Page 91 - Intro to Tensor Calculus

P. 91

form. We observe that the direction cosines can be written as

ˆ ˆ ˆ
` 11 = b e 1 · e 1 =cos θ 11 ` 12 = b e 1 · e 2 =cos θ 12 ` 13 = b e 1 · e 3 =cos θ 13
ˆ ˆ ˆ
` 21 = b e 2 · e 1 =cos θ 21 ` 22 = b e 2 · e 2 =cos θ 22 ` 23 = b e 2 · e 3 =cos θ 23 (1.3.41)
ˆ ˆ ˆ
` 31 = b e 3 · e 1 =cos θ 31 ` 32 = b e 3 · e 2 =cos θ 32 ` 33 = b e 3 · e 3 =cos θ 33
which enables us to write the equations (1.3.39) and (1.3.40) in the form

y i = ` ij y j and y = ` ji y j . (1.3.42)
i

Using the index notation we represent the unit vectors as:

ˆ ˆ
e r = ` pr b e p or b e p = ` pr e r (1.3.43)

where ` pr are the direction cosines. In both the barred and unbarred system the unit vectors are orthogonal
and consequently we must have the dot products

ˆ ˆ
e r · e p = δ rp and b e m · b e n = δ mn (1.3.44)

where δ ij is the Kronecker delta. Substituting equation (1.3.43) into equation (1.3.44) we ﬁnd the direction
cosines ` ij must satisfy the relations:

ˆ ˆ
e r · e s = ` pr b e p · ` ms b e m = ` pr ` ms b e p · b e m = ` pr ` ms δ pm = ` mr ` ms = δ rs
ˆ
ˆ
ˆ
ˆ
and b e r · b e s = ` rm e m · ` sn e n = ` rm ` sn e m · e n = ` rm` sn δ mn = ` rm` sm = δ rs .
The relations
and ` rm ` sm = δ rs , (1.3.45)
` mr ` ms = δ rs
with summation index m, are important relations which are satisﬁed by the direction cosines associated with
a rotation of axes.
Combining the rotation and translation equations we ﬁnd

y i = ` ij y j + b i . (1.3.46)
|{z}
|{z}
rotation translation
We multiply this equation by ` ik and make use of the relations (1.3.45) to ﬁnd the inverse transformation

y = ` ik (y i − b i ). (1.3.47)
k
These transformations are called linear or aﬃne transformations.
Consider the x i axes as ﬁxed, while the x i axes are rotating with respect to the x i axes where both sets
~
i
of axes have a common origin. Let A = A b e i denote a vector ﬁxed in and rotating with the x i axes. We
~ ~
dA dA
~
denote by and the derivatives of A with respect to the ﬁxed (f) and rotating (r) axes. We can
dt dt
f r

86 87 88 89 90 91 92 93 94 95 96