# What is One-forms: Definition and 18 Discussions

In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to

R

{\displaystyle \mathbb {R} }
whose restriction to each fibre is a linear functional on the tangent space. Symbolically,

α
:
T
M

R

,

α

x

=
α

|

T

x

M

:

T

x

M

R

,

{\displaystyle \alpha :TM\rightarrow {\mathbb {R} },\quad \alpha _{x}=\alpha |_{T_{x}M}:T_{x}M\rightarrow {\mathbb {R} },}
where αx is linear.
Often one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials of the coordinates:

α

x

=

f

1

(
x
)

d

x

1

+

f

2

(
x
)

d

x

2

+

+

f

n

(
x
)

d

x

n

,

{\displaystyle \alpha _{x}=f_{1}(x)\,dx_{1}+f_{2}(x)\,dx_{2}+\cdots +f_{n}(x)\,dx_{n},}
where the fi are smooth functions. From this perspective, a one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant tensor field.

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1. ### 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. ### A Understand (k,l) Tensors in Gen. Relativity

In both Wald and Carroll, a type (k,l) tensor has k dual vectors and l vectors, yet a (1,0) tensor is a vector and a (0,1) tensor is a dual vector. I must be missing something simple. Please explain.
3. ### I Difference between vectors and one-forms

Hello! I am reading some introductory differential geometry and they define the vector space associated to a point of a manifold as the tangent plane at that point. Intuitively it makes sense to call these vectors (just as the speed is the tangent to the trajectory), but why are those called...
4. ### I Proving the Wedge Product of 2 One-Forms is a 2-Form

Hello! I was trying to show that the wedge product of 2 one-forms is a 2-form. So we have ## (A \wedge B)_{\mu \nu} = A_\mu B_\nu - A_\nu B_\mu ##. So to show that this is a (0,2) tensor, we need to show that ##(A \wedge B)_{\mu' \nu'} = \Lambda_{\mu'}^\mu \Lambda_{\nu'}^\nu (A \wedge B)_{\mu...
5. ### I Gradient one-form: normal or tangent

Working through Schutz "First course in general relativity" + Carroll, Hartle and Collier, with some help from Wikipedia and older posts on this forum. I am confused about the gradient one-form and whether or not it is normal to a surface. In the words of Wikipedia (gradient): If f is...
6. ### Definition of dx: What is its Domain & Formalization?

Homework Statement http://imgur.com/goozE9f Homework Equations ##(dx_i)_p i= 1,2,3## 3. The Attempt at a Solution [/B] I'm reading Manfredo and Do Carmo's Differential Forms and Applications. This is the very first page Would you mind explaining me what is meant by dx, as highlighted in the...
7. ### Why is a gradient not always a vector

I learned gradient in 3D space. And gradients where always vectors, pointing in the direction of steepest ... and normal to the surface where the functions is constant. But reading one-forms , a gradient of a function is not always a vector and it has something to do with metric... Can you proof...
8. ### One-forms in differentiable manifolds and differentials in calculus

Suppose that we have this metric and want to find null paths: ds^2=-dt^2+dx^2 We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$ dx=\pm dt \to x=\pm t Now switch to the more abstract and rigorous one-forms in differentiable manifolds. Here \mathrm{d}t (v)...
9. ### Find Null Paths in Differentiable Manifolds Using One-Forms

Suppose that we have this metric and want to find null paths: ds^2=-dt^2+dx^2 We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$ dx=\pm dt \to x=\pm t Now switch to the more abstract and rigorous one-forms in differentiable manifolds. Here \mathrm{d}t (v)...
10. ### Definition of One-Forms & Their Action | Clarifying a Concept

Hi, While reading Sean Carroll's book, I came across the following statement: Okay so this has me confused. Perhaps I am nitpicking, but isn't f a scalar function, i.e. a 0-form? So shouldn't he really be saying "why shouldn't df be considered the one-form..."? If f is a scalar, then df (as...
11. ### Christoffel Symbols of Vectors and One-Forms in say Polar Coordinates

Hello all, I've been going through Bernard Schutz's A First Course In General Relativity, On Chapter 5 questions atm. Should the Christoffel Symbols for a coordinate system (say polar) be the same for vectors and one-forms in that coordinate system? I would have thought yes, but If you...
12. ### Understanding Basis Vectors and One-Forms: A Simplified Explanation

Greetings, I have just started studying manifolds, and have come across the idea that the basis vectors can be expressed as: e\mu = \partial/\partialx\mu. I tried to convince myself of this in 2D Cartesian coordinates using a pretty non-rigorous derivation (the idea being to get a...
13. ### Symplectomorphisms preserving tautological one-forms

Symplectomorphisms preserving "tautological" one-forms Homework Statement Let (M,\omega) be a symplectic manifold such that there is a smooth one-form \alpha \in \Omega^1(M) satisfying \alpha = -d\omega . Let v \in \Gamma(TM) be the unique vector field such that \iota_v \omega =...
14. ### Proof set of one-forms is a vector space

Hi, I am currently working through 'Schutz-First course in General Relativity' problem sets. Question 2 of Chapter 3, asks me to prove the set of one forms is a vector space. Earlier in the chapter, he defines: \tilde{s}=\tilde{p}+\tilde{q} \tilde{r}=\alpha \tilde{p} To be...
15. ### Understanding the Difference Between One-Forms and Vectors in GR

I'm aware that this may not necessarily be a Relativity question but since GR seems to be a major area of application for these bits of mathematics, I'm going to go ahead and post it on this forum. I'm trying to understand the fundamental distinction between one-forms and vectors. I thought I...
16. ### Understanding Normal One-Forms for Plane x=0

I'm reading through Schutz's first course in relativity book and am finding question 12 on page 83 a bit problematic. If I understand it correctly an normal one-form to a plane is a one-form that, when operating on a normal vector to the plane, will give the result 0. This seems fairly...
17. ### I've just come across one-forms for the first time

I've just come across one-forms for the first time. Everything I read makes them sound exactly like dual vectors, but nobody mentions them in the same breath. Why? Is it that dual vectors are one-forms, but not all one-forms are dual vectors (e.g. covariant tensors etc) or is the difference...
18. ### Vectors, one-forms and gradients

I'm currently working through Schutz's "A first course in general relativity" as a preparation for a graduate course in General Relativity based on Carroll's notes. I'm a little confused about vectors, one-forms and gradients. Schutz says the gradient is not a vector but a one-form, because...