Study Guide 3
(Math 2423, Calculus III-B, Section 002, Fall 2000)
Homework
Read sections 13.6-13.8, and work the following
problems for the quiz on Tuesday, November 7. YOU MUST WORK ALL PROBLEMS!!
However, you only need to turn in the problems marked by an asterisk.
These are due Tues. Nov. 7 at the beginning of class.
- Section 13:6: 3, 9, 13, 14*, 19, 30*
- Section 13.7: 1, 5, 9, 12, 16*, 29* (turn in a graph of the torus too).
- Section 14.1: 9, 12*, 23, 29, 33, 37, 44, 45*, 53, 57, 63
Overview:
Once you've mastered double integrals, triple
integrals are not much different conceptually. The only problem is
that the regions you are integrating over are three dimensional--so it
can be a challenge getting the limits of integration right. (So you need
to practice!!). Also, with triple integrals, we now have three
different coordinate systems to worry about (rectangular, cylindrical,
and spherical).
Chapter 14 deals with vector fields. These are vector-valued functions of several variables. Section 14.1 defines a bunch of things that we will be using a lot of in the next few weeks. So take the time now to make sure you are comfortable with this terminology. The trickiest skill in 14.1 is finding a potential function for a conservative vector field. (SO BE SURE TO READ EXAMPLE 6, AND WORK PROBLEMS 29,33,44, AND 45).
Key Skills :
- Evaluate a triple iterated integral. (13.6: 3,10, )
- Determine limits of integration that describe a volume in three
dimensions.
- Sketch a volume corresponding to the limits of integration of a triple integral (13.6: 13, 14)
- Change the order of integration of a triple integral (note: this involves changing the limits of integration). (13.6: 13,14)
- Find the volume of a solid using a triple integral. (13.6: 19)
- Find the mass and center of mass of a solid, given its density function.
- Evaluate a triple integrals in cylindrical and spherical coordinates. (13.7: 1,5)
- Sketch a solid corresponding to the limits of integration for a triple integral in either cylindrical or spherical coordinates. (13.7: 9,12)
- Convert a triple integral from rectangular to cylindrical or spherical coordinates (13.7: 16)
- Sketch a vector field. (14.1: 9,12)
- Find the gradient vector field for a scalar function (just calculate the gradient) (14.1: 23)
- Determine whether a vector field is conservative.
(In 2-D, 14.1: 29, 33, in 3-D, 14.1: 44,45)
- Find a potential function for a vector field (Note: you can only do this if the vector field is conservative, so be sure to check if it is conservative first) (In 2-D, 14.1: 29, 33, in 3-D, 14.1: 44,45)
- Find the curl of a vector field. (14.1: 37)
- Find the divergence of a vector field. (14.1; 53, 57, 63)