Page 193 - Intro to Tensor Calculus

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Figure 2.2-1 Tangent, normal and binormal to point P on curve.

Frenet-Serret Formulas
The parametric equations (2.2.1) describe a curve in our generalized space. With reference to the figure
2.2-1 we wish to define at each point P of the curve the following orthogonal unit vectors:

i
T = unit tangent vector at each point P.
i
N = unit normal vector at each point P.
i
B = unit binormal vector at each point P.
These vectors define the osculating, normal and rectifying planes illustrated in the figure 2.2-1.
In the generalized coordinates the arc length squared is

j
2
i
ds = g ij dx dx .
i
Define T = dx i as the tangent vector to the parametric curve defined by equation (2.2.1). This vector is a
ds
unit tangent vector because if we write the element of arc length squared in the form
i
dx dx j i j
1= g ij = g ij T T , (2.2.6)
ds ds
i
we obtain the generalized dot product for T . This generalized dot product implies that the tangent vector
is a unit vector. Differentiating the equation (2.2.6) intrinsically with respect to arc length s along the curve
produces
δT m n m δT n
g mn T + g mn T =0,
δs δs
which simplifies to
δT m
n
g mn T =0. (2.2.7)
δs