What is Differential form: Definition and 57 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:





{\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:









{\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.

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  1. cianfa72

    I Frobenius theorem for differential one forms

    Hi, starting from this old PF thread I've some doubts about the Frobenius condition for a differential 1-form ##\omega##, namely that ##d\omega = \omega \wedge \alpha## is actually equivalent to the existence of smooth maps ##f## and ##g## such that ##\omega = fdg##. I found this About...
  2. cianfa72

    I The Road to Reality - exercise on scalar product

    Hi, I'm keep studying The Road to Reality book from R. Penrose. In section 12.4 he asks to give a proof, by use of the chain rule, that the scalar product ##\alpha \cdot \xi=\alpha_1 \xi^1 + \alpha_2 \xi^2 + \dots \alpha_n \xi^n## is consistent with ##df \cdot \xi## in the particular case...
  3. L

    Vector field and differential form confusion

    Here is a picture of the solution I made : So my question is: Are these right and how do they differ from each other?
  4. Leo Liu

    Gauss' law in differential form

    My book claims that the diff. form of Gauss' law is $$\nabla\cdot\mathbf E=4\pi\rho$$ Can someone tell me why it isn't ##\nabla\cdot\mathbf E=\rho/\epsilon_0##?
  5. E

    Lie derivative of general differential form

    The first two parts I think were fine, I expressed the tensors in coordinate basis and wrote for the first part$$ \begin{align*} \mathcal{L}_X \omega = \mathcal{L}_X(\omega_{\nu} dx^{\nu} ) &= (\mathcal{L}_X \omega_{\nu}) dx^{\nu} + \omega_{\nu} (\mathcal{L}_X dx^{\nu}) \\ &= X^{\sigma}...
  6. K

    A Can we always rewrite a Tensor as a differential form?

    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...
  7. S

    I Differential form of the velocity equation in a non-standard configuration

    I'm reading a text on special relativity (Core Principles of Special and General Relativity), in which we start with the equation for composition of velocities in non-standard configuration. Frame ##S'## velocity w.r.t. ##S## is ##\vec v##, and the velocity of some particle in ##S'## is ##\vec...
  8. D

    Ampere's law in differential form

    According to this image, in the attached files there is the demonstration of the ampere's law in differential form. Bur i have some difficulties in understanding some passages. Probably I'm not understanding how to consider those two magnetic vectors oriented and why have different name. in...
  9. K

    Integral of a differential form

    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...
  10. M

    A How Does the Chain Rule Define the Differential of a Function on a Manifold?

    We define the differential of a function f in $$p \in M$$, where M is a submanifold as follows In this case we have a smooth curve ans and interval I $$\alpha: I \rightarrow M;\\ \alpha(0)= p \wedge \alpha'(0)=v$$. How can I get that derivative at the end by using the definitions of the...
  11. M

    A Differentiability of a function between manifolds

    Hello, let $$M^n \subset \mathbb{R}^N$$ $$N^k \subset \mathbb{R}^K$$ be two submanifolds. We say a function $$f : M \rightarrow N$$ is differentiable if and only if for every map $$(U,\varphi)$$ of M the transformation $$f \circ \varphi^{-1}: \varphi(U) \subset \mathbb{R}^N \rightarrow...
  12. Q

    I What are the insights into the Total Derivative formula?

    I’ve always been confused by the formula for the Total Derivative of a function. $$\frac{df(u,v)}{dx}= \frac{\partial f}{\partial x}+\frac{\partial f }{\partial u}\frac{\mathrm{d}u }{\mathrm{d} x}+\frac{\partial f}{\partial v}\frac{\mathrm{d}v }{\mathrm{d} x}$$ Any insight would be greatly...
  13. Drakkith

    I Quick Differential Form Question

    I've been going through my book learning about differential equations of multiple variables and I have a quick question about differential forms. If you are working a problem and get to the point where you're left with a differential form like ##(y)dx##, does that mean that the change in the...
  14. Drakkith

    I Question About Exact Differential Form

    My book is going through a proof on exact differential forms and the test to see if they're exact, and I'm lost on one part of it. It says: If $$M(x,y)dx + N(x,y)dy = \frac{\partial F}{\partial x}dx + \frac{\partial F}{\partial y}dy$$ then the calculus theorem concerning the equality of...
  15. G

    I Differential form of Gauss' law: All three terms the same value?

