Sites for the Calculus 3 student

• Sites you might want to visit first
• Receiving credit for your work on the Web
• Vectors
• Vector Calculations
• Coordinate Systems
• Quadric Surfaces
• Projectile Motion
• Level Curves and Level Surfaces
• Partial Derivatives
• The Chain Rule
• Tangent Planes
• Gradient
• Maxima, Minima, and Lagrange Multipliers
• Integration
• Centroids and Moments of Inertia
• Green's Theorem
• Stoke's Theorem
• Flux and the Divergence Theorem
• Sites related to Mathematica and Maple
• Computer Graphics
• Applications to Space

Sites you might want to visit first

• A good place to start may be with the story of Isaac Newton 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 Study 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

Vectors

• Vectors -- 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

Vector Calculations

• Vector Cross Product - JAVA Interactive Tutorial -- This interactive site allows you to interactively explore the cross product of two vectors.
• Vector 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 type situations.
• 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 statics.

Coordinate Systems

• Cylindrical Coordinates -- This site contains animations, examples, and problems to test your understanding of the cylindrical coordinate system. Thanks to Bethany Franko
• Spherical Coordinates -- This site also contains animations, examples, and problems to test your understanding of spherical coordinate system. Thanks to Bethany Franko
• Coordinate Systems in Space -- Learn about cartesian, cylindrical, and spherical coordinate systems and the relationship between them. Thanks to Brian Wiltshire
• Coordinate Systems -- Another site offering insight into this important topic.

Quadric Surfaces

• Quadric Surfaces -- Identification - This site introduces ideas on identifying quadric surfaces from the associated equations. Note that this site grew out of a 2350 recitation.
• Mathematica 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.
• Quadric 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
• 16 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
• Liquid Mirror Telescopes -- This page explains the use of the parabolic shape in telescopes like the Hubble Telescope.

Projectile Motion

• Projectile Motion -- An interactive Java script which lets you experiment with projectiles.
• Galileo 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.
• Level Surfaces -- A discussion of level surfaces with ample pictures and examples.
• Flagstaff 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!

Partial Derivatives

• Partial Derivatives -- A helpful discussion on partial derivatives with explanations and illustrations. Thanks to Welton Hong
• A function whose mixed partials are unequal -- A function must satisfy certain conditions in order for the mixed partials to be equal. This site discusses these conditions and provides an example of a function which does not meet them. Thanks to Ali Eftekhari
• A non-differentiable function with partial derivatives everywhere -- This site provides a helpful discussion on this topic and supplies a graph to help with visualization of the ideas. Thanks to Ali Eftekhari
• ADIFOR: Automatic Differentiation of Fortran Codes -- Engineers often use FORTRAN. ADIFOR computes partial derivatives of given functions with specified dependent variables with respect to all of the specified independent variables. Thanks to Marc Kroll

The Chain Rule

• Tutorial for the Chain Rule -- A helpful tutorial of using the Chain Rule from previous semesters of Calculus. Thanks to Brian Barndt
• The Chain Rule and Second Partials -- A page explaining how to find second partials using the Chain Rule. This Web page grew out of a 2350 recitation.
• The Chain Rule for Functions of Two Variables -- Contains helpful discussion, graphics and examples.

Tangent Planes

• Tangent Planes To Surfaces -- This site contains several sections discussing this topic with graphs and questions. Thanks to Jake Klein
• Tangent Planes and Total Differentials -- This site supplements the book's discussion of these topics.
• Tangent Planes:TOC -- This site offers animations to "illustrate the intuitive notion of the tangent plane..." Thanks to Frank Melara

Gradient

• The Gradient and Directional Derivative -- A discussion on gradient and directional derivatives with pictures and an example.
• Temperature Gradients -- See the use of gradients in a temperature map. Apply your knowledge to an example.
• A non-differentiable function with all directional derivatives -- This site provides a helpful discussion on this topic and supplies a graph to help with visualization of the ideas. Thanks to Ali Eftekhari

