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  • $\begingroup$ Thanks! So in this case, the $n_{\theta}$ is some unit vector in the direction of that $\theta$. For the purpose of defining a gradient with unit vectors $n_{\theta_1}$, $n_{\theta_2} and $n_{\theta_3}$, is there a specific way to go about it? I am not sure how to construct a Jacobian moving from Cartesian unit vectors to these three unit vectors. $\endgroup$
    – Stack Exchanger
    Commented Feb 9, 2023 at 23:43
  • $\begingroup$ the $\hat{n}_{\theta_i}$ are unit vectors along the axis of rotation (direction of course determined by right hand rule...). $\endgroup$
    – Amit
    Commented Feb 9, 2023 at 23:45
  • $\begingroup$ Please see also the linked answer. I notice now there is a way to define this with a direct application of the gradient operator, which is used there. $\endgroup$
    – Amit
    Commented Feb 9, 2023 at 23:46
  • $\begingroup$ $\theta$ being an angle has no defined direction in and of itself, I hope that's clear. You need to know which axis each angle refers to, and I suppose that's given in the problem. $\endgroup$
    – Amit
    Commented Feb 9, 2023 at 23:47
  • $\begingroup$ If it happens that the angles are ordered from 1 to 3 as rotations about the x,y,z axis then what I wrote above coincides with $-\nabla U$ w.r.t the angles, but in general it will not $\endgroup$
    – Amit
    Commented Feb 9, 2023 at 23:55