Visualizations

WOLFRAM DEMONSTRATIONS

Stephen Wolfram has a series of more than 20 great interactive demonstrations of numerous multivariable calculus concepts [To engage with them, you need to download the free Wolfram Mathematica Player (get it here).] A video I made of one of the demonstrations (on spherical coordinates) is here.

MIT DEMONSTRATIONS

MIT has a few math applets up for explorations [here]:

  1. Functions of Two Variables
  2. Lagrange Multiplies (Two Variables)
  3. Curves and Vector Fields
  4. Flux Across Circle
  5. Surfaces and Flux in Space

EXPLORING CURL, DIVERGENCE, AND VECTOR FIELDS WITH FLASH [here]

  1. Curl and Divergence in 2-D
  2. Curl and Divergence in 3-D
  3. Surface Demo
  4. Curl and Divergence in 2-D (sound “visualization”)

FLASH MATHLETS [here]

  1. Parametric Curves
  2. Parametric Surfaces
  3. Spherical Coordinates
  4. Parametric Surfaces in Spherical Coordinates
  5. Parametric Surfaces in Cylindrical Coordinates
  6. 3D Function Grapher
DAVID LITTLE’S APPLETS (here)
  1. Investigate Space Curves
  2. Double Integrals
  3. Line Integrals
ROGNESS’ APPLETS (here)
  1. Interactive Gallery of Quadric Surfaces
  2. Parametrized Surface: A Torus
  3. Animated normal vector on the cone — is this a smooth surface?
  4. Continuously Varying Normal Vectors on a Paraboloid
  5. Geometric Interpretation of Partial Derivatives
  6. Directional Derivative Example
  7. Tangent Plane with Vectors
  8. Estimating Double Integrals
  9. Change of Variables: Polar to Rectangular Coordinates
  10. Change of Variables: A Nonlinear Transformation
  11. Stokes’s Theorem: Infinitely many Surfaces with the same Boundary
  12. Lagrange Multipliers
  13. Animated Position and Velocity vectors
  14. Particles in Motion: Parametric Curves with Velocity Vectors
  15. 2D Curvature Examples: parabola; cubic; quartic; cusp; all (large file)
  16. 3D Curvature Examples: helix; elliptical helix; tornado; exponential spiral; twisted cubic; all (large file)
NYKAMP’S APPLETS (here)
  1. Parametrization of a line
  2. Cross Product
  3. Triple Scalar Product
  4. Determining and Parametrizing a plane
  5. Translating, rescaling, and reflecting surfaces
  6. Level Curves
  7. Directional Derivative and Gradient
  8. Arclength
  9. Derivative of a path
  10. Scalar Path Integrals
  11. Path Integrals of Vector Fields
  12. Divergence and Curl
  13. Surfaces of Revolution
  14. Spherical Coordinates
  15. Parametric Surfaces (Helicoid)
  16. A Nonorientable Surface (Moebius Strip)
  17. The Idea of Stokes’s Theorem
  18. Change of Variables: Triple Integrals
VECTOR FIELD VISUALIZATIONS
  1. San Francisco Wind Bay Patterns (a plot of a vector field)
MVT (Mathematical Visualization Toolkit):
The MVT (The Mathematical Visualization Toolkit) can be used online or downloaded, and can graph 2- and 3-D functions, functions in other coordinate systems (e.g. polar, spherical, cylindrical), and vector and gradient fields (including 3-D vector fields!). It also has a number of built in applications — like revolving curves around axes, fourier series approximations, and riemann sums. This is an incredible teaching and visualization tool.

Paul Seedburger’s Multivariable Calculus page with CalcPlot3D Exploration Applet

Bulter Community College (here)