# Normal vector in curvilinear coordinates

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?

## 1 Answer

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

• 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. – Jay May 28 '15 at 12:06
• 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? – Spencer May 28 '15 at 12:49
• 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. – Jay May 28 '15 at 13:13
• @Jay, fantastic! Glad I was able to help. – Spencer May 28 '15 at 13:45