Computer Algebra Systems in Vector Calculus: A radically new approach based on visualization

1. Computer Algebra Systems in Vector Calculus: A radically new approach based on visualization Matthias Kawski Department of Mathematics Arizona State University Tempe, AZ 85287 kawski@asu.edu http://math.la.asu.edu/~kawski Lots of MAPLE worksheets (in all degrees of rawness), plus plenty of other class-materials: Daily instructions, tests, extended projects This work was partially supported by the NSF through Cooperative Agreement EEC-92-21460 (Foundation Coalition) and the grant DUE 94-53610 (ACEPT)

2. Vector Calculus via Linearizations You zoom in calculus I for derivatives / slopes -- Why then don’t you zoom in calculus III for curl, div, and Stokes’ theorem ? • Zooming • Uniform differentiability • Linear Vector Fields • Derivatives of Nonlinear Vector Fields • Animating curl and divergence • Stokes’ Theorem via linearizations • Inverse questions and applications (controllability versus conservative) review: distinguish different kinds of zooming side-track, regarding rigor etc.

3. The pre-calculator days The textbook shows a static picture. The teacher thinks of the process.The students think limits mean factoring/canceling rational expressions and anyhow are convinced that tangent lines can only touch at one point.

4. Multi-media, JAVA, VRML 3.0 ??? Multi-media, VRML etc. animate the process. The “process-idea” of a limit comes across. Is it just adapting new technology to old pictures???

5. Calculators have ZOOM button! Tickmarks contain info about e and d New technologies provide new avenues: Each student zooms at a different point, leaves final result on screen, all get up, and …………..WHAT A MEMORABLE EXPERIENCE! (rigorous, and capturing the most important and idea of all!)

6. Zooming in on numerical tables This applies to all: single variable, multi-variable and vector calculus. In this presentation only, emphasize graphical approach and analysis.

7. Zooming on contour diagram Easier than 3D. -- Important: recognize contour diagrams of planes!!

8. Gradient field: Zooming out of normals! Pedagogically correct order: Zoom in on contour diagram until linear, assign one normal vector to each magnified picture, then ZOOM OUT , put all small pictures together to BUILD a varying gradient field ……..

9. Zooming for line-INTEGRALS of vfs Without the blue curve this is the pictorial foundation for the convergence of Euler’s and related methods for numerically integrating differential equations Zooming for INTEGRATION?? -- derivative of curve, integral of field! YES, there are TWO kinds of zooming needed in introductory calculus!

10. Zooming of the zeroth kind Magnify domain only Keep range fixed Picture for continuity(local constancy) Existence of limits of Riemann sums (integrals) Zooming of the first kind Magnify BOTH domainand range Picture for differentiability(local linearity) Need to ignore (subtract) constant part -- picture can not show total magnitude!!! Two kinds of zooming It is extremely simple, just consistently apply rules all the way to vfs

11. The usual e-d boxes for continuity This is EXACTLY the e-d characterization of continuity at a point, but without these symbols. CAUTION: All usual fallacies of confusion of order of quantifiers still apply -- but are now closer to common sense!

12. Zooming of 0th kind in calculus I Continuity via zooming: Zoom in domain only: Tickmarks show d>0. Fixed vertical window size controlled by e>0

13. Convergence of R-sumsvia zooming of zeroth kind (continuity) Common pictures demonstrate how area is exhausted in limit. The zooming of 0th kind picture demonstrate that the limit exists! -- The first part for the proof in advanced calculus: (Uniform) continuity => integrability. Key idea: “Further subdivisions will not change the sum” => Cauchy sequence.

14. Zooming of the 2nd kind, calculus I Zooming at quadratic ratios (in range /domain) exhibits “local quadratic-ness” near nondegenerate extrema. Even more impressive for surfaces! Also: Zooming out of “n-th” kind e.g. to find power of polynomial, establish nonpol character of exp. Pure meanness: Instead of “find the min-value”, ask for “find the x-coordinate (to 12 decimal places) of the min”. Why can’t one answer this by standard zooming on a calculator? Answer: The first derivative test!

15. Zooming of the 1st kind, calculus I Slightly more advanced, e-d characterization of differentiability at point. Useful for error-estimates in approximations, mental picture for proofs.

16. Uniform continuity, pictorially A short side-excursion, re rigor in proof of Stokes’ theorem Demonstration: Slide tubing of various radii over bent-wire! Many have argued that uniform continuity belongs into freshmen calc. Practically all proofs require it, who cares about continuity at a point? Now we have the graphical tools -- it is so natural, LET US DO IT!!

