# Fundamental Mathematics for Fluid Mechanics

```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
=
∂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 ) = � &middot; 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 ) = � &times; 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&middot;B = B&middot;A
A &times; B = −B &times; A
A &middot; (B &times; C) = B &middot; (C &times; A)
= C &middot; (A &times; B)
�
� Ax Ay Az
�
= �� Bx By Bz
� Cx Cy Cz
(29)
(30)
(31)
(32)
�
�
�
�
�
�
A &times; (B &times; C) = B(A &middot; C) − C(A &middot; B)
(A &times; B) &middot; (C &times; D) = (A &middot; C)(B &middot; D) − (B &middot; C)(A &middot; 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 &middot; B) = A &middot;
+ B&middot;
dx
dx
dx
d
dB dA
&times;B
(A &times; B) = A &times;
+
dx
dx
dx
3.3
(37)
(38)
(39)
Identities with the � Operator
�(f g) = g�f + f �g
(A &middot; �)f = A &middot; �f
(A &times; �)f = A &times; (�f )
�
�
�&middot;
An =
� &middot; An
n
�&times;
�
(40)
(41)
(42)
(43)
n
An
=
n
�
� &times; An
(44)
n
(A &times; �) &middot; B
� &middot; fA
� &times; fA
� &middot; (A &times; B)
� &times; (A &times; B)
�(A &middot; B)
=
=
=
=
=
=
(A &middot; �)A
=
� &times; (�f ) =
� &middot; (� &times; A) =
� &times; (� &times; A) =
4
(36)
A &middot; (� &times; B)
f (� &middot; A) + A &middot; �f
f (� &times; A) + A &times; (�f )
B&middot; (� &times; A) − A &middot; (� &times; B)
A(� &middot; B) + (B &middot; �)A − B(� &middot; A) − (A &middot; �)B
(A &middot; �)B + (B &middot; �)A + A &times; (� &times; B) + B &times; (� &times; A)
1
�A2 − A &times; (� &times; A)
2
0
0
�(� &middot; 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 &middot; en dA = � &middot; 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 &middot; ds =
(� &times; F) &middot; en dA
(56)
A
3
5
Notes
• All identities should be cross&shy;referenced 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).