Multivariable calculus

One of my current interests is in developing material for multivariable calculus. The material in multivariable calculus (our MATH 280) is rich and deep. Much of it can be approached from a more geometric viewpoint than most texts provide. In the last few iterations I have taught of this course, I have been writing handouts (worksheets, problems sets, and exposition) to supplement the common text chosen by the department. My thinking on this topic has been positively influenced by Tevian Dray and Corinne Manogue at Oregon State University through my participation in their Vector Calculus Bridge Project.

Content themes on which I am currently developing material include

• using "total from density'' as the main theme for integration starting with single integrals and continuing through multiple integrals to line and surface integrals (of scalar-valued functions),
• introducing divergence and curl with geometric definitions as flux and circulation densities (following the style of the classic book Div, Grad, Curl, and All That),
• defining line and surface integrals for vector-valued functions geometrically (as opposed to the more common parametrization-based definitions), and
• approaching the divergence theorem and Stoke's theorem as examples of ``total from density''.

Below are links to talks and current versions of relevant course material. You can also check out my course web pages for MATH 280 which provide a bit of context for the course material.

I've also experimented a bit with web-based interactive three-dimensional visualizations, although I have fallen behind in this in comparison to what I do in class using Mathematica. Current versions are in the 3D picture gallery. Most of these were developed starting in Mathematica, then exporting to the JavaView format and refining. In the near future, I hope to explore other options for more efficiently generating this type of visualization.

Talks

• "Density in the calculus sequence", Pacific Northwest Section of the Mathematical Association of America Meeting, Seattle University, April 2010.

Course materials

• Estimating greatest rate of change (Fall 2011 version)
  The first two pages form a handout that I use for an in-class small group activity. I use the remaining pages in debriefing the activity with the class as a whole.
• Components of the gradient vector (Fall 2011 version)
  This handout summarizes the argument I develop in class to go from a geometric definition of gradient to a component expression in cartesian coordinates.
• Gradient vector fields (Fall 2011 version)
  Students work individually on this handout in class.
• Problems on differentials (Fall 2011 version)
  I assign these problems as homework.
• Directional derivatives (Spring 2011 version)
  I have used this as an in-class exercise once and plan to refine it for future use.
• Length density, area density, and volume density (Fall 2011 version)
  This handout gives elementary examples of length, area, and volume densities. I plan to expand this by including more examples of density for quantities other than mass.
• Describing and integrating nonuniform density on a line segment (Fall 2011 version)
  The goal of this handout, with problems included, is to get students thinking about non-uniform density.
• Problems: Total from area density (Fall 2011 version)
  In each of these problems, a student computes a total from a non-uniform area density. Each problem is worded in geometric terms without reference to a coordinate system. Relevant parameters are named but not given specific values. In class, I model an approach that typically includes these elements:
  • Sketch a relevant schematic picture of the physical scenario and then set up a general double integral in a form such as \(M=\int_R\sigma\,dA\).
  • Choose a coordinate system. Then describe the region \(R\) with bounds on the relevant coordinates and find expressions for the density \(\sigma\) and the area element \(dA\) in the terms of the chosen coordinates.
  • Set up an iterated integral equivalent to the double integral (often with some mention of Fubini's theorem as telling us the two different objects have the same value).
  • Evaluate the iterated integral (often "by hand" and occasionally with computing technology). Emphasize factoring integrals whenever relevant to organize and simplify calculations.
  • Do consistency checks on the result, usually including checking units and comparing to some relevant easy-to-compute result such as the total for a uniform density of the maximum value.
  I model this approach for most integration problems such as those below.
• Problems: Total from volume density (Fall 2011 version)
  These problems involve computing a total from a non-uniform volume density. (Note: Many of the problems on this and related handouts involve charge distributions that would be physically difficult to arrange. In future versions, I will replace some uses of charge density with something like number density for a distribution of bacteria. I would also like to work in some probability density examples.)
• The volume element in spherical coordinates (Fall 2011 version)
  This handout serves as a guide for making a geometric argument for the volume element in spherical coordinates. I have students work through this in small groups in class.
• Integration over a curve (Fall 2011 version)
  This handout presents a geometric view of integrating over a curve that is not primarily in terms of a parametrized curve.
• Integration over a surface (Fall 2011 version)
  This handout presents a geometric view of integrating over a surface that is not primarily in terms of a parametrized curve.
• Vector field plots (Fall 2011 version)
  The first two pages form an in-class worksheet. I use the last two pages in debriefing after students have had some time to sketch vector fields independently.
• Integrating a vector field over a curve (Fall 2011 version)
  This handout presents a geometric view of integrating a vector field over a curve. The version students initially get does not include the plots and solution given on the last two pages. I make this version available on-line after students have had time to work on the assignment. I plan to develop a richer set of problems for this material.
• Integrating a vector field over a surface (Fall 2011 version)
  This handout, with problems included, presents a geometric view of integrating a vector field over a surface. I plan to develop a richer set of problems for this material.
• Divergence of a vector field (Fall 2011 version)
  This handout introduce divergence as a volume flux density a la Div, Grad, Curl, and All That. I plan to develop a richer set of problems for this material.
• Curl of a vector field (Fall 2011 version)
  This handout introduce curl as an area circulation density a la Div, Grad, Curl, and All That. I plan to develop a richer set of problems for this material.
• Fundamental theorems of calculus (Fall 2011 version)
  Note: This is just a summary of the fundamental theorems rather than a "total from density" development.

Experimenting with interactive visualizations

I am currently exploring various options for web-based interactive visualizations. My first attempts (using Mathematica's Computable Document Format) are here.