Interior product with differential forms

In summary, the conversation discusses the use of interior products of vectors and differential forms in differential geometry, and how to manipulate them to get the desired result. The speaker also confirms that their understanding is correct, and the expert provides a method for verifying the identity using the fact that dx∧dy=dx⊗dy−dy⊗dx.
  • #1

dyn

770
61
Hi. I'm trying to self-study differential geometry and have come across interior products of vectors and differential forms. I will use brackets to show the interior product and I would just like to check I am understanding something correctly. Do I need to manipulate the differential form to get the differential to be differentiated at the front of the form ? ie
( ∂/∂x , dx∧dy ) = dy and ( ∂/∂x , dy∧dx ) = -dy

( ∂/∂z , dx∧dy∧dz ) = dx∧dy and ( ∂/∂z , dx∧dz∧dy ) = -dx∧dy

Have I got this right ? Thanks
 
Physics news on Phys.org
  • #2
It is really difficult to tell unless you first specify what notation you are using and how you have defined your interior product. As it appears, you have defined the interior product in such a way that ##(X,\omega)(X_1,X_2,\ldots) = \omega(X,X_1,X_2,\ldots)##, where ##\omega## is a tensor of type ##(0,n)##, i.e., essentially using ##X## as the first argument of ##\omega## (seen as a linear map from ##(T_pM)^n## to the real numbers).

You can then use the fact that ##dx \wedge dy = dx \otimes dy - dy \otimes dx## to verify your identity. When you have ##\partial/\partial x## you would get
$$
\left(\frac{\partial}{\partial x}, dy \wedge dx\right) = \underbrace{dy(\partial_x)}_{=0} dx - \underbrace{dx(\partial_x)}_{=1} dy = -dy.
$$
Of course, when you have the differential form expressed in the coordinate differentials, you can generally just anti-commute the relevant coordinate differential to the first position. All terms where it is not in the first position will vanish identically and the interior product will just correspond to anti-commuting the relevant coordinate differential to the front and then removing it from the exterior product.
 
  • #3
Thanks for your reply. I don't understand everything in your post but it sounds like I am right in the calculations .
 

1. What is an interior product with differential forms?

An interior product with differential forms is a mathematical operation that combines a vector field and a differential form to produce a new differential form. It is denoted by the symbol "<i>" and is also known as the inner product or contraction.

2. How is the interior product defined?

The interior product is defined as the contraction of a vector and a differential form, resulting in a new differential form. It is defined using the exterior product and the Hodge star operator.

3. What are the properties of the interior product?

The interior product follows several important properties, such as linearity, associativity, and compatibility with the exterior derivative. It also satisfies the Leibniz rule, which states that the interior product of a differential form with a vector field is equal to the exterior derivative of the interior product of the differential form with the vector field.

4. What is the significance of the interior product in differential geometry?

The interior product plays a crucial role in differential geometry, particularly in the study of manifolds. It allows for the definition of important geometric concepts such as Lie derivatives, Lie brackets, and Lie algebras. It also helps in the formulation of Maxwell's equations in electromagnetism and the study of vector calculus.

5. How is the interior product used in real-world applications?

The interior product has various applications in physics, engineering, and other fields. It is used in the study of fluid dynamics, electromagnetism, and general relativity. It is also used in computer graphics and computer vision to model and manipulate geometric objects. In addition, the interior product is used in optimization problems and machine learning algorithms.

Suggested for: Interior product with differential forms

Replies
13
Views
757
Replies
7
Views
428
Replies
21
Views
2K
Replies
10
Views
596
Replies
5
Views
2K
Replies
4
Views
2K
Replies
4
Views
1K
Replies
14
Views
678
Replies
2
Views
459
Replies
1
Views
2K
Back
Top