b r ¼ SinθCosϕ
ð
Þ b x þ SinθSinϕ
ð
Þ b y þ Cosθ
ð
Þb z
b θ ¼ CosθCosϕ
ð
Þ b x þ CosθSinϕ
ð
Þ b y À Sinθ
ð
Þb z
b
ϕ ¼ À Sinϕ
ð
Þb x þ Cosϕ
ð
Þb y
or in matrix form the latter three equations can be written as
b r
b θ
b
ϕ
0
B
@
1
C
A ¼
SinθCosϕ SinθSinϕ Cosθ
CosθCosϕ CosθSinϕ ÀSinθ
ÀSinϕ
Cosϕ
0
0
B
@
1
C
A
b x
b y
b z
0
B
@
1
C
A:
By forming the following dot products, namely, b r Á b θ, b r Á b
ϕ and b
ϕ Á b θ, it is easy to
show that these unit vectors are orthogonal, that is, b r Á b θ ¼ 0, b r Á b
ϕ ¼ 0 and b
ϕ Á b θ ¼ 0
. Inverting the above matrix we find that
b x
b y
b z
0
B
@
1
C
A ¼
SinθCosϕ CosθCosϕ ÀSinϕ
SinθSinϕ CosθSinϕ Cosϕ
Cosθ
ÀSinθ
0
0
B
@
1
C
A
b r
b θ
b
ϕ
0
B
@
1
C
A:
Derivatives of the Unit Vectors
Later on we will be requiring various derivatives when dealing with the momentum
and continuity equations; accordingly, we obtain the following results from the
previous equations;
∂b r
∂r
¼ 0,
∂ b θ
∂r
¼ 0 and
∂ b
ϕ
∂r
¼ 0
∂b r
∂θ
¼ CosθCosϕ
ð
Þ b x þ CosθSinϕ
ð
Þ b y À Sinθ
ð
Þb z ¼ b θ
∂ b θ
∂θ
¼ À SinθCosϕ
ð
Þ b x À SinθSinϕ
ð
Þ b y À Cosθ
ð
Þb z ¼ Àb r
∂ b
ϕ
∂θ
¼ 0
26
1 Brief Outline of the Equations of Fluid Flow