Space-Age Projects for an Integrated Vector Calculus Matthias Kawski, Department of Mathematics Arizona State University Tempe, AZ 85287, USA http://math.la.asu.edu/~kawski kawski@asu.edu Cats, gymnasts, satellites -- and vector calculus! This work was partially supported by the National Science Foundation through the grant DUE 97-52453 (Vector Calculus via Linearization: Visualization and Modern Applications), through the Cooperative Agreement EEC 92-21460 (The Foundation Coalition), and by an equipment donation from the Intel Corporation.

Foundation Coalition Courses at ASU Team-based learning Technology intensive Integrated Curriculum 1st Year: Intro to Engineering, Calculus 1+2, Physics, Chemistry, English composition 2nd Year: Vector Calculus, Diff Eqns, Mechanics (previously with: Electric Circuits, Linear Algebra, Intro Macroeconomics)

1. Writing on a “blimp” Design and aesthetic aspects as powerful motivators for deeper/better math learning! Calculus content: Parameterizing (“broken”) curves: critical skill for line integrals often hard for students (too many solutions -- how to get one??), and a skill not made obsolete with compute algebra systems. Parameterizing surfaces Composition of the two parameterizations Velocity, speed, acceleration, curvature, Frenet-frame, reparameterize by arc-length (welding robot!) All work is done with a computer algebra system: This project serves as introduction to computers! Students will graphically check their work, keep working until everything matches -- grading trivial. Parameterized word and graph of the curvature

2. Rolling races A small, but crucial modification of a traditional application: The traditional “physics” problem analyzes rolling objects on an inclined plane. It goes as far as asking which object will win the race (compare: D. Drucker’s “Mathematical Roller Derby” in CMJ 11/1992). The calculus link are moments of inertia, i.e. iterated integrals, and a simple separable DE. The problem solution never goes beyond the level of “analysis”. The “engineering” problem goes one, CRITICAL STEP further: We ask the students to “apply” the knowledge gained by DESIGNING and BUILDING a rolling object that will win a race in the class! The results are amazingly fast: Further use of calculus yields an optimal design with J=0.08 ma2 as opposed to J=0.40 ma2 for a solid billiards queue ball!!! PROUD winners and winning design

3. Satellites, falling cats & gymnasts Start with an intriguing question: How does a falling cat / diver / gymnast / satellite “re-orient” itself without violating constraints due to conservation of angular momentum? Lots of motivation (current research literature, WWW, Apollo videos). Mathematical modeling supplies the “needs” (the driving force) for developing MANY aspects of multi-variable/vector calculus ….. In the end: Combine vector calculus (Green’s theorem) and diff eqns to design a shape-change maneuver, test it in computersimulations and on a real robot (under construction, plan for public final exam).

Vector calculus applications: Going beyond E & M and fluid dynamics A recent SIAM-News article - even more motivation: We ask the “inverse questions”, transforming line integrals and Stokes’ theorem from straight- forward calculations into intriguing control problems that have virtual ubiquitous applications. Here gymnasts, divers, falling cats and satellites offer something for very one -- unarguably much more gender neutral than the classical narrow focus applications almost exclusively from E & M and fluid dynamics!

Inverse questions and control Traditional emphasis, physics point of view: Conservative ( = integrable ) vector fields, “closed loops lift to a potential surface” Modern emphasis, engineering point of view: Controllable ( = nonintegrable ) vector fields, “design the closed loop in the base so that the vertical gap of the lifted curve is as desired”. Traditional line integrals: Given F and C find Da (boring w/ computer algebra syst.) Modern line integrals: Given F and Da find C (intelligent, ubiquitous applications)

Search the WWW for current related research JAVA/gif animations from Japanese research labs….

As close as calculus students can get to current cutting-edge research! From R. Murray’s paper in the NRC proceedings! Much additional credibility and motivation: 1996 NRC workshop -- cutting edge applications, but diagrams that exactly match our class/ project materials!!