17. Compare e.g. books by Keith Stroyan A short side-excursion, re rigor in proof of Stokes’ thm. Uniform differentiability, pictorially Demonstration: Slide cones of various opening angles over bent-wire! With the hypothesis of uniform differentiability much less trouble with order of quantifiers in any proof of any fundamental/Stokes’ theorem. Naïve proof ideas easily go thru, no need for awkward MeanValueThm

18. Zooming of 0th kind in multivar.calc. Surfaces become flat, contours disappear, tables become constant? Boring? Not at all! Only this allows us to proceed w/ Riemann integrals!

19. 19 e-d for unif. continuity in multivar. calc. Graphs sandwiched in cages -- exactly as in calc I. Uniformity: Terrific JAVA-VRML animations of moving cages, fixed size.

20. Zooming of 1st kind in multivar.calc. If surface becomes planar (linear) after magnification, call it differentiable at point. Partial derivatives (cross-sections become straight -- compare T.Dick & calculators) Gradients (contour diagrams become equidistant parallel straight lines)

21. e-d for unif. differentiability in multivar.calc. Animation: Slide this cone (with tilting center plane around) (uniformity) Advanced calc: Where are e and d ? Still need lots of work finding good examples good parameter values Graphs sandwiched between truncated cones -- as in calc I. New: Analogous pictures for contour diagrams (and gradients)

22. e-d charact. for continuity in vector calc. Warning: These are uncharted waters -- we are completely unfamiliar with these pictures. Usual = continuity only via components functions; Danger: each of these is rather tricky Fk(x,y,z) JOINTLY(?) continuous. Analogous animations for uniform continuity, differentiability, unif.differentiability. Common problem: Independent scaling of domain / range ??? (“Tangent spaces”!!)

23. Linear vector fields ??? Usually we see them only in the DE course (if at all, even there). Who knows how to tell whether a pictured vector field is linear? ---> What do linear vector fields look like? Do we care? ((Do students need a better understanding of linearity anywhere?)) What are the curl and the divergence of linear vector fields? Can we see them? How do we define these as analogues of slope?

24. Linearity ??? Definition: A map/function/operator L: X -> Y is linear if L(cP)=c L(p) and L(p+q)=L(p)+L(q) for all ….. Can your students show where to find L(p),L(p+q)……. in the picture? [y/4,(2*abs(x)-x)/9] Odd-ness and homogeneity are much easier to spot than additivity We need to get used to: “linear” here means “y-intercept is zero”. Additivity of points (identify P with vector OP). Authors/teachers need to learn to distinguish macroscopic, microscopic, infinitesimal vectors, tangent spaces, ...

25. What is the analogue of “slope” for vector fields?First recall: “linear” and slope in precalc Consider divided differences, rise over run Linear <=> ratio is CONSTANT, INDEPENDENT of the choice of points (xk,yk ) Dy Dx

26. Constant ratios for linear fields Work with polygonal paths in linear fields, each student has a different basepoint, a different shape, each student calculates the flux/circulation line integral w/o calculus (midpoint/trapezoidal sums!!), (and e.g. via machine for circles etc, symbolically or numerically), then report findings to overhead in front --> easy suggestion to normalize by area--> what a surprise, independence of shape and location! just like slope.

27. Algebraic formulas: tr(L), (L-LT)/2 Develop understanding where (a+d), (c-b) etc come from in limit free setting first (x0,y0+Dy) for L(x,y) = (ax+by,cx+dy), using only midpoint rule (exact!) and linearity for e.g. circulation integral over rectangle (x0,y0) (x0-Dx, y0) (x0+Dx,y0) (x0,y0 -Dy) Coordinate-free GEOMETRIC arguments w/ triangles, simplices in 3D are even nicer

28. Telescoping sums Want: Stokes’ theorem for linear fields FIRST! Recall: For linear functions, the fundamental theorem is exact without limits, it is just a telescoping sum!

29. Telescoping sums for linear Greens’ thm. This extends formulas from line-integrals over rectangles / triangles first to general polygonal curves (no limits yet!), then to smooth curves. Caution, when arguing with triangulations of smooth surfaces The picture new TELESCOPING SUMS matters (cancellations!)

30. Nonlinear vector fields, zoom 1st kind The original vector field, colored by rot Same vector field after subtracting constant part (from the point for zooming) If after zooming of the first kind we obtain a linear field, we declare the original field differentiable at this point, and define the divergence/rotation/curl to be the trace/skew symmetric part of the linear field we see after zooming.

