**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]:

- Functions of Two Variables
- Lagrange Multiplies (Two Variables)
- Curves and Vector Fields
- Flux Across Circle
- Surfaces and Flux in Space

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

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

**FLASH MATHLETS** [here]

- Parametric Curves
- Parametric Surfaces
- Spherical Coordinates
- Parametric Surfaces in Spherical Coordinates
- Parametric Surfaces in Cylindrical Coordinates
- 3D Function Grapher

**DAVID LITTLE’S APPLETS**(here)

**ROGNESS’ APPLET**S (here)

- Interactive Gallery of Quadric Surfaces
- Parametrized Surface: A Torus
- Animated normal vector on the cone — is this a smooth surface?
- Continuously Varying Normal Vectors on a Paraboloid
- Geometric Interpretation of Partial Derivatives
- Directional Derivative Example
- Tangent Plane with Vectors
- Estimating Double Integrals
- Change of Variables: Polar to Rectangular Coordinates
- Change of Variables: A Nonlinear Transformation
- Stokes’s Theorem: Infinitely many Surfaces with the same Boundary
- Lagrange Multipliers
- Animated Position and Velocity vectors
- Particles in Motion: Parametric Curves with Velocity Vectors
- 2D Curvature Examples: parabola; cubic; quartic; cusp; all (large file)
- 3D Curvature Examples: helix; elliptical helix; tornado; exponential spiral; twisted cubic; all (large file)

**NYKAMP’S APPLETS**(here)

- Parametrization of a line
- Cross Product
- Triple Scalar Product
- Determining and Parametrizing a plane
- Translating, rescaling, and reflecting surfaces
- Level Curves
- Directional Derivative and Gradient
- Arclength
- Derivative of a path
- Scalar Path Integrals
- Path Integrals of Vector Fields
- Divergence and Curl
- Surfaces of Revolution
- Spherical Coordinates
- Parametric Surfaces (Helicoid)
- A Nonorientable Surface (Moebius Strip)
- The Idea of Stokes’s Theorem
- Change of Variables: Triple Integrals

**VECTOR FIELD VISUALIZATIONS**

- 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 Seeburger’s Multivariable Calculus page with CalcPlot3D Exploration Applet**

**Bulter Community College**(here)