How Can Vector Calculus Help Understand Differential Geometry in Terms of Forms?

In summary, the identity you are looking for is
  • #1
latentcorpse
1,444
0
if [itex]\alpha, \alpha' \in \Omega^1[/itex]. Rewrite the identity,

[itex]d(\alpha \wedge \alpha')=d \alpha \wedge \alpha' - \alpha \wedge d \alpha'[/itex] in terms of vector calculus.

I have absolutely no idea what is going on here. So if anybody could explain to me a) what this is all about and b) how to go about doing it, that would be great.

Cheers.
 
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  • #2
The identity you're looking for is [tex] \nabla \cdot ( \alpha \times \alpha') =(\nabla \times \alpha) \cdot \alpha' - \alpha \cdot (\nabla \times \alpha')[/tex]. Do you know how the exterior derivative is related to div, grad and curl? Do you know how the wedge product is related to the cross and dot products?
 
  • #3
i don't know how the exterior derivative is related to grad div and curl

and i was under the impression that the wedge product was the cross product?
 
  • #4
The wedge product of 1-forms is related to the cross product. The wedge product is a more general thing that is defined in all dimensions, whereas the cross product is only defined in three dimensions.

The gradient of vector calculus corresponds to the exterior derivative of functions, the curl corresponds the exterior derivative of 1-forms, and the divergence corresponds to the exterior derivative of 2-forms.

It is essential that you first learn about these ideas before attempting this problem.
 
  • #5
latentcorpse said:
and i was under the impression that the wedge product was the cross product?

Nooo … the wedge product is a 2-form, and the cross product is a pseudovector.

The dual 1-form (in 3-dimensional space) of the wedge product corresponds to the cross product. :wink:

EDIT: There's a good explanation, and a nice diagram, at http://en.wikipedia.org/wiki/Cross_product#Cross_product_as_an_exterior_product :wink:
 
Last edited:
  • #6
but [itex]\alpha,\alpha' \in \Omega^1[/itex] i.e. they are one forms. if there was an isomorphism (there is one given in the diagram that came with the question) [itex]\Phi_1[/itex] that maps [itex]\Phi_1: \Omega^1 \rightarrow X[/itex] then surely my answer should be

[itex] \nabla \cdot (\Phi_1(\alpha) \times \Phi_1(\alpha'))= \nabla \times \Phi_1(\alpha) \cdot \Phi_1(\alpha') - \Phi_1(\alpha) \cdot \nabla \times \Phi_1(\alpha')[/itex]

note that [itex]\Omega^1[/itex] is the space of 1-forms and [itex]X[/itex] is the space of vector fields in [itex]\maathbb{R}^3[/itex]
 
  • #7
Hi latentcorpse! :smile:

(have an alpha: α and a phi: Φ and a del: ∇ and a dot: · :wink:)

Yes, you should use a different letter, say a, to show that it's a vector (field) …

∇·(a x a') = (∇ x a)·a' - a·(∇ x a') :smile:
 

Related to How Can Vector Calculus Help Understand Differential Geometry in Terms of Forms?

1. What is Differential Geometry?

Differential Geometry is a branch of mathematics that studies the properties of curves and surfaces using calculus and linear algebra. It focuses on understanding the geometric structures and properties of smooth and continuous objects.

2. How is Differential Geometry used in real life?

Differential Geometry has numerous applications in various fields such as physics, engineering, computer graphics, and robotics. It is used to model and analyze the shape and motion of objects, as well as to study the behavior of physical systems.

3. What are the main concepts in Differential Geometry?

The main concepts in Differential Geometry include curves, surfaces, manifolds, tensors, and Riemannian geometry. These concepts are used to study the intrinsic properties of geometric objects, such as curvature, length, and angle.

4. How does Differential Geometry relate to other branches of mathematics?

Differential Geometry is closely related to other branches of mathematics, such as topology, analysis, and algebraic geometry. It uses tools and techniques from these fields to study the geometry of curved spaces and to solve problems in other areas of mathematics and physics.

5. What are some famous theorems in Differential Geometry?

Some famous theorems in Differential Geometry include the Gauss-Bonnet theorem, which relates the curvature of a surface to its topology, and the Nash embedding theorem, which states that any Riemannian manifold can be isometrically embedded into Euclidean space. Other notable theorems include the Poincaré conjecture, the Gauss map theorem, and the Hodge decomposition theorem.

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