To represent a curve on a given surface defined in terms of two parameters uand
v, one can specify how these parameters change. For example, if
r =r (u, v ) = x(u, v )ˆe 1 +y(u, v )ˆe 2 +z(u, v )ˆe 3 (7 .25)
defines a given surface, then
(i) One can specify that the parameters uand vchange as as a function of time
tand write u=u(t)and v=v(t), then the position vector r =r (u, v )becomes a
function of a single variable t
r =r (t) = x(u(t), v (t)) ˆe 1 +y(u(t), v (t)) ˆe 2 +z(u(t), v (t)) ˆe 3 , a ≤t≤b
which sweeps out a curve lying on the surface.
(ii) If one specifies that v is a function of u, say v =f(u), this reduces the vector
r (u, v )to a function of a single variable which defines the curve on the surface.
This surface curve is given by
r =r (u) = x(u, f (u)) ˆe 1 +y(u, f (u)) ˆe 2 +z(u, f (u)) ˆe 3
(iii) An equation of the form g(u, v ) = 0 implicitly defines uas a function of vor vas
a function of uand can be used to define a curve on the surface. The equation
g(u, v ) = 0 , together with the equation (7.25), is said to define the surface curve
implicitly.
(iv) Consider the special curves
r =r (u, v 0 ) v 0 constant
r =r (u 0 , v ) u 0 constant
sketched on the surface for the values
u 0 ∈{ α, α +h, α + 2 h, α + 3 h,... }
v 0 ∈{ β, β +k, β + 2 k, β + 3 k,... }
where α,β,hand khave fixed constant values. These special curves are called
coordinate curves on the surface. The partial derivatives
∂r
∂u and
∂r
∂v evaluated
at a common point (u 0 , v 0 )are tangent vectors to the coordinate curves and the
cross product ∂r
∂u
×∂r
∂v
produces a normal to the surface.
