Download Lecture 2A [pdf]

Math 423 (Day 2)
2/2/10
1
Course Overview (cont.)
Gauss-Bonnet-Hopf Theorem: Let M be a compact surface. Topologically M is characterized by its genius, g, which is the number of holes in M .
Examples: g = 0: Sphere; g = 1: Torus; g = 2: Double Torus.
The Gauss-Bonnet Theorem says that
Z
1
K dA = 2(1 − g)
2π M
where K is the Gauss curvature.
So, the total Gauss curvature is a topologically invariant: It depends only on topology
of M , so if you deform M1 into M2 without tearing it (so as to preserve its genus) then the
total Gauss curvature does not change.
The Gauss-Bonnet-Hopf Theorem relates tangent vector fields on M to the Gauss curvature and the genus.
Let V~ be a vector field on M that is tangent to M at each point then
Z
1
~
κ dA = 2(1 − g)
number of zeros of V =
2π M
where the number of zeros are counted with multiplicity. (Pictures in hand-written notes)
Green’s, Stokes, and Divergence Theorems: These theorems are all versions of the
Fundamental Theorem of Calculus. We will revisit them and then learn about differential nforms which are the mathematical objects we integrate over n-dimensional surfaces. There
is a version of the Fundamental Theorem of Calculus for differential forms that unifies and
generalizes these three theorems.
Fundamental Theorem of Calculus: Let M be (n + 1)-dimensional surface with ndimensional boundary ∂M and let ω be an n-form on M then
Z
Z
dω =
ω
M
∂M
where dω is the derivative of ω.
1
Special Case If M is a surface, then ∂M is a curve, and a 1-form ω on M corresponds to
a vector field V~ . The 2-form dω corresponds to the curl of the vector field, ∇ × V~ . So the
FTC reduces to Stokes Theorem:
Z Z
Z
~=
(∇ × V~ ) · dS
V~ · d~s
M
∂M
Geodesics are curves on a surface M that are the analogues of straight lines in the plane.
They are
• Length minimizing curves (locally)
• “Straight”
• Curves with zero acceleration
To an ant walking on M , a geodesic curve is like a straight line in the plane. The geodesics
on a sphere are the great circles.
2
Calculus on Euclidean Space
2.1
3D Euclidean Space, R3
• Point of R3 is p~ = (p1 , p2 , p3 ) = p
• R3 is a vector space (addition, scalar multiplication, zero)
• R3 is a 3-dimensional vector space with standard basis:
 
 
 
1
0
0





0 , e~2 =
1 , e~3 =
0 
e~1 =
0
0
1
For any p~ ∈ R3 , we have p~ = p1 e1 + p2 e2 + p3 e3 , which is the unique expression for p~
in the standard basis where p1 , p2 , p3 are called coordinates of p~.
We can regard the coordinates as functions on R3 . This will be important later when
we learn about the 1-form dx.
2
Definition 2.1. We define the natural coordinate functions
x : R3 → R by x(~p) = p1
y : R3 → R by y(~p) = p2
z : R3 → R by z(~p) = p3
Alternatively we have x = x1 , y = x2 , z = x3 .
Subtle Distinction
• f = x2 + y 2 + sin(z) → equation involving functions
• f (~p) = x(~p)2 + y(~p)2 + sin(z(~p)) → equation involving numbers
• f (x, y, z) = x2 + y 2 + sin(z) → ambiguous
2.2
Tangent Vectors
A vector field on R2 is a function that assigns a vector ~vp to each point p~ in R2 , V~ : R2 → R2
(V~ (~p) = ~vp ).
In Math 423 we often study vector fields whose domain is a curve or surface in R3 .
Definition 2.2. A tangent vector ~vp to R3 consists of
• point of application p~
• vector ~v at that point
Definition 2.3. Fix p~ ∈ R3 . The tangent space to R3 at p~ is the set Tp R3 of all tangent
vectors ~vp whose point of application is p.
R3 has a different tangent space at each point p~. Tp R3 is a vector space.
Why all these tangent spaces? Well the tangent vector to a curve at a point p is
really a vector at p, i.e., a vector in Tp R3 .
Of course, Tp R3 is a 3D vector space which we can identify with R3 by translating the
point of application p to the origin of R3 .
Definition 2.4. A vector field V on R3 is a function that assigns to each point p~ a tangent
vector, V (~p) to R3 at p~ (i.e. V (~p) ∈ Tp )
3
2.3
Algebraic Operations on Vector Fields
If V, W are vector fields on R3 and f : R3 → R is a function we define V + W , f V , (which
are vector fields on R3 ) by:
• (V + W )(~p) = V (~p) + W (~p) ∈ Tp R3
• (f V )(~p) = f (~p) · V (~p) ∈ Tp R3
Definition 2.5. The natural frame field on R3 is the set consisting of the vector fields
U1 , U2 , U3 on R3 defined by
U1 (~p) = (1, 0, 0)p
U2 (~p) = (0, 1, 0)p
U3 (~p) = (0, 0, 1)p .
At each point p~, {U1 (~p), U2 (~p), U3 (~p)} is the standard basis for Tp R3 .
Lemma 2.6. Let V be a vector field on R3 then there are functions v1 , v2 , v3 : R3 → R so
that
V = v1 U1 + v2 U2 + v3 U3
4