SPHERICAL COORDINATES
Contents
1. Spherical coordinates
1
1. Spherical coordinates
We now briey examine another coordinate system which is sometimes convenient
when computing triple integrals. Spherical coordinates are dened in the following
way: a point
VECTOR-VALUED FUNCTIONS, ARC LENGTH, FUNCTIONS OF
SEVERAL VARIABLES
Contents
1.
2.
3.
4.
Vector-valued functions
Arc length
Functions of several variables
Partial derivatives
1
2
2
3
1. Vector-valued functions
A vector-valued function is a function on R w
VECTOR-VALUED FUNCTIONS, ARC LENGTH, FUNCTIONS OF
SEVERAL VARIABLES
Contents
1. Directional derivatives and the gradient
2. Tangent planes and normal lines
1
2
Partial derivatives not only tell us the rate of change of a function f (x, y ) in either
the x
INTEGRATION OF FUNCTIONS OF SEVERAL VARIABLES
Contents
1. Integration
2. Double integrals
3. Iterated integrals and Fubinis Theorem
1
3
4
1. Integration
Now that the quick review of dierential calculus of several variables is nished,
lets start with the n
INTEGRATION OVER NON-RECTANGULAR REGIONS
Contents
1. A slightly more general form of Fubinis Theorem
1
1. A slightly more general form of Fubinis Theorem
We now want to learn how to calculate double integrals over regions in the plane
which are not necess
INTEGRATION IN POLAR COORDINATES
Contents
1. A review of polar coordinates
2. Integrating using polar coordinates
1
2
1. A review of polar coordinates
There are many situations where we may want to integrate a function over a circular
or elliptical domain
APPLICATIONS OF MULTIPLE INTEGRATION
Contents
1. Physical interpretation of integrals
2. Applications to Probability
1
5
1. Physical interpretation of integrals
Having spent a considerable amount of time studying how to evaluate all sorts
of dierent kinds
SURFACE AREA
Contents
1. Surface area
1
1. Surface area
Consider the part of the surface z = f (x, y ) over the region D in the xy -plane. We
saw how the double integral
f (x, y ) dA
D
represents the signed volume of the region between z = f (x, y ) and D
TRIPLE INTEGRATION
Contents
1. Interchanging order of integration
2. Cylindrical coordinates
1
2
1. Interchanging order of integration
Just like in the two-dimensional case, it is possible to interchange the order of
integration when calculating triple in
A RAPID REVIEW OF VECTORS AND GEOMETRY OF R3
Contents
1.
2.
3.
4.
Vectors
The dot product
The cross product
Lines and planes in R3
1
2
2
4
In todays class we will quickly review the content of Chapter 12 in the text, which
covers vectors and the geometry