31. Check for understanding (important) original v-field is linear subtract constant part at p After zooming of first kind! Zooming of the 1st kind on a linear object returns the same object!

32. A conceptualized interactive microscope Allowing the user to drag the point where to zoom, change magnification factor, and switch between different kinds of zooming & coloring (Compare work by K. Stroyan for single variable calculus.) Proof of concept established in Visual C++ by S.Holland (undergrad student). Final version to be JAVA applet?

33. Student exercise: Limit Instead of ZOOMING, this perspective lets the contours shrink to a point. Do not forget to also draw these contours after magnification! Fix a nonlin field, a few base points,a set of contours, different students set up & evaluate line integrals over their contour at their point, and let the contour shrink. Report all results to transparency in the front. Scale by area, SEE convergence.

34. Typical results obtained in an in-class exercise As the contours shrink to a point the ratios (line integral divided by area) appear to stabilize at the same value independent of the shape used ---->good motivation for definition of divergence as a limit of total flux divided by area………. Independence (in the limit) of the shape used is experienced by the students -> this precedes and motivates the analytic proof of well-definedness. Typical example where students benefit from working as a community (in the same classroom at the same time as opposed to asynchronous distance learning modes), i.e. class-time spent very efficiently!

35. Rigor in the defn: Differentiability Recall: Usual definitions of differentiability rely much on joint continuity of partial derivatives of component functions. This is not geometric, and troublesome: diff’able not same as “partials exist” Better: Do it like in graduate school -- the zooming picture is right! A function/map/operator F between linear spaces X and Z is uniformly differentiable on a set K if for every p in K there exists a linear map L = Lp such that for every e > 0 there exists a d > 0 (indep.of p) such that | F(q) - F(p) - Lp(q-p) | < e | q - p | (or analogous pointwise definition). Advantage of uniform: Never any problems when working with little-oh: F(q) = F(p) + Lp (q-p) +o( | q - p | ) -- all the way to proof of Stokes’ thm.

36. Divergence, rotation, curl Intuitively define the divergence of F at p to be the trace of L, where L is the linear field to which the zooming at p converges (!!). For a linear field we defined (and showed independence of everything): For a differentiable field define (where contour shrinks to the point p, circumference -->0 ) Use your judgment worrying about independence of the contour here…. Consequence:

37. Proof of Stokes’ theorem, nonlinear In complete analogy to the proof of the fundamental theorem in calc I: telescoping sums + limits (+uniform differentiability, or MVTh, or handwaving….). Here the hand-waving version: The critical steps use the linear result, and the observation that on each small region the vector field is practically linear. It straightforward to put in little-oh’s, use uniform diff., and check that the orders of errors and number of terms in sum behave as expected!

38. About little-oh’s & uniform differentiability By hypothesis, for every p there exist a linear field Lp such that for every e > 0 there is a d > 0 (independent of p (!)) such that | F(q) - F(p) - Lp(q - p) | < e | q - p | for all q such that | q - p | < d. The errors in the two approximate equalities in the nonlinear telescoping sum: Key: Stay away from pathological, arbitrary large surfaces bounding arbitrary small volumes, Except for small number (lower order)of outside regions, hypothesize a regular subdivision, i.e. without pathological relations between diameter, circumference/surface area, volume!

39. Trouble w/ surface integrals: “Schwarz’ surface” Pictorially the trouble is obvious. SHADING! Simple fun limit for proof Not at all unreasonable in 1st multi-var calculus Entertaining. Warning about limitations of intuitive arguments, … yet it is easy to fix!

40. Decompositions Preliminary: Review that each scalar function may be written as a sum of even and odd part. Decompose linear, planar vector fields into sum of symm. & skew-symm. part (geometrically -- hard?, angles!!, algebraically = link to linear algebra). (Good place to review the additivity of ((line))integral drift + symmetric+antisymmetric.

41. “CURL”: An axis of rotation in 3d Requires prior development of decomposition symmetric/antisymmetric in planar case. Addresses additivity of rotation (angular velocity vectors) -- who believes that? usual nonsense 3d-field jiggle -- wait, there IS order! It is a rigid rotation! Don’t expect to see much if plotting vector field in 3d w/o special (bundle-) structure, however, plot ANY skew-symmetric linear field (skew-part after zooming 1st kind), jiggle a little, discover order, rotate until look down a tube, each student different axis For more MAPLE files (curl in coords etc) see book: “Zooming and limits, ...”, or WWW-site.

