Let [itex] P(x,y) [/itex] be a point on a unit circle that is centered at (0,0). How to compute exactly the function(adsbygoogle = window.adsbygoogle || []).push({});

[itex] \frac{\partial^2 x}{\partial s^2} [/itex]where [itex] x [/itex] is the x-coordinate of the point [itex] P(x,y) [/itex] and [itex]s[/itex] is the tangent at point [itex] P(x,y) [/itex]. Clearly,

[itex] \frac{\partial x}{\partial s} = t_x = -n_y [/itex] where [itex] t_x [/itex] is the x-component of the tangent at point [itex] P(x,y) [/itex] and [itex] n_y [/itex] is the y-component of the normal to circle boundary at point [itex] P(x,y) [/itex]. I have verified above equation with finite difference. Now how do I obtain an exact expression for

[itex] \frac{\partial }{\partial s }\left(\frac{\partial x}{\partial s}\right) [/itex] to get [itex] \frac{\partial^2 x}{\partial s^2} [/itex]? Thanks for help.

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# Computing tangential derivative d2x/ds2 at a point on a circle.

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