Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

1.3 Coordinate systems and differential operators 7
ˆk
Receiver
R
z
ˆt
ˆr
Figure 1.2: Cylindrical coordinates.
Source ⊗
x θ
r
y
Direction cosines γ 1 = r cos θ √
r
2
+ z 2 = r cos θ
R
γ 2 =
γ 3 =
√ r sin θ
r 2
+ z = r sin θ
2 R
z
√
r
2
+ z 2 = z R
(1.15d)
(1.15e)
(1.15f)
Basis vectors ˆr = î cos θ + ĵ sin θ (1.15g)
ˆt =−î sin θ + ĵ cos θ
ˆk = ˆk
(1.15h)
(1.15i)
Conversion between cylindrical and Cartesian coordinates
u = u x î + u y ĵ + u z ˆk
= u r ˆr + u θ ˆt + u z ˆk (1.16)
⎡ ⎤ ⎡
⎤ ⎡ ⎤
u r cos θ sin θ 0 u x
⎣ u θ
⎦ = ⎣ −sin θ cos θ 0 ⎦ ⎣ u y
⎦
u z 0 0 1 u z
(1.17)
⎡ ⎤ ⎡
⎤ ⎡ ⎤
u x cos θ −sin θ 0 u r
⎣ u y
⎦ = ⎣ sin θ cos θ 0 ⎦ ⎣ u θ
⎦
u z 0 0 1 u z
(1.18)
Nabla operator
∇=ˆr ∂
∂r + ˆt 1 ∂
r ∂θ + ˆk ∂ (1.19)
∂z
Allowing the symbol ⊗ to stand for the scalar product, the dot product, or the cross product,
and considering that