In rectangular coordinates consider the family of surfaces
x=c 1 , y =c 2 , z =c 3 ,
where c 1 , c 2 , c 3 take on the integer values 1 , 2 , 3 ,.... These surfaces intersect in lines
which are the coordinate curves. The vectors
grad x=ˆe 1 , grad y=ˆe 2 , and grad z=ˆe 3
are the unit vectors which are normal to the coordinate surfaces. The vectors
∂r
∂x =
ˆe 1 , ∂r
∂y =
ˆe 2 , ∂r
∂z =
ˆe 3
can also be viewed as being tangent to the coordinate curves. The situation is
illustrated in figure 8-16.
Example 8-15. In cylindrical coordinates (r, θ, z ),the transformation equations
(8.68) become
x=x(r, θ, z ) = rcos θ
y=y(r, θ, z ) = rsin θ
z=z(r, θ, z ) = z
and the inverse transformation (8.69) can be written
r=r(x, y, z) =
√
x^2 +y^2
θ=θ(x, y, z) = arctan yx
z=z(x, y, z) = z.
where the substitutions u=r, v =θ, w =z have been made. The position vector
(8.70) is then
r =r (r, θ, z ) = rcos θˆe 1 +rsin θˆe 2 +zˆe 3.
The curve
r =r (c 1 , θ, c 3 ) = c 1 cos θˆe 1 +c 1 sin θˆe 2 +c 3 ˆe 3 ,
where c 1 and c 3 are constants, represents the circle x^2 +y^2 =c^21 in the plane z=c 3
and is illustrated in figure 8-17. The curve
r =r (c 1 , c 2 , z) = c 1 cos c 2 eˆ 1 +c 1 sin c 2 ˆe 2 +zˆe 3
