Basics of Fluid Mechanics, 2014a

A.1. VECTORS 573
R · S =(xî + y 2 ĵ) · (sin xî + exp(y)ĵ) can be written as
d (R · S)
dt
It can be noticed that
d (R · S)
dt
= d dt
= d ( x sin x + y 2 exp(y) )
dt
dx
dt
(( ) (
))
xî + y 2 ĵ · sin xî + exp(y)ĵ
=
sin x +
d sin x
dt
+ dy2
dt exp(y)+dy2 dt exp(y)
It can be noticed that the manipulation of the simple above example obeys the regular
chain role. Similarly, it can done for the cross product. The results of operations of two
vectors is similar to regular multiplication since the vectors operation obey “regular”
addition and multiplication roles, the chain role is applicable. Hence the chain role
apply for dot operation,
d
dt (R · S) = dR
dt · S + dS
dt · R
(A.23)
And the the chain role for the cross operation is
d
(R × S) =dR
dt dt × S + dS
dt × R
(A.24)
It follows that derivative (notice the similarity to scalar operations) of
d
dt (R · R) =2R dR
at
There are several identities that related to location, velocity, and acceleration. As in
operation on scalar time derivative of dot or cross of constant velocity is zero. Yet, the
most interesting is
d
dU
(R × U) =U × U + R × (A.25)
dt dt
The first part is zero because the cross product with itself is zero. The second part is
zero because Newton law (acceleration is along the path of R).
A.1.3.1
Orthogonal Coordinates
These vectors operations can appear in different orthogonal coordinates system. There
are several orthogonal coordinates which appears in fluid mechanics operation which
include this list: Cartesian coordinates, Cylindrical coordinates, Spherical coordinates,
Parabolic coordinates, Parabolic cylindrical coordinates Paraboloidal coordinates, Oblate
spheroidal coordinates, Prolate spheroidal coordinates, Ellipsoidal coordinates, Elliptic