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I'm learning vector analysis at the moment and having a bit of trouble grasping what's being put in front of me.

My book (Theoretical Physics by Joos) explains it as such:

I understand that d

My book (Theoretical Physics by Joos) explains it as such:

Since this vector (the unit tangentt) is always of unit length, its derivative must always be perpendicular tot, and so must be a vector in the normal plane to the curve. But this derivative, being the vector difference of two consecutive tangent vectors, must lie in theosculating planeformed by the latter, and so its direction is that of the principal normal, which direction we designate by the unit vectorn. In order to calculate the magnitude of the vector dt/ds (s is the arc length), we note that the curve, in the neighbourhood of two consecutive tangents, corresponding to three neighbouring points, may be replaced by the circle of curvature, whose centre M is determined by the intersection of the perpendiculars to the two consectutive tangents. The angle [tex]d\phi[/tex] between these tangents is the same as that between the two radii of the circle of curvature. If [tex]\rho[/tex] is the radius of this circle then, in the limit,

[tex]ds = \rho d\phi[/tex].

On the other hand, |dt| = [tex]d\phi[/tex] whence

|dt/ds| = 1/[tex]\rho[/tex].

I understand that d

**t**/ds is perpendicular to**t**, but the problem I have is where Joos says matter-of-factly that |d**t**| = [tex]d\phi[/tex]. I simply cannot see how he is coming by that result, geometrically or algebraically. Can anyone point me in the right direction? What am I missing?
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