Math 501: Differential Geometry
March 3, 2013
Using the formula
we have the following sequence of equations for paramtrization x : U → S for some regular surface S with V = x(U)
where g =
Let S be the graph of a smooth function f : ℝ2 → ℝ with coordinate mapping
X(u,v) = (u,v,f(u,v)). From this we get that Xu = (1,0,fu) and Xv = (0,1,fv).
(a) Coefficients of The First Fundamental Form
(b) Length of α
Let α(t) be the curve in S with coordinate expression of (t,t) for
0 ≤ t ≤ 1, i.e. α(t) = (u(t),v(t)) where u(t) = t and v(t) = t. With this, the length of α is given by the following.
(c) Area of V
Let V = X(U) for some bounded open set U of ℝ2. Using the equation we
derived in the first problem, we compute the area of V as follows.
Define the following parametrizations of the xy-plane in ℝ3 We can then define the change of coordinate function, h : ℝ2 → ℝ2 by
resulting in the change-of-coordinate functions at the coordinate level of Computing Xu = (1,1,0) and Xv = (1,-1,0) we can then compute Y r and Y θ in terms of Xu and Xv as
The parametrization of the rotation of the regular plane curve, (f(v),g(v), of the
xz-plane with x and z coordinates given by f(v) and g(v), respectively, is
With the above definition of the parametrization, we have the following
differential of X.
Since the curve of rotation is a regular curve then g′(v)≠0 for all v which tells us that the columns of dX(u,v) are linearly
independent and therefore dX(u,v) is injective.
With the above definition of the parametrization we get
from which we get
5 do Carmo Page 109 problem 2
Let φ : S1 → S2 be a local diffeomorphism with S2 orientable.
Then there is some N2 : S2 → ℝ3 that is a differentiable field of normal unit vectors. Hence N1 = N2 ∘ φ
is differentiable in a neighborhood of any q ∈ S1, but since φ(q) ∈ S2, then N2(φ(q)), which is equal to
N1(q), is a unit normal vector. Thus N2 : S1 → ℝ3 is a differentiable field of unit normal vectors, i.e. S1 is