Chapter 6
Curves
6.1
Curves as a map from R to Rn
As weve seen, we can say that a parameterized dierentiable curve is a
dierentiable map from an open interval I = (, ) to Rn . Dierentiability
here means that each of our coordinate functions are dierentiab
Chapter 5
First Fundamental Form
5.1
Tangent Planes
One important tool for studying surfaces is the tangent plane. Given a given
regular parametrized surface S embedded in Rn and a point p S , a tangent
vector to S at p is a vector in Rn that is the tange
Chapter 4
Implicit Function Theorem
4.1
Implicit Functions
Theorem 4.1.1. Implicit Function Theorem Suppose f : Rn Rm
Rm is continuously dierentiable in an open set containing (a, b) and f (a, b) =
0. Let M be the m m matrix Dn+j f i (a, b), 1 i, j m If
Chapter 3
Inverse Function Theorem
(This lecture was given Thursday, September 16, 2004.)
3.1
Partial Derivatives
Denition 3.1.1. If f : Rn Rm and a Rn , then the limit
f (a1 , . . . , ai + h, . . . , an ) f (a1 , . . . , an )
h0
h
Di f (a) = lim
(3.1)
is
Chapter 2
A Review on Dierentiation
Reading: Spivak pp. 15-34, or Rudin 211-220
2.1
Dierentiation
Recall from 18.01 that
Denition 2.1.1. A function f : Rn Rm is dierentiable at a Rn if
there exists a linear transformation : Rn Rm such that
|f (a + h) f (a