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. Spheric
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
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 calculu
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 i
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 in
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 t
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
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 intercha
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