Is a Function from R^n to R^m Considered a 0-Form on R^n?

[quasar987]

Is a function from R^n to R^m for aritrary m a considered a 0-form on R^n, or does 0-form refers only to functions from R^n to R ?

 

[dextercioby]

What's the definition of a p-form on R^n ?

 

[Hurkyl]

Is a function from R^n to R^m for aritrary m a considered a 0-form on R^n, or does 0-form refers only to functions from R^n to R ?

If you have a finite dimensional vector space V with scalar field k, then the space of n-forms is isomorphic to the space of alternating multilinear maps Vⁿ --> k.

In particular, a 0-form is an element of k.



In the typical setting of differential geometry, when analyzing a single point, your scalar field is R and your vector space is the tangent space, so a 0-form would simply be a real number. But more exotic things are possible, and sometimes even fruitful.

 

[quasar987]

In 'Calculus on manifolds', Spivak defines a k-form on R^n as a function w sending a point p of R^n to an alternating multilinear maps (R^n)^k-->R.

This makes sense only for k>0, so he treats the case k=0 separately by saying that by a 0-form we mean a function f.

I was 90% sure he meant a function f:R^n-->R but wanted to make sure.

 

[Hurkyl]

In 'Calculus on manifolds', Spivak defines a k-form on R^n as a function w sending a point p of R^n to an alternating multilinear maps (R^n)^k-->R.

This makes sense only for k>0, so he treats the case k=0 separately by saying that by a 0-form we mean a function f.

I was 90% sure he meant a function f:R^n-->R but wanted to make sure.

Well, it really does make sense for k=0: an A-valued function of 0 variables is the same thing as an element of A, and it's vacuously true that such a thing is alternating and 0-linear.

So a 0-form on the tangent bundle to R^n is, indeed, a map R^n --> R. This agrees with what I said pointwise -- if f is such a thing, then f(P) is a 0-form on the tangent space at P, which is the same thing as an element of R.

 

[quasar987]

Why are forms defined specifically as sending points to alternating tensors? What's wrong with good old arbitrary tensors? Or equivalently, what's so special about alternating ones?

 

[Hurkyl]

Integrating along an opposite orientation should give you the opposite answer -- thus the sign change.

From an algebraic perspective, they are trying to capture first-order differential information -- thus you want dx dx = 0. An immediate consequence of this identity is that differentials must be alternating.

 

[quasar987]

I like your answer :)

 

[Hurkyl]

Oh, and there's a geometric picture too -- given two vectors, you want to combine them to form a bivector that represents the area swept out by your vectors. So this product too should satisfy v v = 0. And since 1-forms are dual to tangent vectors...

 

[mathwonk]

license to steal: salary for answering the same question infinitely many times.