THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 12 Critical Points
Let f : Rn R be a function which is of C2 , i.e., all partial derivatives upto 2nd
order are continuous. The Hessia
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 7 Coordinates
7.1
Cartesian Coordinates
The Euclidean space is dened by giving algebraic and inner product structures to the
underlyin
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 11 Constraint
11.1
A Side Condition
11.1.1
Simple Example
Let us nd the point on the curve cfw_ (x, y) R2 : xy = 1 such that the poin
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 10 Max & Min
10.1
Largest and Smallest
10.1.1
Supremum and Innmum
Let f : Rn R and R is a subset (sometimes called region). We conside
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 6 Limits of a function
6.1
Basics
6.1.1
Denition
Given f : Rn R and a point a (always interior), we say lim f (x) =
xa
if
for every >
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 9b Implicit Functions
9b.1
Implicit Dierentiation
9b.1.1
Recall one variable
Consider the equation y 2 + xy + 4x2 x3 = 6, try to expre
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 5 Multivariable Functions
5.1
Between Euclidean Spaces
As we have seen, a curve is indeed a function from J R to Rn . The most general
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 8 Dierentiable Functions
8.1
8.1.1
Dierentiability
Revisit one variable
Usual denition. A function f : J R R is dierentiable at a J if
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 9a Composition of Functions
9a.1
Dierentiating Composition
The chain rule is about composition of functions. Let f : Rn
Rm and
g:
R
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 4 Curves
4.1
1-Dimensional Objects
The simple geometric objects in Rn are hyperplanes (lines and planes are included) and
quadric hype
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 5a Topological Terminologies
5a.1
Situations in Rn
In R, we usually work on intervals (a, b) or [a, b]. These subsets are important an
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 3b Quadric Surfaces
Standard Forms
A quadric surface in R3 is given by the following set,
(x, y, z) R3 : Ax2 + By 2 + Cz 2 + 2P xy + 2
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 3 Quadratic Objects
3.1
Important Objects
The simplest objects in Rn are the lines, planes, k-dimensional hyperplanes. What are
the ne
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 2 Euclidean Geometry
2.1
Euclidean Spaces
The Euclidean space actually involves a set with algebraic and geometric structures.
2.1.1
n
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 1 Multivariable Preview
1.1
Review
There is certain similarity in the study of 1-variable and multivariable dierential calculus. For
t
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2010A (First Term, 20112012)
Advanced Calculus I
Notes 3a More about Quadratic
Be careful
Consider an object S dened by a quadratic expression, for example, in R2 or R3 ,
cfw_
S = (x, y) R