    Hi. Is the Maxwell equation $$\nabla\cdot\vec{E}=\frac{\rho}{\varepsilon_0}$$ even true in the stronger form $$\frac{\partial E_i}{\partial x_i}=\frac{\rho}{3\cdot\varepsilon_0}\enspace ?$$ I guess not, since I haven't found a source suggesting this. But shouldn't the isotropic electric field...
  16. D

    I Lie derivative of a differential form

    Hello, I have a maybe unusual question. In a paper, I recently found the equation $$\mathcal{L}_v(v_i dx^i) = (v^j \partial_j v_i + v_j \partial_i v^j) dx^i$$ Where v denotes velocity, x spatial coordinates and \mathcal{L}_v the Lie derivative with respect to v. Now I'm an undergraduate who...
  17. F

    I Prove what the exterior derivative of a 3-form is....

    I am trying to prove the following: $$3d\sigma (X,Y,Z)=-\sigma ([X,Y],Z)$$ where ##X,Y,Z\in\mathscr{X}(M)## with M as a smooth manifold. I can start by stating what I know so it is easier to see what I do wrong for you guys. I know that a general 2-form has the form...
  18. 1

    The restriction of differential form

    Assume M=xdy -ydx+dz ∈ Ω1(R^3). What's the restriction of M to the plane {z=2}? I think it's xdy-ydx. Is that right?
  19. 1

    A question about differential form

    Suppose x ∈ Ω^(n−1)(Rn \{0}) is closed and the integral of x on S^(n-1) equals to 1. I am stuck on how to show there does not exist an n − 1 form y ∈ Ω(n−1)(R^n) with y|R^n\{0} = x.
  20. M

    References for Self Study in de Rham Cohomology

    I've been trying to self study the section on de Rham cohomology in Guillemin and Pollack's book Differential Topology. The section is in a sense hands on: most of the results are presented as exercises scattered throughout the section, and some hints are given. I've hit a road block in a few of...
  21. D

    Electric field inside and around a hollow sphere

    Hi everyone, I am wondering if anybody could help me out. For my study I got the following question but I got stuck in part C (see image below). I Found at A that due to symmetry all components which are not in Ar direction will get canceled out I found at B that there is only charge density at...
  22. Q

    Gauss' law in differential form

    I'm trying to understand how the integral form is derived from the differential form of Gauss' law. I have several issues: 1) The law states that ## \nabla\cdot E=\frac{1}{\epsilon 0}\rho##, but when I calculate it directly I get that ## \nabla\cdot E=0## (at least for ## r\neq0##). 2) Now ##...
  23. M

    Differential Form - Notation Help

    Hi there, I was reading up on Holonomic constraints and came across this equation on the Wikipedia page: The page says it is a differential form. Can anyone explain the notation for me or provide a link or two to documents or pages which explain this notation? Thank you very much, Geoff
  24. V

    MHB Differential form surface integral

    Question: Evaluate the surface integral $$J = 2xzdy \land dz+2yzdz \land dx-{z}^{2}dx \land dy$$ where S \subset {\Bbb{R}}^{3} is the rectangle parametrised by: $$x(u,v) = 1-u,\ y(u,v) = u,\ z(u,v) = v,\ \ 0\le u, v \le 1$$ so far I have: \begin{array}{}x = u\cos v, &dx = \cos v\, du -...
  25. R

    Ampere's law in differential form

    Homework Statement A long cylindrical wire of radius R0 lies in the z-axis and carries a current density given by: ##j(r)= j_0 \left( \frac{r}{R_0} \right)^2 \ \hat{z} \ for \ r< R_0## ##j(r) = 0 \ elsewhere## Use the differential form of Ampere's law to calculate the magnetic field B inside...
  26. R

    Gauss's Law (Differential Form)

    Homework Statement Find the electric field inside and outside a sphere of radius R using the differential form of Gauss's law. Then find the electrostatic potential using Poisson's equation. Charge density of the sphere varies as ##\rho (r) = \alpha r^2 \ (r<R)## and ##\rho(r)=0 \...
  27. B

    Differential form of gauss's law.

