Interior Product: Definition & Understanding

In summary, the interior product is defined as a transformation from r-form to (r-1)-form on a manifold, where X is a vector on the manifold and Ω^r(M) is the vector space of r-form at a point p. This transformation works by taking an r-form ω and applying it to r-1 vector fields, including X, to produce a new (r-1)-form. This may seem contradictory, but it is consistent with the definition and the number of arguments required for each form.
  • #1
Silviu
624
11
Hello! The interior product is defined as ##i_X:\Omega^r(M)\to \Omega^{r-1}(M)##, with X being a vector on the manifold and ##\Omega^r(M)## the vector space of r-form at a point p on the manifold. Now for ##\omega \in \Omega^r(M) ## we have ##i_X\omega(X_1, ... X_{r-1}) = \omega (X,X_1, ... X_{r-1})##. I am not sure I understand it. Based on this definition, ##i_X## acts on an r-1 form and it turns it into an r-form, by making it acts on X, too. But by the definition in the first line, ##i_X## should act the other way around. What am I reading wrong here? Thank you!
 
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  • #2
There is no contradiction. Say ##\omega \in \Omega^r(M)##. Then ##\omega## applies to ##r## vector fields. Now we want to define ##i_X(\omega) \in \Omega^{r-1}(M)##, i.e. by its action on ##r-1## vector fields ##X_1,\ldots , X_{r-1}##. We do this by ##(i_X(\omega))(X_1,\ldots ,X_{r-1}) := \omega(X,X_1,\ldots ,X_{r-1})##.
 
  • #3
To put it slightly differently, ##i_X\omega## is an ##r-1##-form so it takes ##r-1## arguments as you can see on the LHS of your expression. On the RHS of the expression you have only ##\omega##, which is an ##r#-form and therefore takes ##r## arguments. Clearly, there are ##r-1## ##X_i## on the LHS and the RHS therefore needs the extra argument ##X##, which is the ##X## of the interior product.
 

1. What is the definition of Interior Product?

The interior product, also known as the inner product, is a mathematical operation used in multilinear algebra and differential geometry to define the notion of a directional derivative of a multivector in a vector space.

2. How is the interior product different from the dot product?

The interior product is a generalization of the dot product, which is defined for vectors in Euclidean space. The dot product only takes two vectors as input, while the interior product can take multiple vectors as input and output a scalar or multivector.

3. What is the purpose of using the interior product in differential geometry?

In differential geometry, the interior product is used to define the Lie derivative, which is a way of measuring the change of a tensor field along a given vector field. It is also used to define the exterior derivative, which is a generalization of the concept of a derivative to differential forms.

4. Can the interior product be used in other fields besides mathematics?

Yes, the interior product has applications in physics, specifically in the study of electromagnetism and general relativity. It is also used in computer graphics to calculate lighting and shading effects on 3D objects.

5. How can understanding the interior product benefit scientists?

Understanding the interior product can benefit scientists by providing a deeper understanding of concepts in multilinear algebra and differential geometry. It can also be a useful tool in various fields such as physics, computer graphics, and engineering. Additionally, the interior product can be used to solve complex problems and make calculations more efficient.

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