Page 194 - Intro to Tensor Calculus

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m
The equation (2.2.7) is a statement that the vector δT m is orthogonal to the vector T . The unit normal
δs
vector is defined as
1 δT i 1 δT i
i
N = or N i = , (2.2.8)
κ δs κ δs
i
where κ is a scalar called the curvature and is chosen such that the magnitude of N is unity. The reciprocal
of the curvature is R = 1 , which is called the radius of curvature. The curvature of a straight line is zero
κ
while the curvature of a circle is a constant. The curvature measures the rate of change of the tangent vector
as the arc length varies.
The equation (2.2.7) can be expressed in the form

i
j
g ij T N =0. (2.2.9)
Taking the intrinsic derivative of equation (2.2.9) with respect to the arc length s produces

δN j δT i j
i
g ij T + g ij N =0
δs δs
or
δN j δT i j i j
i
g ij T = −g ij N = −κg ij N N = −κ. (2.2.10)
δs δs
The generalized dot product can be written

i
j
g ij T T =1,
and consequently we can express equation (2.2.10) in the form

δN j i j i δN j j
i
g ij T = −κg ij T T or g ij T + κT =0. (2.2.11)
δs δs
Consequently, the vector
δN j j
+ κT (2.2.12)
δs
j
i
i
is orthogonal to T . In a similar manner, we can use the relation g ij N N = 1 and differentiate intrinsically
with respect to the arc length s to show that
δN j
g ij N i =0.
δs
This in turn can be expressed in the form

j
δN j
i
g ij N + κT =0.
δs
This form of the equation implies that the vector represented in equation (2.2.12) is also orthogonal to the
i
unit normal N . We define the unit binormal vector as
i
1 δN i 1 δN i
i
B = + κT or B i = + κT i (2.2.13)
τ δs τ δs
where τ is a scalar called the torsion. The torsion is chosen such that the binormal vector is a unit vector.
The torsion measures the rate of change of the osculating plane and consequently, the torsion τ is a measure