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Suppose $f$ is a infinite continuously differentiable map: $R^n\to R$, and $x,x_0 \in R^n$, then we have the following second order Taylor expansion of $f(x)$ at $x_0$:

$$f(x)\approx f(x_0)+(x-x_0)^T\nabla f(x_0)+\dfrac{(x-x_0)^T \nabla^2f(x_0)(x-x_0)}{2} $$

What is the next iterm? Do I need to use tensor? what is a simple representation?

and how to conduct the multiplication between tensors and matrices and vectors?

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I think multi-indices are the typical way to expand this out further.

See Wikipedia

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  • $\begingroup$ thank you very much. I have only some basic knowledge of the Taylor series; but what I really want to know is, how to represent the further iterms in matrices and vectors(probably tensors). The equatons in Wiki does not answer my puzzle $\endgroup$ – LCFactorization Jan 9 '14 at 14:28
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    $\begingroup$ What you are asking about sounds like a Taylor expansion of a function of multiple variables. The partial derivative operator is more complicated with multiple variables. You must sum all possible combinations of partial derivatives that have the order of the term you are adding. Check out the book "Calculus in Vector Spaces" by Corwin and Szczarba, pg. 191: books.google.com/… $\endgroup$ – Dan Jan 9 '14 at 20:07

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