## Math 501: Differential Geometry

Homework 5
Lawrence Tyler Rush

<me@tylerlogic.com>

March 3, 2013

http://coursework.tylerlogic.com/courses/math501/homework05
1

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 =

2

Let S be the graph of a smooth function f : ℝ^{2} → ℝ with coordinate mapping
X(u,v) = (u,v,f(u,v)). From this we get that X_{u} = (1,0,f_{u}) and X_{v} = (0,1,f_{v}).

(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 ℝ^{2}. Using the equation we
derived in the first problem, we compute the area of V as follows.

3

Define the following parametrizations of the xy-plane in ℝ^{3} We can then define the change of coordinate function, h : ℝ^{2} → ℝ^{2} by
with inverse

resulting in the change-of-coordinate functions at the coordinate level of

Computing X_{u} = (1,1,0) and X_{v} = (1,-1,0) we can then compute Y _{r} and Y _{θ} in terms of X_{u} and X_{v} as
follows

4

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

(a)

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.

(b)

With the above definition of the parametrization we get
and

from which we get

5 do Carmo Page 109 problem 2

Let φ : S_{1} → S_{2} be a local diffeomorphism with S_{2} orientable.
Then there is some N_{2} : S_{2} → ℝ^{3} that is a differentiable field of normal unit vectors. Hence N_{1} = N_{2} ∘ φ
is differentiable in a neighborhood of any q ∈ S_{1}, but since φ(q) ∈ S_{2}, then N_{2}(φ(q)), which is equal to
N_{1}(q), is a unit normal vector. Thus N_{2} : S_{1} → ℝ^{3} is a differentiable field of unit normal vectors, i.e. S_{1} is
orientable.