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My question is how to calculate the taylor series of a function in non-cartesian coordinates. For orders of two or less, this is answered in another question (taylor expansion in cylindrical coordinates).

For second order taylor expansion this is (Quote from: user):

The Taylor expansion of a scalar function $f(\vec{x})$ around $\vec{x_0}$ can be written as:

$$ f(\vec{x})=f(\vec{x_0}) +(\nabla f)(x_o) \cdot (\vec{x}-\vec{x_0}) + \frac{(\vec{x}-\vec{x_0})^T(Hf(\vec{x_0}))(\vec{x}-\vec{x_0})}{2!}\ + ... $$

What would the third, fourth order, and so on terms look like?
Obviously third order, fourth order,.. tensors would come into play describing all of the partial derivatives. In addition there would have to be outer products of $\vec{x}-\vec{x_0}$ that would give a third, fourth order tensor. What I don't understand is, how these are combined in general for a n-th order expansion.

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