Page 92 - Intro to Tensor Calculus

87

~ i
dA dA i d b e i d b e i
write, with respect to the fixed axes, that = b e i + A .Note that is the derivative of a
dt dt dt dt
f
vector with constant magnitude. Therefore there exists constants ω i , i =1,... , 6 such that
d b e 1 d b e 2 d b e 3
= ω 3 b e 2 − ω 2 b e 3 = ω 1 b e 3 − ω 4 b e 1 = ω 5 b e 1 − ω 6 b e 2
dt dt dt
e
e
db 2 db 1
i.e. see page 80. From the dot product b e 1 · b e 2 = 0 we obtain by differentiation b e 1 · + · b e 2 =0
dt dt
which implies ω 4 = ω 3 . Similarly, from the dot products b e 1 · b e 3 and b e 2 · b e 3 we obtain by differentiation the
~
additional relations ω 5 = ω 2 and ω 6 = ω 1 . The derivative of A with respect to the fixed axes can now be
represented
~ i ~
dA dA dA
~
= b e i +(ω 2 A 3 − ω 3 A 2 ) b e 1 +(ω 3 A 1 − ω 1 A 3 ) b e 2 +(ω 1 A 2 − ω 2 A 1 ) b e 3 = + ~ω × A
dt dt dt
f r
~
where ~ω = ω i b e i is called an angular velocity vector of the rotating system. The term ~ω × A represents the
~ i
dA dA
velocity of the rotating system relative to the fixed system and = b e i represents the derivative with
dt dt
r
respect to the rotating system.
Employing the special transformation equations (1.3.46) let us examine how tensor quantities transform
when subjected to a translation and rotation of axes. These are our special transformation laws for Cartesian
tensors. We examine only the transformation laws for first and second order Cartesian tensor as higher order
transformation laws are easily discerned. We have previously shown that in general the first and second order
tensor quantities satisfy the transformation laws:

∂y j
A i = A j (1.3.48)
∂y
i
i j ∂y i
A = A (1.3.49)
∂y j
mn ∂y m ∂y n
A = A ij (1.3.50)
∂y i ∂y j
∂y i ∂y j
A mn = A ij (1.3.51)
∂y m ∂y n
m i ∂y m ∂y j
A n = A j (1.3.52)
∂y i ∂y n
For the special case of Cartesian tensors we assume that y i and y ,i =1, 2, 3 are linearly independent. We
i
differentiate the equations (1.3.46) and (1.3.47) and find
∂y
∂y i j ∂y k ∂y i
= ` ij = ` ij δ jk = ` ik , and = ` ik = ` ik δ im = ` mk .
∂y ∂y
k k ∂y m ∂y m
Substituting these derivatives into the transformation equations (1.3.48) through (1.3.52) we produce the
transformation equations
A i = A j ` ji
i j
A = A ` ji
mn ij
A = A ` im ` jn
A mn = A ij ` im ` jn
m i
A = A ` im ` jn .
n j