Fundamental Mathematics for Fluid Mechanics
Daniel Enderton
March 20, 2004
Herein A, B, C, and D are vectors, and f and g are scalar quantites.
1 Coordinate Systems and Transformations
• Cartesian
rP = xex + yey + zez
= xi + yj + zk
(1)
(2)
rP = rer + θeθ + zez
�
rc =
x2 + y 2
y
θ = tan−1
x
x = rc cos θ
y = rc sin θ
er = cos θ ex + sin θ ej
eθ = − sin θ ex + cos θ ej
(3)
• Cylindrical
(4)
(5)
(6)
(7)
(8)
(9)
• Spherical
rP = rer + θeθ + φeφ
�
rs =
x2 + y 2 + z 2
(10)
(11)
(12)
• Earth Centered Coordinates
2 Operators
• Gradient:
grad(f ) = �f
=
∂f
∂f
∂f
ex +
ey +
ez
∂x
∂y
∂z
(Cartesian)
(13)
Figure 1: Coordinates and transformations between Cartesian and cylindrical (left panel) and
Cartesian and spherical (right panel).
1
=
=
=
∂f
1 ∂f
∂f
er +
eθ +
ez
∂r
r ∂θ
∂z
(Spherical)
(Earth)
(Cylindrical)
(14)
(15)
(16)
• Divergence:
div(f ) = � · A
=
=
=
=
∂Ay
∂Az
∂Ax
+
(Cartesian)
+
∂x
∂y
∂z
∂Az
1 ∂(rAr ) 1 ∂Aθ
+
+
(Cylindrical)
r ∂r
r ∂θ
∂z
(Spherical)
(Earth)
(17)
(18)
(19)
(20)
• Curl:
curl(f ) = � × A
�
�
� ex e y ez �
�
�
� ∂
∂
∂ �
= � ∂x
∂y
∂x �
�
�
� Ax Ay Az �
� e
�
ez �
� r
eθ
r �
� r
� ∂
∂
∂ �
= � ∂r
∂θ
∂z ��
�
� Ar rAθ Az �
=
=
(Cartesian)
(Cylindrical)
(Spherical)
(Earth)
(21)
(22)
(23)
(24)
• Laplacian:
�2 f
=
=
=
=
3
3.1
∂2f
∂2f
∂2f
+
+
(Cartesian)
∂y 2
∂y 2
∂x2
�
�
∂f
1 ∂2f
∂2f
1 ∂
r
+ 2 2 + 2
(Cylindrical)
r ∂r
∂r
r ∂θ
∂z
(Spherical)
(Earth)
(25)
(26)
(27)
(28)
Vector Identities
Vector Products
A·B = B·A
A × B = −B × A
A · (B × C) = B · (C × A)
= C · (A × B)
�
� Ax Ay Az
�
= �� Bx By Bz
� Cx Cy Cz
(29)
(30)
(31)
(32)
�
�
�
�
�
�
A × (B × C) = B(A · C) − C(A · B)
(A × B) · (C × D) = (A · C)(B · D) − (B · C)(A · D)
2
(33)
(34)
(35)
3.2
Differentiation with Respect to a Scalar
Herein, A = A(x), B = B(x), and f = f (x).
dA dB
d
(A + B) =
+
dx
dx
dx
d
dA
df
(f A) = f
+ A
dx
dx
dx
d
dB
dA
(A · B) = A ·
+ B·
dx
dx
dx
d
dB dA
×B
(A × B) = A ×
+
dx
dx
dx
3.3
(37)
(38)
(39)
Identities with the � Operator
�(f g) = g�f + f �g
(A · �)f = A · �f
(A × �)f = A × (�f )
�
�
�·
An =
� · An
n
�×
�
(40)
(41)
(42)
(43)
n
An
=
n
�
� × An
(44)
n
(A × �) · B
� · fA
� × fA
� · (A × B)
� × (A × B)
�(A · B)
=
=
=
=
=
=
(A · �)A
=
� × (�f ) =
� · (� × A) =
� × (� × A) =
4
(36)
A · (� × B)
f (� · A) + A · �f
f (� × A) + A × (�f )
B· (� × A) − A · (� × B)
A(� · B) + (B · �)A − B(� · A) − (A · �)B
(A · �)B + (B · �)A + A × (� × B) + B × (� × A)
1
�A2 − A × (� × A)
2
0
0
�(� · A) − �2 A
(45)
(46)
(47)
(48)
(49)
(50)
(51)
(52)
(53)
(54)
Integration Theorems
• Gauss’ theorem or divergence theorem. Herein, F is a vector function, V is any volume, and
A is the area that encloses the volume:
�
�
F · en dA = � · F dV
(55)
A
V
• Stokes’ theorem. Herein, F is a vector function and A is any area with s the line which
bounds it and en normal to the area:
�
�
F · ds =
(� × F) · en dA
(56)
A
3
5
Notes
• All identities should be cross­referenced in a couple other books.
• References to proofs of some or all of the identities.
• Introduction, description?
• What are core Fluid mechanics texts, including and excluding GFD.
• Assuming mixed partials equal?
• Spherical, Earth coordinates, description (notation, order).
• grad() versus �.
• Align text to right of equations.
• Quick references.
• Include long versions of curl() operator?
• Script version of Laplacian operator, i.e. curl().
• Expand to include things such as Taylor series, Euler’s formula, hyperbolic function defini­
tions
• Finish spherical and earth centered coordinates. Look up common notation.
• Define volume integrals in different coordinate systems?
References
[1] Granger, Robert A. “Fluid Mechanics.” Dover Publications, Inc. 1995.
4