I'm trying to write the second Taylor expansion of a field defined on a plane using cartesian and polar coordinates. The coordinate system independent equation is: $$f({\bf x})=f({\bf x_o})+({\bf x-x_o}) \cdot \nabla f|_{\bf x_o}+\frac 1 2[({\bf x-x_o})({\bf x-x_o}):\nabla\nabla]f|_{\bf x_o}.$$
So in cartesian coordinate I manage to derive this, is it correct: $$f({\bf x})=f({\bf x_0})+\sum_i(x-x_o)_i \frac {\partial f} {\partial x_i}|_{x_o}+\frac 1 2 \sum_{i,j} (x-x_o)_i(x-x_o)_j \frac {\partial f} {{\partial x_I}{\partial x_j}}|_{x_o},$$ where $i,j$ can be $1$ or $2$.
Instead I don't know how to do it in polar coordinates, any suggestions?
If your $f$ is directly given as a function of $\rho,\theta$, the formula is exactly the same, with $x_1=\rho$ and $x_2=\theta$.
If $f$ is given in Cartesian coordinates, develop
$$g(\rho,\theta):=f(\rho\cos\theta,\rho\sin\theta)$$
using the chain rule.