3d curvature equations I'm looking to make a 3-dimensional equation for a dataset that I have. My data in excel has an $x, y,$ and $z$ column and there are about $130$ data points for each of these. When I make a 3d scatter plot, I can also extract a 3d regression model and get something like $Z = 0.08 - 0.05X - 1.5Y$.
Here is where I'm having trouble. I am trying to find twist and curvature values. The curvature in the x direction, curvature in y direction and twist can be found by the following:
x, y and twist equations
So what I need to do is find an equation from my data that will allow me to find the second order partial $xy$ derivative of twist. I know there is an equation because I can visibly see twist on my 3d data. I'm assuming I'll need an equation like: $z = 3xy + 2y^2  x$ or something along those lines.
 A: I would start by finding a Taylor series approximation, and then take twist derivatives of the approximation. From Wikipedia, Taylor series may be generalized to functions of more than one variable with
$T(x_{1},\ldots ,x_{d})=\sum _{n_{1}=0}^{\infty }\cdots \sum _{n_{d}=0}^{\infty }{\frac {(x_{1}-a_{1})^{n_{1}}\cdots (x_{d}-a_{d})^{n_{d}}}{n_{1}!\cdots n_{d}!}}\,\left({\frac {\partial ^{n_{1}+\cdots +n_{d}}f}{\partial x_{1}^{n_{1}}\cdots \partial x_{d}^{n_{d}}}}\right)(a_{1},\ldots ,a_{d})\\=f(a_{1},\ldots ,a_{d})+\sum _{j=1}^{d}{\frac {\partial f(a_{1},\ldots ,a_{d})}{\partial x_{j}}}(x_{j}-a_{j})+{\frac {1}{2!}}\sum _{j=1}^{d}\sum _{k=1}^{d}{\frac {\partial ^{2}f(a_{1},\ldots ,a_{d})}{\partial x_{j}\partial x_{k}}}(x_{j}-a_{j})(x_{k}-a_{k})\\\qquad \qquad +{\frac {1}{3!}}\sum _{j=1}^{d}\sum _{k=1}^{d}\sum _{l=1}^{d}{\frac {\partial ^{3}f(a_{1},\ldots ,a_{d})}{\partial x_{j}\partial x_{k}\partial x_{l}}}(x_{j}-a_{j})(x_{k}-a_{k})(x_{l}-a_{l})+\cdots $
or, more compactly, using multi-index notation
${\displaystyle T(\mathbf {x} )=\sum _{|\alpha |\geq 0}{\frac {(\mathbf {x} -\mathbf {a} )^{\alpha }}{\alpha !}}\left({\mathrm {\partial } ^{\alpha }}f\right)(\mathbf {a} )}$
That being said, your notation $-\dfrac{\partial^2w}{\partial x^2}$ as $K_x$ seems a little odd to me. Typically, something like $K_x$ would be a first order partial derivative, corresponding to $\dfrac{\partial k}{\partial x}$ or $\dfrac{\partial w}{\partial x}$, whereas $\dfrac{\partial^2w}{\partial x^2}$ would be a second order partial derivative corresponding to $K_{xx}$ or $W_{xx}$. Similarly for $K_y$ and $\dfrac{\partial^2w}{\partial y^2}$.
And, as Barry Carter pointed out, we may be misleading you with our suggestions. If you need something else instead, please say so.
