Sites for the Calculus 3 student
Many of the sites in this page were found by students studying Calculus 3.
They were asked to find sites which would help future Calculus 3 students.
It is our hope that their efforts might open new avenues of learning and
understanding for you as a Calculus student. We also hope you can see
applications of your work and as one student commented see that "...there
is a reason we've lugged around the heaviest book in the bookstore for
This site contains the following sections:
Sites you might want to visit first
- A good place to start may be with the story of
who was one of the founders of Calculus. As one student noted, "Newton
gives us an example of why we should study Calculus...to solve real
world problems and situations." Thanks to Lisa Thurston
- Next, if you need to review Calculus concepts from previous semesters
visit the site
Finite Mathematics and Calculus Applied the the Real World. The site contains
many links. You may find the links regarding Calculus helpful.
Thanks to Brian Barndt
- Finally, a good site to bookmark for the semester is the
Guide for Vector Calculus is an on-line study guide from Oregon
State University. The study guide parallels much of the Calculus 3
course at CU. You may want to visit it often for additional insight
and help. The site contains many graphics, examples, and applications
to the concepts you will be learning. Thanks to Jennifer Anthony
and Welton Hong
-- This site contains a basic introduction to vectors and vector
arithmetic. Thanks to Bryan Ploetz
- Vectors and
Velociraptors -- This site describes how vectors were used to create
the dinosaurs in Jurassic Park. The site also discusses the use
of vectors in art.
- Apollo 13
and Vector Calculus -- See how Vector Calculus played a part in the
Apollo 13 mission. The astronauts used Calculus concepts when it was
important to do the math correct, as it was life or death. Thanks to
Mouji Barash and Matt Morris
NASA Shuttle Web: Shuttle Sightings -- How can you sight a Space
Shuttle in orbit? This page discusses how and gives a link to actual
launch and orbit information for a shuttle. The link with state vectors
shows the mass of the shuttle, time of launch, and directional vectors for
the shuttle's expected path. Thanks to Kyle Scoby
Cross Product - JAVA Interactive Tutorial -- This interactive site
allows you to interactively explore the cross product of two vectors.
Utilities -- An interactive site that calculates a unit vector, dot
product, scalar product, and triple scalar product.
- Lesson 8 -
Scalar Product - This site is a lesson on finding the dot product.
The page also gives two examples of using the dot product in engineering
- Lesson 9 -
Triple Product -- This sites is a lesson on finding a Triple Scalar
Product. The page also gives an example of using such calculations in
Coordinates -- This site contains animations, examples, and problems to test your
understanding of the cylindrical coordinate system. Thanks to Bethany Franko
Coordinates -- This site also contains animations, examples, and problems to test your
understanding of spherical coordinate system. Thanks to Bethany Franko
Systems in Space -- Learn about cartesian, cylindrical, and spherical
coordinate systems and the relationship between them. Thanks to Brian Wiltshire
Systems -- Another site offering insight into this important topic.
Surfaces -- Identification - This site introduces ideas on identifying
quadric surfaces from the associated equations. Note that this site grew
out of a 2350 recitation.
3.0 and Quadric Surfaces -- This site introduces the process of
graphing quadric surfaces with Mathematica 3.0. Note, this site grew out
of a 2350 recitation.
Surfaces - This site is a helpful table showing the equations of the
quadric surfaces discussed in class and the associated surface.
Thanks to Bryan Browne
- My Humble
Graphics Page: The Quadrics -- a helpful page on quadrics. Raytraced
quadrics are presented as well as a helpful table at the end. Thanks
to Peter Loftsgordon and Nicholas Saucier
Quadrics -- This site contains actually 17 possible kinds of quadric
surfaces including imaginary ones (if you like to use your
imagination.)Thanks to Bryan Browne
Mirror Telescopes -- This page explains the use of the parabolic
shape in telescopes like the Hubble Telescope.
Motion -- An interactive Java script which lets you experiment with
and the Mathematics of Motion -- This site discusses how Galileo
studied the motion of the canon ball and derived the formulas for
projectile motion. Thanks to Brian Frenkel
Level Curves and Level Surfaces
- Surfaces and
Level Curves -- A discussion of level curves and their uses. The
page contains ample pictures and examples.