    I don't understand what charge density is meant in the equation: div E = constant times charge density. I have the derivation in front of me and the last step follows from accepting that the rate of change of the integral of the field divergence per change in volume is the same as the rate of...
  28. B

    Lie derivative of contraction and of differential form

    Hello. I'm learning about Lie derivatives and one of the exercises in the book I use (Isham) is to prove that given vector fields X,Y and one-form ω identity L_X\langle \omega , Y \rangle=\langle L_X \omega, Y \rangle + \langle \omega, L_X Y \rangle holds, where LX means Lie derivative with...
  29. Q

    What exactly is differential form?

    I'm reading about multlinear algebra and I'm stuck at differential form and outer product. The definitions involve tensor product and quotient set and I really cannot grab the concrete idea of the differential form. Can someone explain in a somewhat layman term what is differential form...
  30. W

    Differential Form on Product Manifold

    Hi, I have an exercise whose solution seems too simple; please double-check my work: We have a product manifold MxN, and want to show that if w is a k-form in M and w' is a k-form in N, then ##(w \bigoplus w')(X,Y)## , for vector fields X,Y in M,N respectively, is a k-form in MxN. I am...
  31. O

    Differential form of Gauss' theorem with dielectrics

    Hi all, I'm stuck on this incompatibility within the differential form of Gauss' thearem (or Maxwell's first equation) with dielectrics. \vec{\nabla}\cdot\vec{E}=\frac{\rho_{free}+\rho_{bound}}{\epsilon_{0}} \rho_{bound}=-\vec{\nabla}\cdot\vec{P} But with a linear, homogeneous...
  32. N

    How to solve magnetic field using differential form?

    is there any general way? i mean i know how to do it in integra form but the course I am taking right now requires me to do it in differential method
  33. E

    Deriving poroelastic equation in differential form

    Hi, I'm trying to fill in the gaps in my notes - looking at a poroelastic model of tissue. We have the simple spherical model below. The centre sphere is where liquid is produced, then the two following spheres are brain tissue with different permeabilities, and the final sphere is an...
  34. Z

    The Relation between the integral and differential form of Amperes Law

    The integral form of Ampere's law in vacuum is ∫B\cdotdl=μ_{0}I (a) Using the relation between I and J, obtain the differential form of Ampere's law. You may ignore any displacement current. (b)Define the displacement current density J_{d} in terms of the displacement field D and show...
  35. G

    Solve Spherical Shell Gauss Problem with Differential Form Only

    Homework Statement A hollow spherical shell carries charge density \rho=\frac{k}{r^2} in the region a<=r<=b, where a is the inner radius and b is the outer radius. Find the electric field in the region a<r<b. I'm not allowed to use integral form of Gauss's law, must use differential...
  36. K

    Differential Form: Closed/Exact

    1 & 3 I have this differential form: ##\omega = F_1 dx + F_2 dy + F_3 dz## And I concluded that ##\omega## is closed because I calculated the partials and found out that ##\displaystyle \frac{\partial F_i}{\partial x_j} = \frac{\partial F_j}{\partial x_i}##. Also, ##F_1## contains only...
  37. S

    Integrating a differential form on a manifold without parametric equations

    I am sure that this can be done, but I haven't been able to figure it out, Is there a way to integrate a differential form on a manifold without using the parametric equations of the manifold? So that you can just use the manifold's charts instead of parametric equations? If you a function...
  38. H

