Send me email if you'd like
a hard or electronic copy of a worksheet on any topic covered
in our MA3160 course.
Worksheets for Multivariable Calculus
Tamara Olson
Michigan Technological University
The postscript documents are printer-ready.
If you'd like to adapt the worksheets for your course,
download the LaTeX version and the accompanying
Mathematica notebook, if any.
There are instructions for
how to view, print, and edit
these documents on the Sun Rays in Fisher 330/331.
Section numbers refer to the relevant section(s) in
CALCULUS, Single and Multivariable
(third edition)
by Hughes-Hallet et alli.
- 12.1: Distances and Spheres (for overhead slide)
- 12.1-2: Introduction to Functions of Two Variables
- 12.3: Contour Diagrams and Graphs of Functions of Two Variables
- 12.5: Functions of Three Variables
- Chapter 13: Vectors Review
- 14.1: The Partial Derivative
- 14.2: Partial Derivative (symbolic)
- 14.3: Tangent Planes and Approximations
- 14.4: Gradients and Directional Derivatives
- 14.5: Gradients in Space (for overhead slide)
- 14.6-7: The Chain Rule and Second Derivatives
- 15_1: Critical Points and Local Extrema (for overhead slide)
- 15_3: Constrained Optimization (Graphical)
- 15_3: Constrained Optimization (Symbolic)
- 16.2: Double Integrals
- 16.3: Triple Integrals
- 16.4: Double Integrals in Polar Coordinates
- 16.5: Cylindrical and Spherical Coordinates
- 16.5: Cylindrical Coordinates
- 16.5: Spherical Coordinates
- 17.1: Parametrizing Circles and Lines
(check solutions using
this Mathematica notebook)
- 17.1-2: Parametrizing Circles and Lines
- 17.2: Position Vectors and Velocity
I distribute these problems and ask students to answer on
the board: (1) is this a circle or line or neither?
(2) if this is a position vector, what is the velocity vector?
(3) what is the velocity of the object at time t=0?
- 17_3: Vector Fields
- 17.1,17.3,18.2:
Parameterized Curves, Vector Fields, and Line Integrals
- 18.4: Path-Dependent Vector Fields and Green's Theorem
- 19.1: Flux Integrals
(examples where either the normal component of F
is constant, or the surface is parallel to a coordinate plane)
- 19.2: Flux Integrals on Spheres, Cylinders, and Graphs
(not available yet)
- 20.1, 20.3:: Visualizing Divergence and Curl
- 20.2, 20.4:: The Divergence Theorem and the Curl Theorems
(Stokes' Theorem)
- That's all I have right now!
- section: Title
All of these worksheets are under continual revision.
If you'd like an an updated version, just
send me email.
If you have any suggestions for corrections, additions,
or improvements, please let me know.
trolson@mtu.edu.
Tamara Olson
Dept. of Mathematical Sciences
Michigan Technological University