- Forums
- Mathematics
- Differential Geometry
- Thread starterdavi2686
- Start date
- Tags
- Forms
In summary, the conversation discusses the use of differential forms, specifically in the context of Gauss's Law. It is mentioned that 0 can be considered a 0-form, and there is a discussion on the notation and meaning of d\omega=0. The use of musical isomorphism is also mentioned. However, there is a discrepancy in the notation used in the steps, and it is clarified that it should be d(\star \vec E ^\flat)=\frac{\rho}{\epsilon_0} dV.
- #1
davi2686
- 33
- 2
if i have [itex]\int_{\partial S} \omega=0[/itex] by stokes theorem [itex]\int_{S} d \omega=0[/itex], can i say [itex]d \omega=0[/itex]? even 0 as a scalar is a 0-form?
Physics news on Phys.org
- #2
ShayanJ
Insights Author
Gold Member
- 2,810
- 605
Consider [itex] d\omega=x^3 dx [/itex] integrated over [itex] S=(-a,a) [/itex]. The integral gives zero but the integrand is zero in only one point of the region of integration. So this is a counterexample to [itex] \int_S d\omega=0 \Rightarrow d\omega=0 [/itex].
- #3
davi2686
- 33
- 2
Shyan said:
Consider [itex] d\omega=x^3 dx [/itex] integrated over [itex] S=(-a,a) [/itex]. The integral gives zero but the integrand is zero in only one point of the region of integration. So this is a counterexample to [itex] \int_S d\omega=0 \Rightarrow d\omega=0 [/itex].
thanks, but have no problem with 0 is a 0-form and [itex]d\omega[/itex] a k-form? so can i work with something like [itex]d\omega=4 [/itex]?
- #4
HallsofIvy
Science Advisor
Homework Helper
- 42,988
- 975
I have no idea what "[itex]d\omega= 4[/itex]", a differential form equal to a number, could even mean. Could you please explain that?
- #5
davi2686
- 33
- 2
HallsofIvy said:
I have no idea what "[itex]d\omega= 4[/itex]", a differential form equal to a number, could even mean. Could you please explain that?
my initial motivation is in Gauss's Law, [itex]\int_{\partial V} \vec{E}\cdot d\vec{S}[/itex]=[itex]\int_V \frac{\rho}{\epsilon_0}dV[/itex], i rewrite the left side with differential forms, [itex]\int_{\partial V} \star\vec{E}^{\flat}=\int_V \frac{\rho}{\epsilon_0}dV[/itex] which by the Stokes Theorem [itex]\int_{V} d(\star\vec{E}^{\flat})=\int_V \frac{\rho}{\epsilon_0}dV\Rightarrow d(\star\vec{E}^{\flat})=\frac{\rho}{\epsilon_0}[/itex], if i don't make something wrong in these steps, in left side we get a n-form and right side a 0-form, and that i don't know if i can do.
- #6
ShayanJ
Insights Author
Gold Member
- 2,810
- 605
davi2686 said:
thanks, but have no problem with 0 is a 0-form and [itex]d\omega[/itex] a k-form? so can i work with something like [itex]d\omega=4 [/itex]?
Its correct that 0 is a 0-form but by a zero 1-form we actually mean [itex] \omega= 0 dx [/itex] and write it as [itex] \omega= 0[/itex] when there is no chance of confusion.
davi2686 said:
my initial motivation is in Gauss's Law, [itex]\int_{\partial V} \vec{E}\cdot d\vec{S}[/itex]=[itex]\int_V \frac{\rho}{\epsilon_0}dV[/itex], i rewrite the left side with differential forms, [itex]\int_{\partial V} \star\vec{E}^{\flat}=\int_V \frac{\rho}{\epsilon_0}dV[/itex] which by the Stokes Theorem [itex]\int_{V} d(\star\vec{E}^{\flat})=\int_V \frac{\rho}{\epsilon_0}dV\Rightarrow d(\star\vec{E}^{\flat})=\frac{\rho}{\epsilon_0}[/itex], if i don't make something wrong in these steps, in left side we get a n-form and right side a 0-form, and that i don't know if i can do.
You missed something. You should have written [itex] d(\star \vec E ^\flat)=\frac{\rho}{\epsilon_0} dV [/itex].(What's [itex] \flat [/itex] anyway?)
- #7
davi2686
- 33
- 2
You missed something. You should have written [itex] d(\star \vec E ^\flat)=\frac{\rho}{\epsilon_0} dV [/itex].
Thanks i did not know that.
What's [itex] \flat [/itex] anyway?
That is musical isomorphism [itex]\flat:M \mapsto M^*[/itex], in fact i understand it works like a lower indice, [itex]\vec{B}^{\flat}[/itex] give me a co-variant B or it related 1-form.
Related to Can \( d \omega = 0 \) Be Concluded from \( \int_{\partial S} \omega = 0 \)?
1. What is the difference between k and p forms?
The main difference between k and p forms is their mathematical properties. K forms represent objects that can be described using vectors and have a fixed orientation in space, while p forms represent objects that can be described using planes and have a variable orientation in space.
2. How does equality between k and p forms affect physics?
Equality between k and p forms is essential in physics, particularly in the study of electromagnetism and relativity. It allows for a better understanding of the relationship between space and time, and helps in the development of mathematical models to describe physical phenomena.
3. Can you give an example of a physical concept that uses both k and p forms?
Maxwell's equations, which describe the behavior of electric and magnetic fields, use both k and p forms. The electric field is represented by a k form, while the magnetic field is represented by a p form.
4. How does the concept of equality between k and p forms relate to differential geometry?
Equality between k and p forms is a fundamental concept in differential geometry, as it helps to define the geometric structures of a space. It is also used to define the concept of curvature, which is crucial in understanding the behavior of objects in curved spaces.
5. Is there a practical application for the concept of equality between k and p forms?
Yes, there are many practical applications for this concept, particularly in the fields of physics and engineering. It is used in the development of mathematical models for electromagnetic devices, such as antennas and transformers, and in the study of fluid dynamics and general relativity.
Similar threads
IResources on the Derivation of generalized Stokes' theorem
- Differential Geometry
- Replies
- 11
- Views
- 750
IFrobenius theorem for differential one forms
- Differential Geometry
- Replies
- 6
- Views
- 795
IConformal flatness of ellipsoid
- Differential Geometry
- Replies
- 0
- Views
- 293
IManifold hypersurface foliation and Frobenius theorem
- Differential Geometry
2
- Replies
- 54
- Views
- 1K
IIntegration of differential forms
- Differential Geometry
- Replies
- 5
- Views
- 3K
IDarboux theorem for symplectic manifold
- Differential Geometry
- Replies
- 4
- Views
- 2K
AConservation Laws from Continuity Equations in Fluid Flow
- Classical Physics
- Replies
- 2
- Views
- 991
Fourier transform of ##e^{-a |t|}\cos{(bt)}##
- Calculus and Beyond Homework Help
- Replies
- 2
- Views
- 1K
IIs My Understanding of the Generalized Stokes' Theorem Correct?
- Differential Geometry
- Replies
- 1
- Views
- 1K
IVector valued integrals in the theory of differential forms
- Differential Geometry
- Replies
- 4
- Views
- 2K
- Forums
- Mathematics
- Differential Geometry