Recent content by orion

I see it. Thanks!

Thanks to both of you. I need to digest all this before I can say whether I have any other questions, but I do have a rather simple, silly question that came up in lavinia's first reply because I've seen it stated in textbooks as well. Given ##T_pM## why can we say that ##T_pM## is homeomorphic...

This is very appealing to me except I have one question. Take, for example, the simple real-valued function ##f(x)=x^2##. Now this function associates the number 3 with the number 9. How do I know? Because the function says take the number given and square it. I know exactly the prescription...

Ok. thanks guys. You have been a big help.

I understand, but where does the information reside? In ##\pi## or in ##v## itself? To complicate matters, in the book by JEFFREY Lee, he defines a vector at a point ##p## by ##v_p:=(p,v)##.

Thank you for your answers. They are very helpful. I think maybe your answers might cover my next question but I have to think about it some more. My next question concerns ##\pi^{-1}(p)##. This should be ##T_pM## as I am told. Suppose ##v \in T_pM## for some ##p \in M##. How exactly is ##p##...

I don't know how to create a commutative diagram here so I'd like to refer to Diagram (1) in this Wikipedia article. I need to discuss the application of this diagram to the tangent bundle of a smooth manifold because there are some basic points that are either glossed over or conflict in the...

No, it has to be written the way I wrote it. Otherwise, the Einstein summation convention does not work and also there is a need to distinguish contravariant components from covariant components. I realize that I posted in a calculus forum but that was because I wanted input on a derivative...

Thanks, fresh 42. I'm sorry I'm late in responding, but I forgot I wrote this question. It turns out that after I wrote this, I realized a mistake I was making in the proof and you are right, the gradient works. Thanks again.

If I have a scalar function of a variable ##x## I can write the derivative as: ##f'(x)=\frac{df}{dx}##. Now suppose ##x## is no longer a single variable but a vector: ## x=(x^1, x^2, ..., x^n)##. Then of course we have for the derivative ##(\frac{\partial f}{\partial x^1}, ..., \frac{\partial...

Forget it. That's not what I did.

Thank you, lavinia. That was clear.

This is very interesting. Coordinate free derivation. And I agree that it is a good example. I need to think about this and what this example shows besides coordinate free derivations and how it sheds light on my original question because to me this is a very special case of a derivation. But I...

You did not use coordinates here. But you didn't do any derivations also.

Thank you! That clears a lot up. This kind of thing is what I am looking for. I never said the various definitions aren't equivalent, but I admit that I get heavily invested with one definition and don't step back to see the larger picture. Somehow this all became about definitions, but my...