Why are you allowed to define the unit vector of the angle $\theta$ in either cylindrical or spherical coordinates to be a straight line? Shouldn't the unit vector be curved?
2026-05-16 07:23:00.1778916180
Why is the unit vector of the angle in spherical or cylindrical coordinates a straight line?
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No. Vector are straight lines. In fact the polar coordinates is just a rotated regular Cartesian coordinate system. The unit vector in the $\theta$ direction is a tangent to the circle/sphere of radius 1.