Sites for the Calculus 3 student

• Sites you might want to visit first
• 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
• Additionally, you may want to examine the numerous contributions of Women in Mathematics. This is a collection of the biographies of many of the women who played an important role in the development of Mathematics and Calculus. Learn about Maria Agnesi, and see if you recognize the curve which made her famous - the "Witch of Agnesi". Thanks to Julie Burland
• 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
• Fairly complete sets of notes for similar courses taught at UBC can be found here and here

Vectors

• Vectors -- This site contains a basic introduction to vectors and vector arithmetic. Thanks to Bryan Ploetz
• 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

Vector Calculations

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

Coordinate Systems

• Cylindrical Coordinates -- Examine cylindrical coordinates in more detail.Thanks to Alex Lombardia
• Spherical Coordinates -- Examine spherical coordinates in more detail.Thanks to Alex Lombardia
• Coordinate Systems in Space -- Learn about cartesian, cylindrical, and spherical coordinate systems and the relationship between them. Thanks to Brian Wiltshire

Quadric Surfaces

• 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.

Level Curves and Level Surfaces

• 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

• 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

Tangent Planes

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

Gradient

• 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

• 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

• The Integrator -- An on-line integrator as part of Wolfram Research: Makers of Mathematica. Thanks to Cory Hooper
• 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

• 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 -- 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

Flux and the Divergence Theorem

• 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

• 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
• History of Computer Graphics. Thanks to Gary Ting

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