Modeling and mathematization In agreement with Murray’s NRC presentation consider most simple model that will exhibit the amazing features --- a “planar robot” of 3 linked rigid bodies, actuated at the joints. (Thee full 3D-models require grad school math, yet the planar model is perfect for 3rd semester calculus!) ((For practical purposes, don’t consider the 3 body- assembly in space, but with frictionless support at center of central link allowing free rotation.)) The model is characterized by 3 state variables: 2 internal angles:Q1 and Q2 , and angle a with respect to a (inertial) reference frame They are related by a differential constraint, conservation of angular momentum. da - F1(q1, q2) dq1 - F2(q1, q2) dq2 = 0 ((depending on comfort level of students write as system of ODEs or as a “vector field” F(q1,q2) = [ F1 (q1,q2), F2 (q1,q2) ] in the q1q2-plane))

Exploratory computer “games” T = 3 Dt T = 7 Dt q2 q1 Interactive animated simulations (currently in MAPLE, MATLAB, Working Model, soon also in JAVA) allow students to understand how “shape changes”correspond to parameterized curves in the q1q2-plane. T = 9 Dt q2 . . . . . . Later, they help develop an intuitive feeling how the shape changes affect the overall orientation, reconcile theory with personal experiences (skaters). “Computer games” are encouraged, but number of parameters in good project makes “trial and error approach” unappealing - eventually.students will ask for theory and more powerful tools! Da q1 T = 25 Dt

Project drives development of almost entire calc III Already had: understanding several variables and interpreting parameterized curves in the q1q2-plane. Explicit formulae for F(q1,q2) require iterated integrals (moments of inertia), and are greatly simplified using vector analysis (parallel axis theorem). Conservation law or differential constraint provides strong link to differential equations (“exactness”, “first integrals”) and interpretation as vector fields. The attitude changes are calculated via line integrals. To understand how to design the shape changes ( = curve = control), use Green’s theorem to rewrite the line integral as a double integral. None of the expressions are difficult -- but they are messy and almost require computer algebra systems, in particular for simplifying partial derivatives. The final design is obvious upon interpretation of double integral as signed volume under graph. Typical solution in terms of parameterized families of closed loops (naïve homotopies).

Details that make a great project Observe the sharp peaks and pits very close to each other: Random loops result in “disappointing” attitude changes Da, while understanding of Green’s theorem & interpretation of double integral as signed volume under the graph gives immediate idea for design of amazingly effective maneuvers. Bonus: Translate these loops into actual shape changes, and understand physics behind the location of peaks and pits! Graph of rot(F)(x,y) or dw(x,y) for typical data

Student reactions, and future plans Too much work: “Never before in my life did I work as hard for a class” “This project provided us with an excellent opportunity to apply the knowledge that we have learned about iterated integrals in a real-life situation. The project also allowed us to learn more about the powerful applications that Maple can handle. It was nice to be able to put our equations into the appli- cation and see if we were doing the iterated integrals correctly. The project was also fun because we were able to build the object that we were testing with Maple. Having the solid object is much better than having the abstract picture on a computer because we felt like we had accomplished something. Hopefully, when we race our object all of our work will have been well worth it.” “When we first started this project, our task seemed to be next to impossible…..” First year projects in integrated curriculum are widely perceived as the most fun part, and the glue (or umbrella) that links the courses together. The “falling cats” project still needs to be fine-tuned -- next time we will spread it out over the entire course. Students were excited that as sophomores they were able to get that close to NASA applications, excitedly showed of their discoveries from WWW-searches….. Typical illustration in students reports (project 2): Student artwork -- graphics in final reports exhibits great PRIDE, and much care for aesthetic aspects!

Summary Real applications from modern world Provide stronger motivation by using applications of 1990s and 2000s Integrate physics and engineering, diff. equations and vector calculus Complex, multi-step problems foster superior teamwork Be ambitious, challenge the students, have & instill “can-do-attitude” Inverse questions: Trial and error possible, but unsatisfactory “Analysis” is demonstrably much more powerful (This is far from trivial, yet important: Many old problems are no longer good examples for power of theory as brute-force calculations have often become more efficient than learning, understanding, and applying theory!) Motivation for developing and understanding theory (entire course) Employ aesthetics and beauty as driving forces for better work! Computer technology is integral part, can’t do without