Page 195 - Intro to Tensor Calculus

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of the twisting of the curve out of a plane. The value τ = 0 corresponds to a plane curve. The vectors
i
i
i
T ,N ,B , i =1, 2, 3 satisfy the cross product relation
i
B = ijk T j N k .
If we differentiate this relation intrinsically with respect to arc length s we find

δB i ijk δN k δT j
= T j + N k
δs δs δs
= ijk [T j (τB k − κT k )+ κN j N k ] (2.2.14)
i
= τ ijk T j B k = −τ ikj B k T j = −τN .
The relations (2.2.8),(2.2.13) and (2.2.14) are now summarized and written

δT i i
= κN
δs
δN i i i
= τB − κT (2.2.15)
δs
δB i i
= −τN .
δs
These equations are known as the Frenet-Serret formulas of differential geometry.

Velocity and Acceleration

Chain rule differentiation of the generalized velocity is expressible in the form

i
dx i dx ds i
i
v = = = T v, (2.2.16)
dt ds dt
i
i
where v = ds is the speed of the particle and is the magnitude of v . The vector T is the unit tangent vector
dt
to the trajectory curve at the time t. The equation (2.2.16) is a statement of the fact that the velocity of a
particle is always in the direction of the tangent vector to the curve and has the speed v.
By chain rule differentiation, the generalized acceleration is expressible in the form
δv r dv r δT r
r
f = = T + v
δt dt δt
r
dv r δT ds
= T + v (2.2.17)
dt δs dt
dv 2
r
r
= T + κv N .
dt
The equation (2.2.17) states that the acceleration lies in the osculating plane. Further, the equation (2.2.17)
indicates that the tangential component of the acceleration is dv , while the normal component of the accel-
dt
2
eration is κv .