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Is it true that the normal vector, or, $\ddot{\mathbf r}$ always vanishes for:

  • a helix in cylindrical coordinates
  • a loxodrome in spherical coordinates
  • a torus knot in toroidal coordinates

When does $\ddot{\mathbf r}$ vanish for a curve in curvilinear coordinates?

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If $\ddot{\vec{r}}$ vanishes in one coordinate system then it will vanish in all coordinate systems. This is a fundamental property of vectors.

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  • $\begingroup$ then if in one coordinate system $\ddot{\mathbf r}$ is nonzero, it will be nonzero in all. But $\ddot{\mathbf r}$ is related to the curvature of the curve through $|\ddot{\mathbf r}|$. Wouldn't the curvature depend on the manifold in which it is observed? Say, the helix has no curvature in cylindrical coordinates, but is curved in Cartesian. $\endgroup$ – Jay May 28 '15 at 12:06
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    $\begingroup$ Changing coordinates doesn't change the manifold. Coordinates are just different charts describing the same manifold. Have you calculated the helix to have no curvature in cylindrical coordinates? $\endgroup$ – Spencer May 28 '15 at 12:49
  • $\begingroup$ Wonderful! I just calculated the curvature in both Cartesian and cylindrical coords and it is the same. What tripped my thinking is the misunderstanding that curvilinear coordinates meant looking at things in a curved space. $\endgroup$ – Jay May 28 '15 at 13:13
  • $\begingroup$ @Jay, fantastic! Glad I was able to help. $\endgroup$ – Spencer May 28 '15 at 13:45

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