Tensors for Physics

2.7 Differentiation of Vectors and Tensors with Respect to a Parameter 29

The parity operation is also timeless. Thus it commutes with the differentiation
with respect to time. Consequently, the time derivativeA ̇of a tensor has the same
parity as the original tensorA.
In short: neither the property of a physical quantity being a tensor, nor its parity
behavior are affected by differentiating it with respect to time.

2.7.2 Trajectory and Velocity

The trajectory of a mass point or of the center of mass of any solid object is described
by the time dependence of its position vectorr=r(t), or equivalently,rμ=rμ(t),
μ= 1 , 2 ,3. The velocityvis defined by

vμ=

d

dt

rμ≡ ̇rμ. (2.61)

The velocity is a polar vector, just as the position vector.
The unit vector
̂vμ=v−^1 vμ=(r ̇νr ̇ν)−^1 /^2 r ̇μ, (2.62)

points in the direction of the tangent of the curve describing the trajectory. It is
referred to astangential vector.
Two simple types of motion are considered next.

1. Motion along a straight line. The trajectory is determined by

rμ(t)=rμ^0 +f(t)eμ,

whererμ^0 and the unit vectoreμare constant. The differentiable functionf(t)is

assumed to be equal to zero fort=0, thenrμ( 0 )=rμ^0 .Forr^0 μ=0, the line runs

through the origin. The resulting velocity is

vμ(t)=

df

dt

eμ.

Here, one haŝvμ=eμ=const. andv=f ̇.Forastraight uniform motion, not

only the direction of the velocity, but also the speedvis constant. Thenf(t)=vt

hold true.

2. Motion on a circle. The motion on a circle with the radiusRand the angular
   velocitywis described by

r 1 =Rcos(wt), r 2 =Rsin(wt), r 3 = 0 ,

where, obviously, the circle lies in the 1–2-plane. The origin of the coordinate

system is the center of the circle. At timet=0, the position vector points in the