Quadratic Taylor Polynomial in Multiple Variables

Question

I know of two different formulas for the determination of the Quadratic Taylor Polynomial:

• One with the Multi Index Notation:
  $$f(x_0 + \zeta) = \sum_{\vert\alpha\vert \le 2} \frac{1}{\alpha!} \, \partial^{\alpha} \, f(x_0) \, \zeta^{\alpha}$$

• One without:
  $$ f(x_0 + \zeta) = f(x_0) + \sum_{i=1}^{n} \partial_{i} \, f(x_0) \, \zeta_i + \frac{1}{2}\sum_{i,\,j=1}^{n} \partial_{i} \, \partial_{j} \, f(x_0) \, \zeta_i \, \zeta_j$$

Let's pretend $f = f(x,y)$.

I'm curious about how often the term containing $\partial_{x} \partial_{y}$ appears. With the Multiple Index Notation this would be the tuple (1, 1), therefore the term appears exactly once $\frac{1}{2} \partial_{x} \partial_{y} \,f(x_0) \zeta_x \, \zeta_y$. Sadly the second sum in the later formula contains both of the cases ($i = 1, j = 2$ and $i = 2, j = 1$), which results in two terms $\frac{1}{2} \partial_{x} \partial_{y} \, f(x_0) \zeta_x \, \zeta_y$.

Which is the right calculation of the Quadratic Taylor Polynomial? And what am I doing wrong/missing in my "wrong" formula?

Answer 1

In multi index notation $\alpha !=1$ when $\alpha=(1,1)$

Answer 2

Not sure what to tell you; the cleanest way to remember it is, with column vector $\zeta,$ $$ f( x_0 + \zeta) \approx f(x_0) + \nabla f(x_0) \cdot \zeta + \frac{1}{2} \zeta^T H \zeta, $$ where $H$ is the symmetric Hessian matrix of second partial derivatives, evaluated at $x_0.$ Oh, $\nabla f$ is the gradient vector of $f.$

I cannot tell whether this is part of your difficulty, but mixed partials commute once the function is smooth enough. That is a quick application of Fubini's Theorem.