Parallel Transport
Outline
1. Vector Fields Along Curves
Let S be a surface, and let : [a, b] S be a curve on S . A vector eld along is a
function that assigns to each point of a tangent vector to the surface at that point. That is,
a vector eld along is
Derivatives of Vector-Valued Functions
Outline
1. Components
Consider a function general vector-valued function f : Rm Rn . Such a function can be
written as
f(x1 , . . . , xm ) = f1 (x1 , . . . , xm ), . . . , fn (x1 , . . . , xm ) ,
where each fi : Rm R
The First Fundamental Form
Outline
1. Bilinear Forms
Let V be a vector space. A bilinear form on V is a function V V R that is linear in
each variable separately. That is, , is bilinear if
u + v, w = u, w + v, w
and
u, v + w = u, v + u, w
for all u, v, w
Geodesics
Outline
1. Geodesics
A curve on a surface S is called a geodesic if
1. has unit speed, and
2. is normal to S at each point.
For example:
The geodesics on a sphere are great circles.
The geodesics on a cylinder are straight lines, circular cros
Pullbacks, Isometries & Conformal Maps
Outline
1. Pullbacks
Let V and W be vector spaces, and let T : V W be an injective linear transformation.
Given an inner product , on W, the pullback of , is the inner product T ,
on V dened by
T v, v = T (v), T (v
The Second Fundamental Form
Outline
1. Normal Acceleration
Let S be a surface, and let be any curve on S . The normal acceleration of is the
quantity
n = N,
where N is the unit normal vector at each point of S . If is unit-speed, then n is the same
as the
Tangent Vectors, Normal Vectors & Orientation
Outline
1. Tangent Vectors
A vector v is said to be tangent to a surface S at a point p if there exists a curve on S whose
tangent vector at p is v.
For example, if : U S is a surface patch, then the vector pa
Surfaces: Basic Denitions
1. Surface Patches
A surface patch for a surface S is a one-to-one parametrization : U S of a portion of S .
Surface patches are required to be one-to-one, and the domain U must be an open subset
of R2 .
We use the letters (u, v
Final Exam Practice Problems
Math 352, Fall 2011
1. Let S be the portion of the paraboloid z = x2 + y 2 lying below the plane z = 4.
(a) Find the surface area of S .
(b) Evaluate
1 + 4z dA.
S
2. Let P be the plane 5x + 4y + 3z = 36, and let f : P R be the
Functions on Surfaces
Outline
1. Smooth Functions
Let S be a smooth surface, and let f : S Rn be a real-valued function on S
Denition. We say that f is smooth if the composition f is smooth for every smooth
surface patch : U S .
This denition involves eve
Area and Jacobians
Outline
1. Jacobians
Let f : R2 R2 be a smooth map from the uv -plane to the xy -plane. The Jacobian of f is
the absolute value of the determinant of the derivative matrix:
x y x y
u v v u
J f = | det(D f )| =
If R is any region in the
Final Exam Practice Problems
Math 352, Fall 2011
1. Let : (0, 2) (0, 2 ) S be the surface patch
(r, ) = (r cos , r sin , r2 ).
Then the image of is almost all of S , and the Jacobian of is
r
= r
2
2
(a) The surface area is
0
2
2
(b)
0
(r sin , r cos , 0)
Critical Points
Outline
1. Critical Points: Simple Cases
If f : R R is a dierentiable function, a critical point for f is any value of x for which
f (x) = 0. There are two simple generalizations of this denition:
If : R Rn is a dierentiable curve, a crit
Curves on Surfaces
Outline
1. The Darboux Frame
Let S be a surface, and let be a curve on S . At each point on , consider the following
three vectors:
The unit normal vector N to the surface.
The unit tangent vector t to the curve .
The tangent normal