Blaga P. Lectures on the differential geometry of - tiera.ru

4.2. Surfaces 117
Explicit representation. If f : U → R is a smooth function, where U ⊂ R 2 is a
domain, then its graph, S = {(x, y, z) ∈ R 3 | z = f (x, y)} is a simple surface. Indeed, we
have the global parameterization r : U → R 3 , r(u, v) = (u, v, f (u, v)).
Implicit representation. Let F : V → R, with V ⊂ R 3 an open set, be a smooth
function. We denote by S = {(x, y, z) ∈ R 3 | F(x, y, z) = 0} the 0-level set of F. If, at
each point of S , the vector
grad F =
� ∂F
∂x
�
∂F ∂F
, ,
∂y ∂z
is different from zero, then S is a surface. Indeed, for instance, at (x0, y0, z0) ∈ S , we
have F ′ z(x0, y0, z0) � 0, then, from the implicit function theorem, there is an open (in the
topology of R 3 ) neighbourhood M of (x0, y0, z0) such that the set M∩S (which is an open
neighbourhood of (x0, y0, z0), this time in the topology of S ) is the graph of a smooth
function z = f (x, y), therefore, according to the previous paragraph, there is a local
parameterization of S around the point (x0, y0, z0). We note that this parameterization is
global for M ∩ S but, generally, it is not so for the entire S . Even if F ′ z is different from
zero on the entire S , it is still not sure that the function f can be defined globally. Thus,
in general, unlike that case of the surfaces given explicitly, the implicitly given surfaces
are not simple.
Examples. 1. The plane Π passing through r0 and having the direction given by the
vectors a and b, with a×b � 0 is a simple surface with the global parameterization
r(u, v) = r0+ua+vb. In projection on the coordinate axes, the parametric equations
of the plane will be
⎧
x = x0 + uax + vbx
⎪⎨
y = y0 + uay + vby
⎪⎩ z = z0 + uaz + vbz
, (u, v) ∈ R 2 .
2. Revolution surfaces
Let C be a curve in the plane xOz, which does not intersect the z-axis, and S – the
subset of R 3 obtained by rotating C around the z-axis. Let v be the rotation angle
of the plane xOz and a ′ – the point of S obtained by rotating the point a ∈ C by an
angle v0. Let (I, ρ = ρ(t)) be a local parameterization of the curve C around the
point a, ρ(t) = (x(t), z(t)). The we get the local parameterization os S around a ′ ,
r(t, v) = (x(t) cos(v + v0), x(t) sin(v + v0), z(t)),
�
.
defined on the domain U = I × � − π π
2 , 2