Multivariable Taylor series reading 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.
 A: 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
