1. Regular value Theorem
Let U be an open subset of R3. Suppose f : U → R is a smooth function. A point p ∈ U is a critical point of f if ∇f (p) = 0. If p is a critical point of f, we say f (p) is a critical value of f. A number a ∈ Im f is called a regular value of f if a is not a critical value.
Theorem 1.1. Let f : U → R be smooth and a is a regular value of f. Then f−1(a) = {(x, y, z) ∈ U : f (x, y, z) = a}
is a regular surface in R3.
Proof. Denote S = f−1(a). To show that S is a regular surface, we need to show that for each p ∈ S, there is an open neighborhood W of p such that U ∩ S is a parametrized surface in R3.
Since a is a regular value of f, for each p ∈ S, ∇f (p) 6= 0. Assume that fx(p) 6= 0. Define a function
F : U ⊂ R3 → R3, (x, y, z) 7→ (f (x, y, z), y, z).
Then F is a smooth map on U. Moreover, the Jacobian of dFp is nonzero:
J (F )(p) =
fx(p) 0 0 fy(p) 1 0 fz(p) 0 1
= fx(p) 6= 0.
By inverse function theorem, there is a neighborhood W of p contained in U and a neighbor- hood V of F (p) such that the mapping F : W → V is a diffeomorphism. Let G : V → W be its inverse map. Assume that G(u, v, w) = (g1(u, v, w), g2(u, v, w), g3(u, v, w)), where g1, g2, g3 are smooth functions on V. Then F ◦ G = idR3 implies that
(f ◦ G)(u, v, w) = u, ∀(u, v, w) ∈ V.
The open subset V ∩ {(u, v, w) ∈ R3 : u = a} is homeomorphic to an open subset D of R2. More precisely, let us define a map
j : R2 → R3, (v, w) 7→ (v, w, a).
Then j is a homeomorphism onto its image. The set
D = j−1(V ∩ {(u, v, w) ∈ R3 : u = a})
is open in R2. Let X : D → R3 be the map X(v, w) = G(a, v, w) for (v, w) ∈ D. Since (f ◦ G)(u, v, w) = u for (u, v, w) ∈ V,
f (X(v, w)) = a, ∀(v, w) ∈ D.
This implies that the image of X lies in f−1(a). Since G : V → W is a homeomorphism, G : V ∩ {(u, v, w) ∈ R3 : u = a} → W ∩ f−1(a)
is a homeomorphism. Since j : D → V ∩ {(u, v, w) ∈ R3 : u = a} is a homeomorphism and the composition of homeomorphisms is again a homeomorphism,
X = G ◦ j : D → W ∩ f−1(a) is a homeomorphism. For each q ∈ D,
dXq= dGj(q)◦ djq.
Since djq has rank 2 and dGj(q) is invertible, dXq has rank 2. We find that X : D → W ∩ f−1(a) is a parametrization around p. We complete the proof of our assertion.
1
2
If a is a regular value of a smooth function f : U ⊂ R3 → R, then the level surface f−1(a) is a closed subset of U by the continuity of f.
Example 1.1. The sphere S2 = {(x, y, z) ∈ R3 : x2+ y2+ z2 = 1} is a compact surface in R3.
Let us define a function f (x, y, z) = x2+ y2+ z2. Then 1 is a regular value of f. Hence S2 = f−1(1) is a regular surface which is also closed in R3. Since S2 is bounded and closed in R3, by Heine-Borel theorem, S2 is compact.
