,x^n)}=\frac{\partial{f(x^1,...,x^n)}}{\partial{x^1}}dx^1+...+\frac{\partial{f(x^1,...,x^n)}}{\partial{x^n}}dx^n
Is there (especially in General Relativity) differential of higher orders, like \partial^2{f(x^1,...,x^n)}? If so, how is it...
Homework Statement
Let ##V^{3}(t)## be a compact region moving with the fluid.
Assume that at ##t=0## the vorticity ##2##-form ##\omega^{2}## vanishes when restricted to the boundary ##\partial V^{3}(0)##; that is, ##i^{*}\omega^{2}=0##, where ##i## is the inclusion of ##\partial V## in...
Homework Statement
Euler's equations can be written using vector calculus as
##\displaystyle{\frac{\partial v_{i}}{\partial t}+v^{j}\left(\frac{\partial v_{i}}{\partial x^{j}}\right) = -\left(\frac{1}{\rho}\right)\frac{\partial p}{\partial x^{i}}+f_{i}}.##
Euler's equations can also be...
The electromagnetic action can be written in the language of differential forms as
##\displaystyle{S=-\frac{1}{4}\int F\wedge \star F.}##
The electromagnetic action can also be written in the language of vector calculus as
$$S = \int \frac{1}{2}(E^{2}+B^{2})$$
How can you show the...
Let
##0##-form ##f =## function ##f##
##1##-form ##\alpha^{1} =## covariant expression for a vector ##\bf{A}##
Then consider the following dictionary of symbolic identifications of expressions expressed in the language of differential forms on a manifold and expressions expressed in the...
Consider a curve ##C:{\bf{x}}={\bf{F}}(t)##, for ##a\leq t \leq b##, in ##\mathbb{R}^{3}## (with ##x## any coordinates). oriented so that ##\displaystyle{\frac{d}{dt}}## defines the positive orientation in ##U=\mathbb{R}^{1}##. If ##\alpha^{1}=a_{1}dx^{1}+a_{2}dx^{2}+a_{3}dx^{3}## is a...
Suppose we are given this definition of the wedge product for two one-forms in the component notation:
$$(A \wedge B)_{\mu\nu}=2A_{[\mu}B_{\nu]}=A_{\mu}B_{\nu}-A_{\nu}B_{\mu}$$
Now how can we show the switch from tensor products to wedge product below...
Hello,
I would like to know if anybody here has used the book "Differential Forms and the Geometry of General Relativity" by Tevian Dray and how they found it.
Thanks!
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...