3-2 ORTHOGONAL COORDINATE SYSTEMS 141

used to describe them, the solution of a practical problem lengths, areas, and volumes. In Cartesian coordinates a
can be greatly facilitated by the choice of a coordinate differential length vector (Fig. 3-8) is expressed as
system that best fits the geometry under consideration. The
following subsections examine the properties of each of the dl = x dlx + y dly + z dlz = x dx + y dy + z dz, (3.34)
aforementioned orthogonal systems, and Section 3-3 describes where dlx = dx is a differential length along x, and similar
how a point or vector may be transformed from one system to interpretations apply to dly = dy and dlz = dz.
another. A differential area vector ds is a vector with magnitude ds
equal to the product of two differential lengths (such as dly
and dlz ), and direction specified by a unit vector along the third
3-2.1 Cartesian Coordinates direction (such as x). Thus, for a differential area vector in the
The Cartesian coordinate system was introduced in Section 3-1 yz plane,
to illustrate the laws of vector algebra. Instead of repeating dsx = x dly dlz = x dy dz (yz plane), (3.35a)
these laws for the Cartesian system, we summarize them in
Table 3-1. Differential calculus involves the use of differential

Table 3-1 Summary of vector relations.

Cartesian Cylindrical Spherical
Coordinates Coordinates Coordinates
Coordinate variables x, y, z r, , z R, ,
Vector representation A = xAx + yAy + zAz A + zAz
rAr + A +
RAR + A
  
+
Magnitude of A |A| = A2x + A2y + A2z + A2 + A 2 + A 2
r z
+ A2 + A 2 + A 2
R

Position vector

OP = xx1 + yy1 + zz1 , rr1 + zz1 , RR1 ,
1
for P (x1 , y1 , z1 ) for P (r1 , 1 , z1 ) for P (R1 , 1 , 1 )
Base vectors properties x x = y y = z z = 1 r r =

= z z = 1 R R =

= =1
x y = y z = z x = 0 r
= z = z r = 0 R
= R = 0
=
x y = z = z
r R =

y z = x z = r

= R
z x = y
z r = R =

Dot product AB = Ax Bx + A y By + A z Bz Ar Br + A B + A z Bz A R BR + A B + A B
     
 x y z   r
z   R


     
Cross product A B =  Ax Ay Az   Ar A Az   A A A 
     R 
 Bx By Bz   Br B Bz   B B B 
R

Differential length dl = x dx + y dy + z dz r d + z dz
r dr + R d +
R dR + R sin d
Differential surface areas dsx = x dy dz dsr = rr d dz dsR = RR 2 sin d d
dsy = y dx dz dr dz
ds = R sin dR d
ds =
dsz = z dx dy dsz = zr dr d R dR d
ds =
Differential volume d v = dx dy dz r dr d dz R 2 sin dR d d