What's a 0-form and what's not?


by quasar987
Tags: 0form
quasar987
quasar987 is offline
#1
Aug12-07, 02:41 PM
Sci Advisor
HW Helper
PF Gold
quasar987's Avatar
P: 4,768
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 ?
Phys.Org News Partner Science news on Phys.org
SensaBubble: It's a bubble, but not as we know it (w/ video)
The hemihelix: Scientists discover a new shape using rubber bands (w/ video)
Microbes provide insights into evolution of human language
dextercioby
dextercioby is offline
#2
Aug12-07, 02:56 PM
Sci Advisor
HW Helper
P: 11,866
What's the definition of a p-form on R^n ?
Hurkyl
Hurkyl is offline
#3
Aug12-07, 03:27 PM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,101
Quote Quote by quasar987 View Post
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
quasar987 is offline
#4
Aug12-07, 03:48 PM
Sci Advisor
HW Helper
PF Gold
quasar987's Avatar
P: 4,768

What's a 0-form and what's not?


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
Hurkyl is offline
#5
Aug12-07, 04:08 PM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,101
Quote Quote by quasar987 View Post
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
quasar987 is offline
#6
Aug17-07, 11:52 PM
Sci Advisor
HW Helper
PF Gold
quasar987's Avatar
P: 4,768
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
Hurkyl is offline
#7
Aug18-07, 12:22 AM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,101
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
quasar987 is offline
#8
Aug18-07, 12:41 AM
Sci Advisor
HW Helper
PF Gold
quasar987's Avatar
P: 4,768
I like your answer :)
Hurkyl
Hurkyl is offline
#9
Aug18-07, 01:19 AM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,101
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
mathwonk is offline
#10
Aug18-07, 11:19 PM
Sci Advisor
HW Helper
mathwonk's Avatar
P: 9,428
license to steal: salary for answering the same question infinitely many times.


Register to reply

Related Discussions
Polar form to Cartesian form Calculus & Beyond Homework 1
Question about the General form to normal form of Diff Eq Calculus & Beyond Homework 1
changing from parametric form to algebraic form Introductory Physics Homework 1
Having troubles putting matrices in another form, linear combination form i think. Introductory Physics Homework 14
6th form Academic Guidance 4