Calc III modeling projects - part 2

The blog Continuous everywhere and differentiable nowhere, (CEDN) by a high school math teacher in Brooklyn, New York, has a list of Calc III projects.  Here are the ones I felt were modeling projects.  See the blog site for more details, including the overall assignment sheet.

• Volume of an n-dimensional sphere:  Define a sphere in 2D, 3D, 4D, etc.  Find the formula for the volume of an n-dimensional sphere.  Strangely, as the dimension increases, the volume of given radii decreases.  This is related to the problem with data modeling in multidimensions when the measure is a distance.  The higher the dimension, a larger percentage of points are more than a fixed distance from a given point.  Study this weirdness.  Resource: Rogawski, Section 15.3.
• Fluid dynamics and hurricane modeling:  This blogger recommends studying material from Anton, pp. 1883-7, on fluid dynamics and how that study can be used to model hurricanes.
• Lissajous Curves:  The prospect is to study Lissajous curves in 2D, and then consider them in 3D.  Maybe this would be a good 3D printing project.
• River navigation problem:  The blogger describes a navigation problem where the boat has limited maneuvering capabilities and there are safe and unsafe places to land on the shore of a river.  It sounds like an interesting problem that can be modified to consider more variables, such as the difference in current along the shore versus the center of a river.
• Maxwell's Equations:  Research Maxwell’s Equations and explain them. Resource: Daniel Fleisch’s A Student’s Guide to Maxwell’s Equations.
• Ideal Gas Law:  Investigate the Ideal Gas Law (PV = nRT) focusing on what the 3D plotting of P vs. V vs. T looks like, given constants n and R.  The CEDN blog has a spreadsheet that can help, more details for the project, and some online resources. Murphy's note:  Students often have difficulty interpreting plots such as found on the CEDN website, and so I am particularly interested in this project.
• Mathematica Demonstrations:   If you know a thing or two about programming or are willing to learn some basic Mathematica, explore the Wolfram Demonstration Projects for Multivariable Calculus here. Find a topic in Multivariable Calculus that hasn’t been illustrated yet, or one that might be poorly illustrated, and create your own Demonstration. 
• Five Intersecting Tetrahedra:  This is Activity 11 of Thomas Hull's "Project Origami."  There is a problem in the activity that asks the reader to solve a multivariable optimization problem.
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The CEDN blogger mentions harmonographs, which are covered in a previous post in this blog.

Again, these ideas came from the blog Continuous everywhere and differentiable nowhere, and more details, links, and resources can be found there.