In linear algebra, a oneform on a vector space is the same as a linear functional on the space. The usage of oneform in this context usually distinguishes the oneforms from higherdegree multilinear functionals on the space. For details, see linear functional.
In differential geometry, a oneform on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a oneform 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,
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{\displaystyle \alpha :TM\rightarrow {\mathbb {R} },\quad \alpha _{x}=\alpha _{T_{x}M}:T_{x}M\rightarrow {\mathbb {R} },}
where αx is linear.
Often oneforms are described locally, particularly in local coordinates. In a local coordinate system, a oneform is a linear combination of the differentials of the coordinates:
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{\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 oneform has a covariant transformation law on passing from one coordinate system to another. Thus a oneform is an order 1 covariant tensor field.
Hi, starting from this old PF thread I've some doubts about the Frobenius condition for a differential 1form ##\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...
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.
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...
Hello! I was trying to show that the wedge product of 2 oneforms is a 2form. 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...
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 oneform and whether or not it is normal to a surface.
In the words of Wikipedia (gradient):
If f is...
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...
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 oneforms , a gradient of a function is not always a vector and it has something to do with metric... Can you proof...
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 oneforms in differentiable manifolds.
Here \mathrm{d}t (v)...
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 oneforms in differentiable manifolds.
Here \mathrm{d}t (v)...
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 0form? So shouldn't he really be saying "why shouldn't df be considered the oneform..."?
If f is a scalar, then df (as...
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 oneforms in that coordinate system?
I would have thought yes, but If you...
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 nonrigorous derivation (the idea being to get a...
Symplectomorphisms preserving "tautological" oneforms
Homework Statement
Let (M,\omega) be a symplectic manifold such that there is a smooth oneform \alpha \in \Omega^1(M) satisfying \alpha = d\omega . Let v \in \Gamma(TM) be the unique vector field such that \iota_v \omega =...
Hi,
I am currently working through 'SchutzFirst 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...
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 oneforms and vectors. I thought I...
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 oneform to a plane is a oneform that, when operating on a normal vector to the plane, will give the result 0. This seems fairly...
I've just come across oneforms 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 oneforms, but not all oneforms are dual vectors (e.g. covariant tensors etc) or is the difference...
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, oneforms and gradients.
Schutz says the gradient is not a vector but a oneform, because...