Recent content by CMoore

No...It's not a scalar product because the result is a vector, not a number; it's a mapping from R^m to V.

Hubbard's & Hubbards "Concrete To Abstract" Function On page 215 of H&H Vector Calculus, Linear algebra and Differential Forms text the authors define something they call the "concrete to abstract function". It is defined as follows: Let (x_1, ..., x_m) be a point in R^m and let {v_1, ...

Hi Davin, In my opinion, you could do no better in your choice of calculus texts. I frequently reference Apostol's calculus texts to remind myself of things I have forgotten. When you have worked your way thoroughly through them, you will be well-prepared to takle more much more advanced...

Defining the term "connected" Some authors use the word "connected" as applied to a set S to mean that for any two points A and B in S there exits a (piecewise) smooth path in S from A to B. This is, for instance, the approach taken by Apostol in Calculus Vol II (p 332) and the approach taken...

Thanks, but that's really not what I'm looking for. What I'm looking for is a logical sequence of definitions and theorems that begins with, say, the Peano postulates, works its way through the construction of real numbers and lists theorems on sequences, limits, derivatives, etc. The idea of...

Can anyone recommend notes or a book that contains only the theorems and definitions of real analysis in a logical sequence to which the Moore Method could be applied? Thank You.

For an excellent introduction to elementary analysis, I would suggest https://www.amazon.com/gp/product/0521288827/?tag=pfamazon01-20 by K. G. Binmore. The author provides numerous examples, his proofs are clear and answers to ALL problems are in the back of the book. It is thus a great choice...

I too like Dover books as a general rule; I'm not quite sure why Dover is getting knocked so much on here. Of course Dover, like any other publisher, has its share of stinkers but it also provides many excellent volumes at a reasonable price. For example, here's three good ones: 1. Advanced...

Indeed; that is what the author does. He uses the standard coordinate functions for this purpose. But how can such a selection be justified? The relationship should hold not just for particular choices of f but for any f. Obviously, I'm being dense here, but I just can't see it...

Hello, I am trying to understand how the geometric tangent space \mathbb{R}^n_a given by \begin{displaymath} \mathbb{R}^n_a = \{(a,v) | v \in \mathb {R}^n\} \end{displaymath} is isomorphic to the space of all derivations of C^{\infty}(\mathbb{R}^n) at a, denoted by...

Thank you SA, that was a very coherent and informative explanation.

John, Thanks for your reference. Unfortunately, however, that thread did little to clear things up. I don't know enough about string theory to know how valid it may or may not be. I do know that any reasonable scientific theory must have some means of experimental verification; however, I...

The reference at the superstringtheory website is excellent; thank you. I did, however, receive a disouraging email. The author claimed that string theory was a sinking ship and by the time someone learned enough math to understand it that it would be dead. I have read threads on here that...

Please forgive me if this question has been posted before, but I was wondering if anyone could provide a semi-detailed and sequential mathematical syllabus that, once mastered, would allow one to follow development of string theory. So, assuming basic undergraduate mathematics such as real...

Thank you for your response; it was very helpful. Every time I tried the problem in the past I would end up with a jumbled up mess. I approached the it by attempting to apply the definition of the inverse mapping directly, i.e. by setting the composition of p and p^{-1} to the identity map. The...