Surfaces -- A discussion of level surfaces with ample pictures and
USGS Shaded Relief Map -- We usually think of starting with a
surface and obtaining the level curves (contours) from it. This site
gives examples of shaded relief maps which start with a contour map (a
map of the level curves) and mathematically reconstruct the surface
from the level curves!
Maxima, Minima, and Lagrange Multipliers
- Volume --
A helpful site on finding volume with multiple integrals. The site contains graphics,
animations, and example problems. Thanks to Bethany Franko
Mathematica -- This page discusses how to find multiple
integrals with Mathematica. The page was authored by a 2350 teaching
assistant and contains examples from the 2350 textbook. Thanks to
- The Integrator -- An
on-line integrator as part of Wolfram
Research: Makers of Mathematica. Thanks to Cory Hooper
Integrals -- This page discusses double integrals. The page begins
with integration over a rectangular region and progresses to more general
regions. Applications of double integrals are also discussed. Thanks
to Brian Mears.
- Double Integrals
in Polar Coordinates -- This page discusses double integrals using polar
coordinates. There are several pictures to aid the discussion.
Integrals -- This page discusses triple integrals. The page begins
with integration over a box-liked region and progresses to more general
regions. Applications of triple integrals are also discussed. Thanks
to Brian Mears.
- Triple Integrals
in Cylindrical and Spherical Coordinates -- This pages contains a helpful
discussion on finding multiple integrals using cylindrical and spherical coordinates.
Multiple Integrals using Maple -- A helpful site if you would like to use Maple to find
double or triple integrals. The site also covers the topic of centroids and centers of mass.
Thanks to Jim Gerend
graphics -- This page contains many graphics to enhance the teaching
and learning of Calculus concepts. In particular, this link provides a
discussion regarding the ideas of volume calculations in 3 dimensions.
Thanks to Chris Staab
Centroids and Moment of Inertia
- Area Moment of
Inertia -- This web site discuses moments of inertia, center of mass,
and radius of gyration. In particular, the page discusses why an engineer
need by concerned with moments of inertia. Pictures and examples are
included. Thanks to Erin Coffman
and Moment of Inertia -- A helpful discussion on this topic with ample
pictures and examples.
of Inertia Calculator -- An interactive site allowing you to click a
figure with your mouse to choose a horizontal axis about which the object
will be rotated. Then the computer returns the moment of inertia of the
object about that axis.
Bending, and Boundary Conditions: Centroids -- This site contains many
questions to deepen your understanding of the centroid of an object.
Flux and the Divergence Theorem
Sites related to Mathematica and Maple
The following sites contain notebooks and labs that may aid in your studies during the semester.
The following sites apply the topics of Calculus 3. The first several sites
apply vector Calculus to computer graphics. The last site applies
Calculus to a proposed theory for the origin of the universe.
Note, the sites may not explicitly mention or cover Calculus
Several students found sites which apply Calculus to space related field.
Note, the sites may not explicitly mention or cover Calculus
- POV-Ray - the Persistence of
Vision Raytracer -- Raytracing and 3D graphics are a real world
example of how Calculus 3 may be applied. It is all too easy to forget
that raytracing is simple vector calculus used billions of times over.
Thanks for Peter Loftsgordon
- The Animation Process -- Learn more about the animation process in computer graphics.
This site discusses how animation is done in the movie Toy Story.
Thanks to Gary Ting
- Toy Story
-- Toy Story was the first full feature computer animated film. Computer graphics are an active application of Vector Calculus. This site also contains the
History of Computer Graphics. Thanks to Gary Ting
Character Studio -- is an innovative character animation system. The site
contains ample information about the package and its unique approach to computer
graphics and even includes downloading options. To view a humorous example of an animation
created with this software that is not linked from the page,
Sculpting Software -- This computer software allows interactive
computer-assisted editing and free-form sculpting of three-dimensional
(3D) polygonal mesh surface models. The field of computer graphics
applies many facets of vector Calculus.
- Bryce 2 -- A 3D
terrain graphics package called Bryce. Although the actual math the
program itself uses is high level, the student can imagine using multiple
integrals, vectors, splines, ect on the objects. Thanks to Mike
- Lunar Rover Initiative --
This page contains extensive information on the lunar rover. In particular,
the Current Design page contains many uses of mathematics. Thanks to
- The Scale
Expanding Cosmos -- This site shows the application of calculus to
interpret a new theory as to the origin of the universe in opposition to
the Big Bang Theory. Thanks to Brad Bailey
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