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geordief

Tensors

52 posts in this topic

Posted (edited)

Ha! So I am fired, in the nicest possible way! *wink*

Not at all. I acknowledge letting my enthusiasm exceed my concentration, discipline, and ability to make this a priority. Your expectations were reasonable and your disappointment perfectly understandable.

 

I simply want to get us both out of that loop. No expectations. I do want to get back to your posts. I'm stuck on one piece of notation as I'll get to.

 

I seek only to go slowly. I've always been interested in this material. Just no expectations about my pace, which might be arbitrarily small but not zero.

 

Do not feel bad, wtf. Differential geometry is a hard subject, as you would see if you had all 5 volumes of Michael Spivak's work.

What struck me was the depth of the detailed examples and calculations he did. He has fifty pages of explicit calculations of various manifolds before he defines the differentiable structure. So you and I are flying way way above the territory.

 

The good news is I can undertand things in there. The bad news is I will never have the storehouse of examples he gives. I'm not sure where you're suppose to learn those.

 

I do not pretend to have his depth of knowledge - I merely took a college course. Moreover his reputation as a teacher is extremely high, whereas mine is ....... (do NOT insert comment here!)

Writing clear exposition is challenging for everyone.

 

Regarding applications, all I can say is that I am neither an engineer nor a physicist, so as far as bridge bolts etc. you would need to ask somebody else.

I was thinking some of the engineering-oriented people here probably know. Studiot perhaps.

 

On the other hand, it is not possible to study differential geometry without at some point encountering tensor fields, especially metric fields and the curvature fields that arise from them. These are the principle objects of interest in the General Theory of Gravitation.

I can definitely imagine associating a tensor at each point of a manifold, by analogy with a vector field. I know what vector fields are.

 

If I offered to give guidance on this subject, it would be strictly as an outsider, an amateur.

Fine by me. I'm really curious about the [math]\frac{df}{da}[/math] notation. That's the exact spot I got stuck. Is that a typo or a feature? Would you write [math]\frac{df}{d\pi}[/math]?

 

ps -- I am finding this article most enlightening. https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors. I'm going to work through it. It explains why some components of the tensor are vectors and others covectors. It depends on which way the coordinates of a vector transform under a change of basis. Contravariant and covariant. This is a big piece of the puzzle.

Edited by wtf
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I'm really curious about the [math]\frac{df}{da}[/math] notation. That's the exact spot I got stuck. Is that a typo or a feature?

Neither. If I say for some arbitrary [math]a \in \mathbb{R}[/math] that [math]f(a) \in \mathbb{R}[/math] I am entitled to ask how the image varies as the argument varies. In other words, [math]a[/math] is a variable. For reasons I gave in another post, I can write this as [math]\frac{df}{da}[/math] it being understood this is unambiguous shorthand for [math]\frac{df(a)}{a}[/math].

 

You seem to believe that [math]x[/math] is the only possible label I can attach to an arbitrary Real number - it's not

Would you write [math]\frac{df}{d\pi}[/math]?

Of course not. [math]\pi[/math] is a constant. You cannot differentiate with respect to a constant

 

ps -- I am finding this article most enlightening. https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors. I'm going to work through it. It explains why some components of the tensor are vectors and others covectors. It depends on which way the coordinates of a vector transform under a change of basis. Contravariant and covariant. This is a big piece of the puzzle.

Interesting - I don't like it one bit. But let's not go there......
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