- #1
NanakiXIII
- 392
- 0
I'm aware that this may not necessarily be a Relativity question but since GR seems to be a major area of application for these bits of mathematics, I'm going to go ahead and post it on this forum.
I'm trying to understand the fundamental distinction between one-forms and vectors. I thought I understood them until I noticed the following.
[tex]\nabla f = \frac{df}{dx} \^i + \frac{df}{dy} \^j + \frac{df}{dz} \^k[/tex]
Here we have a one-form (the gradient) which seems to be expressed as a linear combination of vectors (the base Carthesian vectors). Something's wrong. If I write
[tex]v = x \^i + y \^j + z \^k[/tex]
then v is a vector. But when the components are derivatives, we suddenly get a one-form.
I can think of several things that might be the issue, but I don't know which, if any, is true. First, it may be that the base vectors in the expression for the gradient are not base vectors, but rather one-forms. In Euclidean space, there's not much difference. Second, it may be that somehow the fact that the components are derivatives turns the whole thing into a one-form. I'm not sure, however, how that would work. Last, it may just be that the way I look at it is faulty.
If anyone can offer any insight, I would appreciate it.
I'm trying to understand the fundamental distinction between one-forms and vectors. I thought I understood them until I noticed the following.
[tex]\nabla f = \frac{df}{dx} \^i + \frac{df}{dy} \^j + \frac{df}{dz} \^k[/tex]
Here we have a one-form (the gradient) which seems to be expressed as a linear combination of vectors (the base Carthesian vectors). Something's wrong. If I write
[tex]v = x \^i + y \^j + z \^k[/tex]
then v is a vector. But when the components are derivatives, we suddenly get a one-form.
I can think of several things that might be the issue, but I don't know which, if any, is true. First, it may be that the base vectors in the expression for the gradient are not base vectors, but rather one-forms. In Euclidean space, there's not much difference. Second, it may be that somehow the fact that the components are derivatives turns the whole thing into a one-form. I'm not sure, however, how that would work. Last, it may just be that the way I look at it is faulty.
If anyone can offer any insight, I would appreciate it.