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I am having trouble understanding these indices, and exactly what this is saying. I learned the multivariable Taylor series from Lang and it is presented in completely different manner. The highest he displays in this sort of format is of order 2, and he displays some order 3 terms using $D_i$ type notation. The second order terms and beyond is where I'm getting lost.

Could someone point me to a source so that I can unravel this formula and really interpret it, or perhaps explain it? If you were to explain it could you please give examples. Perhaps this can be translated into this form, $((H \cdot \nabla)^rf)(p)$ or vice versa.

This is paraphrased from a book which drops this out on page 4, page three being the introductory paragraphs. Reading onward it is not really explained, and I assume assumed that the reader understand all of it. I typed the formula exactly as it appears. This is from Tu's, An Introduction to Manifolds.

"The function f $\ldots$ it's Taylor series at p: $$f(x) = f(p) + \sum_{i} \frac{\partial f}{\partial x^{i}} (p) (x^{i}-p^{i}) + \frac{1}{2!}\sum_{i,j} \frac{\partial^2 f}{\partial x^{i} \partial x^{j}}(p)(x^i-p^i)(x^j-p^j) \\+ \cdots +\frac{1}{k!}\sum_{i_1, \ldots, i_k} \frac{\partial^k f}{\partial x^{i_1} \cdots \partial x^{i_k}}(p)(x^{i_1}-p^{i_1}) \cdots (x^{i_k}-p^{i_k})+ \cdots,$$ in which the general term is summed over all $ 1 \le i_1, \ldots i_k, \le n.$"

Thank you for your time.

smokeypeat
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  • Maybe this will help --- First, read my answer to How to expand $(a_0+a_1x+a_2x^2+...a_nx^n)^2$? Although the question is about squaring multinomials, my answer also discusses in detail how to cube a multinomial, which gives more intuition as to what goes on when raising multinomials to higher powers. Then look at my discussion for 3. [Multi-Variable Taylor Expansion] on pages 3-5 of this May 1999 take-home test. – Dave L. Renfro Jan 10 '18 at 09:47
  • I ran out of room in my previous comment for what I'd recommend looking at after the first two things I mentioned, which is to look at the Wikipedia article Multinomial theorem. After all this, you should be in a position to make sense of the expansion you're dealing with. For what it's worth, unless one takes an upper level undergraduate combinatorics or "finite mathematics" course, I suspect most pure mathematics' students exposure to this seemingly algebraic nightmare expansion is in a stiff advanced calculus course or manifolds. – Dave L. Renfro Jan 10 '18 at 09:58
  • Did you teach this course with your lecture notes or a textbook? I'm curious; and would like to have a look at the material. – smokeypeat Jan 10 '18 at 10:08
  • The course used a standard elementary calculus text (1994 1st edition of Finney, Thomas, Demana, Waits Calculus: Graphical Numerical Algebraic, Chapters 9-14, infinite series to multiple integrals), but I supplemented this material a lot. I taught this course 4 times in 3 years at this particular school, and while the students overall were well above average, the students who took this particular course were truly phenomenal --- see my 2nd paragraph comments (begins with As for good) here. – Dave L. Renfro Jan 10 '18 at 10:26
  • I need to leave for the gym now (I go very early in the mornings, it being 4:30 a.m. my time now), but I think I'll post my earlier comments (expanded slightly) as an answer since there's probably enough in them to merit this. – Dave L. Renfro Jan 10 '18 at 10:29
  • To follow-up with my earlier comment about what was used for the course in response to your question about lecture notes or a textbook, I don't have any lecture notes for the course, other than very rough handwritten notes and various handouts I wrote, a very few of which I've managed to post online, such as this (see this recent comment about it) (continued) – Dave L. Renfro Jan 10 '18 at 16:34
  • and this and this and this. These last three are 3 of a total of 9 sequences and series handouts I wrote for that course. The 3 that I've posted are listed in order, but there are others that came before and between and after them. – Dave L. Renfro Jan 10 '18 at 16:38

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Maybe this will help --- First, read my answer to $(a_0+a_1x+a_2x^2+...a_nx^n)^2$? Although the question I answered is about squaring multinomials, my answer also discusses in detail how to cube a multinomial, and this will give you more intuition as to what goes on when raising multinomials to higher powers. Then look at my discussion for 3. [Multi-Variable Taylor Expansion] on pages 3-5 of this May 1999 take-home test. Finally, look at the Wikipedia article Multinomial theorem. After all of this, you should be in a position to make sense of the expansion you're dealing with. For what it's worth, I suspect that unless one takes an upper level undergraduate combinatorics or "finite mathematics" course, most pure mathematics' students exposure to this seemingly algebraic nightmare expansion would be in a stiff advanced calculus course or in a beginning differentiable manifolds course such as yours.

Incidentally, for some references at the “stiff advanced calculus” level for what you’re dealing with, and references that will also likely be useful for other “advanced calculus” things that might turn up in your course, I recommend the following:

Wendell Fleming, Functions of Several Variables

Carl H. Edwards, Advanced Calculus of Several Variables

Michael Spivak, Calculus on Manifolds

Dave L. Renfro
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