PS I do wish that members would not ask questions where either they they are not equipped to understand the answers or have no real interest in the subject they raise
I take this to heart and plead guilty. As my philosophy prof once said: The spirit is willing but the flesh is weak
. I have the math skills but my interest is drifting. The good news is that your posts enabled me to read parts of Spivak (*) and reading Spivak enabled me to understand parts of your posts. Learning has taken place and this has been valuable. You've moved me from point A to point B and I am appreciative.
I have not given up. I'm going far more slowly than I thought I would. I'll post specific questions if I have any. For the record you have no obligation to post anything. I regret encouraging any expectations that have led to disappointment. No one is more disappointed than me.
(*) Michael Spivak, A Comprehensive Introduction to Differential Geometry, Volume I, Third Edition. PDF here
Now, all that said ... I have four specific comments, all peripheral to the main line of your exposition. Regarding the main line of your exposition, I pretty much understand all of it, but not well enough to turn it around and say something meaningful in response. The concepts are in my head but can't yet get back out. You should not be discouraged by that. Your words are making a difference.Question 1) Definition of differentiable structure on
Yes, but at no pint did I assert that a continuous function needs to be differentiable. Rather I asserted the converse - a differentiable function must be continuous.
First I stipulate that this issue is unimportant and if we never reach agreement on it, I'm fine with that.
However this remark was in response to my pointing out that you need the map
to be differentiable order to define the differentiability of
. It's the only possible thing that can make sense. And yes of course by
, sorry if that wasn't clear earlier.
For whatever reason you seem to have forgotten this. It's true that we think of
as having a differentiable structure. But we have to define it as I've indicated. I verified this in Spivak. Homeomorphism can't be enough because there's no differentiability on an arbitrary manifold till we induce it.
Your not agreeing with this puzzles me. And your specific response about differentiable implying continuous doesn't apply to that at all.
As I say no matter on this issue but wanted to register my puzzlement.* Question 2) Definition of
In response to this issue you wrote:
Maybe I did not make myself clear. I said that the property for a function "subsumes" the property. If we attach the obvious meaning to the in we will say that a function is continuous to order zero, a function is continuous to order one..... a function is continuous to order
I am sorry if my language was not sufficiently clear.
I apologize but you are still not clear. What does subsume mean? You can't mean subset, because the inclusions go the other way. If a function is
-times continuously differentiable then it's certainly
. So subsume doesn't mean subset.
Of course it does mean that a
function is conntinuous. Differentiable functions are continuous, we all agree on that (is this what you were saying earlier?) So you are saying that a
function must be continous. Agreed, of course. That's "subsumed."
However this seems to be missing the point. The point is that there exists a differentiable function whose derivative is not continuous
Therefore it's not good enough to say that
is all the differentiable functions. It's all the differentiable functions whose derivative is continuous
. There's no way I can fit "subsumes" into this.
Again like I say, trivial point, not important, we can move on. But I wanted to be as clear as I could about my own understanding, since like any beginner I must be picky.* Question 3) The notation
Earlier you wrote:
So recall from elementary calculus that, given a function with then is a Real number.
I have never seen this notation.
is a constant. I asked about this earlier and did not understand your response. If
would you write
? I would write
, which you seem to think are radically different. Or even
. I'm confused on this minor point of notation.* Question 4) The real thing I want to know
After glancing through Spivak I realized that I am never going to know much about differential geometry. Perhaps looking at Spivak was a mistake
I'm trying to refocus my search for the clue or explanation "like I'm 5" that will relate tensors in engineering, differential geometry, and relativity, to what I know about the tensor product of modules over a commutative ring in abstract algebra.
What I seek, which perhaps may not be possible, is the 21 words or less -- or these days, 140 characters or less -- explanations of:
- How a tensor describes the stresses on a bolt on a bridge; and
- How a tensor describes the gravitational forces on a photon passing a massive body; and
- Why some components of these tensors are vectors in a vector space; and why others are covectors (aka functionals) in the dual space.
And I want this short and sweet so that I can understand it. Like I say, maybe an impossible dream. No royal road to tensors.
Ok that is everything I know tonight.