Maxima, Minima, and Lagrange Multipliers

• Maxima and Minima of Functions of Two Variables -- This site also comes from the Study Guide for Vector Calculus at Oregon State University. It contains graphics and examples.
• Lagrange Multipliers -- This site also comes from the Study Guide for Vector Calculus at Oregon State University. It contains graphics and examples.
• Lagrange Multipliers Lab -- This site discusses what Lagrange multipliers are and applications of their use.
• A function with a single critical point, which is a local minimum but not a global minimum -- See the graph of a function with a single critical point which is a local minimum but not a global minimum. Thanks to Ali Eftekhari

Integration

• Volume -- A helpful site on finding volume with multiple integrals. The site contains graphics, animations, and example problems. Thanks to Bethany Franko
• Integration and 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 Mark Werner
• The Integrator -- An on-line integrator as part of Wolfram Research: Makers of Mathematica. Thanks to Cory Hooper
• Double 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.
• Triple 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
• Calculus 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
• Centroids and Moment of Inertia -- A helpful discussion on this topic with ample pictures and examples.
• Moment 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.
• Beams, Bending, and Boundary Conditions: Centroids -- This site contains many questions to deepen your understanding of the centroid of an object.

Green's Theorem

• Green's Theorem -- A general discussion on Green's Theorem. Thanks to LuAnna Sago
• Green's Theorem -- Use Green's Theorem on a Triangle. Note, the page uses Maple.
• Areas with Gauss-Green Formula -- Find area using Gauss-Green's Theorem. Note, the page uses Mathematica. Thanks to Ryan Kerr

Stoke's Theorem

• Lecture Notes on Stoke's Theorem --This site contains pictures and a few examples on Stoke's Theorem. Thanks to Woody Cain
• Stoke's Theorem -- Pictures, discussion and examples regarding Stoke's Theorem.

Flux and the Divergence Theorem

• Divergence and Curl of Vector Fields -- A short page containing explanation and examples of Divergence and Curl of a vector field.
• Flux -- A discussion of flux with ample pictures and an example.
• The Divergence Theorem -- A discussion of the Divergence Theorem with pictures and an example.
• Surface Integrals, Divergence Theorem, and the Del Operator -- This site contains pictures and a few examples. Thanks to Woody Cain
• Flows of Vector Fields -- An interactive Java applet regarding the flow of vector fields.
• Earth's Gravitational Field -- See how the Divergence Theorem and Flux apply to the work at NASA. Thanks to Shane Murphy

Sites related to Mathematica and Maple

• Computational Labs in Mathematica for Vector Calculus -- a large index of labs in Mathematica to aid in the study of Vector Calculus.
• Laboratory Manual for Calculus -- this site includes a lab manual for mathcad and Maple V as they can be used for calculus. It gives instructions on how to use these programs to solve Calculus problems. Thanks to Matt Lutters
• Calculus&Mathematica Home Page -- This site presents a course which combines computers and mathematics in the teaching of Calculus. Thanks to Mark Krom
• Multivariable Calculus in the Lab -- a large index of pages with examples of Maple code in Vector Calculus.
• Advanced Mathematics for Applications Mathematica Notebooks -- a series of Mathematica notebooks to download related to Vector Calculus and Differential Equations.
• Labs and Projects for Calculus using Maple -- This page contains several Maple labs. Thanks to Jim Gerend

Computer Graphics

• 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, click here
• Computer-aided 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 Micucci

Applications to Space

• 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 David D'Silva
• 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

Receiving Credit for your work on the Web

Before you begin the form, copy the name of the site you visited. (It was the name you clicked which sent you to the site you visited.) You will be asked for this information in the form.

When you are ready, click here to submit your work.

http://amath-www.colorado.edu/appm/student/chartier/Calc3/calc3site.html
Last modified: May, 1997
visits this semester.