by Matt
Last Updated September 11, 2019 15:20 PM

I have a scalar-valued function, f, defined on a 2N-dimensional Euclidean space. I want to Taylor expand this function about a point $P$. I need to be able to explicitly write all terms in the expansion of at least 2nd order.

If I were working in Cartesian coordinates, I would define a basis such that $P = (x_1^\prime,y_1^\prime,x_2^\prime,y_2^\prime,...,x_N^\prime,y_N^\prime)$, and the Taylor expansion would be given by $$f(x_1,y_1,...) = f(x_1^\prime,y_1^\prime,...) + \sum_{i=1}^N \Big[(x_i-x_i^\prime)\frac{\partial f}{\partial x_i}|_{x_1^\prime,y_1^\prime,...} + (y_i-y_i^\prime)\frac{\partial f}{\partial y_i}|_{x_1^\prime,y_1^\prime,...}\Big] + \\ \frac{1}{2!}\sum_{i=1}^N \sum_{j=1}^N \Big[ (x_i-x_i^\prime)(x_j-x_j^\prime)\frac{\partial^2 f}{\partial x_i \partial x_j}|_{x_1^\prime,y_1^\prime,...} + (x_i-x_i^\prime)(y_j-y_j^\prime)\frac{\partial^2 f}{\partial x_i \partial y_j}|_{x_1^\prime,y_1^\prime,...} + (y_i-y_i^\prime)(x_j-x_j^\prime)\frac{\partial^2 f}{\partial y_i \partial x_j}|_{x_1^\prime,y_1^\prime,...} + (y_i-y_i^\prime)(y_j-y_j^\prime)\frac{\partial^2 f}{\partial y_i \partial y_j}|_{x_1^\prime,y_1^\prime,...} \Big] + ...$$

However, I want to work in polar coordinates, $(r_1,\theta_1,r_2,\theta_2,...)$. So, I should define $P = (r_1^\prime,\theta_1^\prime,...)$, and the Taylor expansion, written explicitly to first order, looks like the following (if I have this correct).

$$\require{enclose} \enclose{horizontalstrike}{f(r_1,\theta_1,r_2,\theta_2,...) = f(r_1^\prime,\theta_1^\prime,...) + \sum_{i=1}^N \Big[ (r_i-r_i^\prime)\frac{\partial f}{\partial r}|_{r_1^\prime,\theta_1^\prime,...} + r_i(\theta_i-\theta_i^\prime)\frac{\partial f}{\partial \theta_i}|_{r_1^\prime,\theta_1^\prime,...} \Big] + ...}$$

$$f(r_1,\theta_1,r_2,\theta_2,...) = f(r_1^\prime,\theta_1^\prime,...) + \sum_{i=1}^N \Big[ (r_i-r_i^\prime)\frac{\partial f}{\partial r}|_{r_1^\prime,\theta_1^\prime,...} + (\theta_i-\theta_i^\prime)\frac{\partial f}{\partial \theta_i}|_{r_1^\prime,\theta_1^\prime,...} \Big] + ...$$

I feel like this formula should be written somewhere, but I cannot find it. I know the second order terms can be written as a tensor product $x^i H_{ij} x^j$, where $H_{ij}$ is the Hessian matrix (tensor), which would be helpful if I could find an explicit formula for the Hessian in a polar coordinate basis.

Can anyone write the second-order terms in the Taylor expansion, or equivalently, provide the elements of the Hessian in a polar basis? Please keep in mind that I am an engineer, so I am ideally looking for an answer written explicitly using the polar coordinates, rather than covariant gradients, Levi-Civita symbols, etc. Though any help achieving progress toward the explicit formula is much appreciated.

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