    Problem understanding the differential form of the circulation law

    I've encountered a problem in learning about the curl of a vector field. (My learning material is the "Div, Grad, Curl and all that" from H.M. Shey.) Introduction to problem: The curl of a field F is defined as: ∇×F = i (∂Fz/∂y - ∂Fy/∂z) + j(∂Fx/∂z - ∂Fx/∂x) + k(∂Fy/∂x - ∂Fx/∂y)...
  39. ShayanJ

    Differential form of gauss's law and surface charge distributions

    Consider the integral form of gauss's law \oint \vec{E}.\vec{d \sigma}=\frac{q}{\epsilon_0} Then let's write q in terms of the volume and surface charge densities \rho and \delta and let's assume that the surface charge is distributed over the gaussian surface of the above integral \oint...
  40. E

    Difficulty in differential form of gauss law

    hi all, im new at electromagnetics and vector calculus, so facing these trivial problems. The differential form of gauss's law for empty space states ∇E = ρ/ε .The right side refers to charge density at the point of calculation of divergence? If a point charge(or a small uniformly charged...
  41. A

    Differential form of distances and some other doubts

    1) In a lot of instances i see distances and volumes written in the differential form. For instance dV = dxdydz Why not just write it as V = xyz (or any other letters, but not in the differential form)? In the image below, dx seems to be the inital length in x axis, and dy in the y axis...
  42. C

    Confusion about Gauss' law differential form

    http://einstein1.byu.edu/~masong/emsite/S1Q50/EQMakerSL1.gif Hi guys, I have a little confusion about the Gauss' law in differential form over here, obviously, many textbook wrote it in the above form, but actually, the only place at which divE is not zero is at the locations where the...
  43. B

    Gauss law differential form.

    Homework Statement Space distribution of electric charge is limited by to planes. Charge has the same intensity in planes parallel to these planes, but in dependence of the x coordinate, charge density is being distributed like: \rho (x)=A\cdot x\cdot (d-x) d is distance between those to...
  44. O

    Differential form of the first law of thermodynamics

    Hi everyone, I'm new here. I'm an italian student who has an exam in applied thermodynamics soon. Through the whole course as well as the Physics one I have faced a lot of equations expressed with differentials, basically all of them. I have never been taught how to use them though. For...
  45. C

    Proving h is a Differential Form

    Hello, I try to understand differential forms. For istance i want to prove that h=e_1\wedge e_2 + e_3\wedge e_4 is a differential form, where e_1,..,e_4 are elements of my basis. Do you have a idea, why this is a differential form? Regards
  46. A

    Differential form, alternating

    Suppose the standard coordinates in R^4 are x,y,x,w. We have a differential 2-form a= z dx \wedge dy. Trying to evaluate Alt(a) I am trying to see this form as a bilinear form that acts on a pair of vectors so that I can apply the Alt operator formula. I am able understand the formula for...
  47. L

    How the compute the pullback of a differential form

    If there's a function, \phi(t,u,v)=((v+t)u,t,u,v) =: (w,x,y,z). How do I compute the pullback of dw: \phi^*(dw)? I think what you do is: \phi^*(dw)=\frac{\partial}{\partial t}((v+t)u)dt+\frac{\partial}{\partial u}((v+t)u)du+\frac{\partial}{\partial v}((v+t)u)dv. Is that correct? This is...
  48. C

    Gauss' law on differential form.

    The differential form of Gauss' law states that \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}. So the divergence of the electric field is the chargedensity divided by epsilon zero. I just wondered.. since divergence is a local or "point" property. Is the chargedensity in this law also...
  49. W

    Using Gauss' Law (differential form) to get Coulomb's law.

    I don't really know much about serious business electrostatics. I've only taken the AP Physics C E&M exam (using the integral form), but I was looking at wikipedia and I was curious about the differential form of Gauss' Law. I don't understand what I'm doing wrong with this. I'm trying to...
  50. A

    Differential form of Gauss's Law

    Homework Statement In a model of an atomic nucleus, the electric field is given by: E = α r for r < a where α is a constant and a is the radius of the nucleus. Use the differential form of Gauss's Law to calculate the charge density ρ inside the nucleus. 2. The attempt at a...