Curvilinear Coordinate Systems 141vectors u> and ro, the vector w x r 0 is the velocity of the point with po-
sition vector ro, rotating with constant angular speed u — ||u>|| about a
fixed axis; the vector w is the angular velocity of the rotation, see page
- The vector field v = u> x r therefore corresponds to the rotation of all
points in space with the same angular velocity CJ. Direct computations in
cartesian coordinates show that curlw = 2u>, establishing the connection
between the curl and rotation: if F = F(P) is interpreted as the velocity
field of a continuum of moving points, then curl F measures the rotational
component of this velocity; the field F is called irrotational in a domain
Gif curlF = OinG.
3.1.3 Curvilinear Coordinate Systems
Recall that a vector field (P,F(P)) is a collection of vectors with different
starting points, and, to study the vector field, we need a local coordinate
system, that is, a set of basis vectors, at every point P. So far, we mostly
worked with cartesian coordinates, where this local coordinate system is
the parallel translation of (i, j, k) to the corresponding point P. More
generally, these local coordinate systems are not necessarily parallel trans-
lations of one another. For example, the polar coordinates r, 6 in R^2 give
rise to the unit vectors r,8, see (1.3.22) on page 35 or Figure 3.1.1 below.
At every point, r A. 6, but the corresponding vectors r have, in general,
different directions at different points. For various applications, it is natural
to use cylindrical or spherical coordinates, and then it becomes necessary
to express all the operations of differentiation (gradient, divergence, curl,
Laplacian) in these coordinate systems.
The objective of this section is to study general orthogonal curvilinear
coordinate systems, where the basis vectors at every point are mutually or-
thogonal, but their direction changes from point to point. Our first step
is to establish the general relation between the coordinates and their ba-
sis vectors. For notational convenience, we denote by X — (£1,0:2,0:3) the
cartesian coordinates, with the correspondence x\ = x, X2 = y, X3 = z. For
non-cartesian coordinates Q = {qi,q2,q?), each qk is a given function of
xi,X2,Xz: qk = qfc(a:i,a:2,a;3), k = 1,2,3. The correspondence between the
X and Q coordinates must be invertible, so that each Xk is a correspond-
ing function of q\,q2,qz'- xk = Xk{qi,q2,qz), k = 1,2,3. FOR EXAMPLE,
in cylindrical coordinates, with qi = r, q^ — 6, qz = z, we have
r = \Jx\ + x\, 9 = tan-1 (0:2/0:1), z = 0:3, so that 0:1 = rcosO, o: 2 =
rsin#, X3 = z. This example also shows that there could be some points