Lecture XIV
Multiple Integrals
1
Integrals of onevariable functions
For a realvalued function f dened on an interval [a, b], the integral of f over
b
[a, b], denoted by a f dx, is dened as follows:
n
b
f dx =
a
lim
n,max xi 0
f (x )xi ,
i
i=1
where a = x1
Lecture XIII
TwoVariable Test
Constrained MaximumMinimum Problems
1
The twovariable test
Recall that by the Critical Point Theorem, onlyif the gradient of a function f
at P is 0 (i.e. P is a critical point for f ), can P be an extreme point of f .
Let f (
Lecture XII
Terminology for PointSets in Euclidean Spaces and
MinimumMaximum Theorems
First let us take a short look at a problem that was on the exam. We are
given a level curve (in E2 ) or a surface(in E3 ) and a point P on that curve or
surface. How to
Lecture XI
Chain Rule: Elimination Method
Let w = f (x, y ) be a dierentiable function of x and y . The linear approximation
of f is given by
fapp = fx (x, y )x + fy (x, y )y.
We introduce a new notation, the dierential notation for the increments f,
x, y
Lecture X
Linear Approximation
Chain Rule
1
Linear Approximation; Gradient
We say that a function has a linear approximation on a domain D if it has a
linear approximation at any point P D.
Theorem 1 If f has a linear approximation on a domain D, then f i
Lecture IX
Linear Approximation
1
Onevariable Functions
Let f be a onevariable function on a domain D. If f is dierentiable at x = c,
we say f has a linear approximation, which we dene in the following way.
Denition 1 If f is a function dierentiable at c,
Lecture VIII
Scalar Fields
Cylindrical Coordinates
1
Scalar Fields
Denition 1 Let D be a subset of E3 . A function f that associates each point
P in D to a real number f (P ) is called a scalar eld. D is called the domain
of f .
Denition 2 Let f be a scal
Lecture VII
Paths and Curves
First we go through several basic notions about paths. Let R(t) on [a, b] be a
given path.
Denition 1 R(t) is called elementary if for every pair (t1 , t2 ), with t1 and t2
distinct in [a, b], R(t1 ) = R(t2 ).
Denition 2 R(t)
Lecture VI
Calculus of Vector Functions
dR
dt
2
denotes the rst-order derivative of R(t), and that d tR ded2
notes the second-order derivative of R(t). We introduce new notations for these
2
i
j
functions: dR = R(t) and d R = R(t). Let R(t) = a1 (t) + a2
Lecture V
Calculus of OneVariable Functions
Let us rst review some denitions in calculus on real numbers.
In order to dene limit on the real numbers we will use the concept of funnel
functions.
Denition 1 A function (t) on [0, d] is called a funnel functi
Lecture IV Analytic Geometry in E2 and E3
First we review some basic facts of analytic geometry in E2 . Let us consider a Cartesian coordinate system in E2 . We denote by F [x, y ] an algebraic formula in the variables x and y . Any equation of the form F
Lecture III
Vector Algebra in Cartesian Coordinates
Let us construct a Cartesian coordinates system in E3 . First we choose a
point O, called the origin. Then we chose three mutually perpendicular rays
starting from O. These rays are called the positive x
Lecture II
Vectors and Vector Algebra
A set S of rays is called a direction if it satises the following laws:
(1) Any two rays in S have the same direction.
(2) Every ray that has the same direction as some member of S is in S .
A vector A consists of a n
18.022 Lecture notes
The course is divided into 6 parts:
Part 1 (Lectures I VII): Euclidean Spaces and Vector Algebra
Part 2 (VIII XIII): Differential Calculas for Scalar Fields and Functions of Several Real
Variables.
Part 3 (XIV XVII): Multiple Integral
Lecture XXXIV
Subspaces
In the previous lecture we have seen that there are two methods for nding
the inverse of a square matrix A. In the rst method, we use the fact that if A
is nonsingular, then the rowreduced form of [A : I ] is [I : A1 ]. In the seco
Lecture XXXIII
Determinants; Matrix Algebra
1
Determinants
For a square matrix A, the determinant of A has the following properties:
1. Interchanging two rows of the matrix multiplies the value of the determi
nant by 1.
2. If there exists two identical ro
Lecture XXXII
Row Reduction; Determinants
1
Row Reduction
Recall the 3 elementary operations we will use to solve the system of equations
AX = D:
() multiplying an equation by a nonzero scalar;
( ) adding to an equation some multiple of a dierent equation
Lecture XXXI
Linear Equation Systems
As we saw in the previous lecture, we can multiply m n matrices by column
nvectors. Consider the rows of an m n matrix A to be nvectors:
1 C
r
c1
1
r
2
2 C
c2
r
r
=
C =
then
A
A = . , C .
.
.
.
.
.
.
.
m C
r
Lecture XXX
nVectors and Matrices
1
nVectors
We dene En to be the ndimensional Euclidean space, and Rn to be the set
of points in En . Hence Rn is the set of all ordered ntuples of real numbers
(x1 , . . . , xn ). Ordered ntuples allow repetitions, i.e. x
Lecture XXIX
Mathematical Applications
1
Leibnitzs Rule
Leibnitzs Rule : Let f (x, t) be a C 1 function dened for a x b. Then
b
db
f
f (x, t)dx =
ft (x, t)dx, where ft (x, t) =
.
dt a
t
a
In other words, if we dene g (t) =
b
A
f (x, t)dx, then
dg
dt
=
b
a
Lecture XXVIII
Measures; Irrotational elds
1
Circulation and ux measures
Let us rst see what the theorems from the past lectures say about the circula
tion and ux measures. From Greens theorem, we get that, if F is a C 1 vector
eld on D in E2 , then
F (R)
Lecture XXVII
Stokess Theorem
In the previous lecture, we saw how Greens theorem deals with integrals on
elementary regions in E2 and their boundaries. We also saw how the divergence
thoerem deals with elementary regions in E3 and their boundaries. We wil
Lecture XXVI
The Divergence Theorem
In this lecture, we will dene a new type of derivative for vector elds on E3 ,
called divergence. Let F be a vector eld dened on a domain D. Let us start
by dening the divergence of F on interior points of D, i.e. point
Lecture XXV
Greens Theorem
Let us dene a new type of derivative, called rotational derivative, applicable to
vector elds in E2 . For such a vector eld F on a domain D in E2 , let us dene
the rotational derivative at interior points of D. Here, a point P i