What is the relationship between differential forms and degree?

[KarateMan]

I have a quetion about the forms.

When we say, "differential forms of degree one (or more)" rather than degree zero, the algebra is now mixed with topological properties. Am I correct?

I am simply trying to find my way to understand this.

 

[astros]

Sorry,what do you mean by "mixed with topological properties" ?

 

[KarateMan]

uh, it is hard to express in words, though. I will do my best.

Algebra and topology are distinct mathematical concepts because they have their own definitions. We can develop mathematical theorems with each of those independently. When we talk about the differential forms of degree ZERO, it is purely a matter of algebra. But now it is degree one or higher, then it is now the concept of differential kicks in. Can I say in this case that two distinct mathematical concepts "work together" to form whatever comes out?

please ask me again if unclear...

 

[astros]

I think that one can even study the topological aspect of functions i.e: forms of degree 0 ! Furthermore one can also find algebraic structures for the algebra of forms as a whole (the graded algebra)...
>> Can I say in this case that two distinct mathematical concepts "work together" to form whatever comes out?
Yes, hey give what is called algebraic topology, which has for principal idea to associate to a topological space an algebraic object (vector-space, group, ...) which is topological invariant, i.e: which remains unchanged if one performs a continuous deformation of the topological space: a homeomorphism , the well-known example is the Euler number...
But why do you say that degree 0 is purely a matter of algebra and that the concept of differentials implies automatically topological structures?

 

[KarateMan]

But why do you say that degree 0 is purely a matter of algebra and that the concept of differentials implies automatically topological structures?

I am only guessing. If you say I am wrong, that is simply because I am not that good yet. I found these concepts very difficult to organise in my head. I am trying to find the best way to understand the differential forms of degree TWO or higher.

Regarding "topological invariant", does this mean that topological aspects are not so obvious? In other words, one can study other mathematical concepts without paying much attention to it?

 

[astros]

Hi,
I think that one can study other mathematical concepts independently of topology, but I'm not an expert on the subject, For this reason, I will content myself to give you this link, It has served me in the past ... But warn me when you'll get it, So I remove it! I hope that it will serve you.
http://astrosurf.com/bouzid-dz/For%20Karateman/Differential%20Geometry.rar

 

[wofsy]

Differential forms are just tensors that are antisymmetric and map k-tuples of tangent vectors into the base field. They can reflect the topology of the underlying manifold but on each tangent space they are purely algebraic.

If one has a metric then forms can be thought of as defining infinitesimal areas, volumes, or hypervolumes on each tangent space.

 

[pmb_phy]

Differential forms are just tensors that are antisymmetric and map k-tuples of tangent vectors into the base field.

I don't see how you arrived at that assertion. A 1-form is a map from vectors into real numbers and as such is a tensor of first rank and as such antisymmetric has no meaning for such a tensor.

Please clarify your comment so that I may better understand what you're referring to. Thanks.

Pete

 

[Doodle Bob]

I don't see how you arrived at that assertion. A 1-form is a map from vectors into real numbers and as such is a tensor of first rank and as such antisymmetric has no meaning for such a tensor.

Please clarify your comment so that I may better understand what you're referring to. Thanks.

Pete

Wofsy is completely correct. A 1-form is antisymmetric by default: Since there is only one slot to do any switching of variables, it is vacuously true that a 1-form is antisymmetric.

You can also look at it as simply convention. We define a 0-form to be a function; a 1-form to be a 1-tensor; and a k-form to be an anti-symmetric k-tensor for k>1.