42. Proposed class outline Assuming multi-variable calculus treatment as in Harvard Consortium Calculus, with strong emphasis on Rule of Three, contour diagrams, Riemann sums, zooming. • What is a vector field: Pictures. Applications. Gradfields <-->ODEs. • Constant vector fields. Work in precalculus setting!.Nonlinear vfs. (Continuity). Line integrals via zooming of 0th kind.Conservative <=>circulation integrals vanish <=> gradient fields. • Linear vector fields. Trace and skew-symmetric-part via line-ints.Telescoping sum (fluxes over interior surfaces cancel etc….),grad<=>all circ.int.vanish<=>irrotational (in linear case, no limits) OPPOSITE: nonintegrable (not exact) <==> “controllable” • Nonlinear fields: Zoom, differentiability, divergence, rotation, curl.Stokes’ theorem in all versions via little-oh modification of arguments in linear settings. Magnetic/gravitat. fields revisited.

43. Animate curl & div, integrate DE (drift) Color by rot: red=left turn green=rite turn divergence controls growth

44. Spinning corks in linear / magnetic field Period indep.of radius compare harmonic oscillator - pendulum clock Always same side of the moon faces the Earth -- one rotation per full revolution. Irrotational (black = no color). Angular velocity drops sharply w/increasing radius.

45. Tumbling “soccer balls” in 3D-field Need to see the animation! At this time: User supplies vector field and init cond’s or uses default example. MAPLE integrates DEs for position, calculates curl, integrates angular momentum equations, and creates animation using rotation matrices. Colored faces crucial!

46. Stokes’ theorem & magnetic field Do your students have a mental picture of the objects in the equn? Homotop the blue curve into the magenta circle WITHOUT TOUCHING THE WIRE (beautiful animation -- curve sweeping out surface, reminiscent of Jacob’s ladder). 3D=views, jiggling necessary to obtain understanding how curve sits relative to wire.More impressive curve formed from torus knots with arbitrary winding numbers, ...

47. Don’t ask the old questions! “Many traditional exercises -- which anyhow never had any intrinsic value -- have been trivialized with the advent of modern computer/calculator technology. Not only did they loose their appeal, but they may actually be harmful by hindering the students to develop the desired understanding and appreciation of the subject.” Example: “Use calculus to find the local max / sketch the graph of y=x3-3x2+5x-7”. “Technology clearly is much more efficient and reliable. Calculus is inappropriate! This exercise was useful in the days before technology as an example to point out the relation between monotonic behavior and the signs of derivatives, or demonstrate the use of the Fermat test to find critical points. To develop, and check for understanding of these still central calculus topics, one needs to ask new questions.” The first attempt is to ask the inverse questions, e.g. “Find formula for graph”, or “For which values of the parameter k does the graph of y=x3-kx2+5x-7 have a local maximum, is it increasing on the interval (-,0] , …… ?”

48. Asking inverse questions in DE and VC natural leads to problems in “control” The parameters in calculus are constants. In differential equations, the parameters are “functions of time”, i.e. controls! In vector calculus, the natural inverse questions also lead to control problems. Example: Line integrals Old: Given a vector field and a curve, find the value of the line integral. (Trivial w/ computer technology at hand. No lasting learning experience.) New: Given a vector field, two endpoints, and a desired value for the line integral, find a suitable (!) between these endpoints…….

49. Value of the inverse questions • Appropriate contemporary exercises to understand classical topics, practice still important lines of reasoning • Not trivialized by technology, but rather inviting technology for exploration (special cases) to develop intuition, and to validate the answers • Often open-ended problems w/ multiple solutions; inviting the formulation of additional criteria for best “solution” • Often intimately related to much more compelling modern applications that are not contrived, and are accessible only via the use of computer algebra systems…...

50. Green’s theorem and inverse questions • Example: “Given a vector field, find a suitable curve connecting two given endpoints such that the line integral has a desired value.” This problem requires the understanding of several aspects of line integrals as well as Green’s theorem for an effective solution. Moreover, it is easily interpreted as an“open-loop” control problem with three states and one constraint (i.e. two controls):A typical mechanical interpretation is that of a “planar skater”: Three planar rigid bodies coupled by actuated rotational joints, withconservation of angular momentum constraint.Compare most recent literature on:“Falling cats, gymnasts, and satellites” 1  “Choose q1(t), q2(t) from(0,0) to (0,0) such that Dais the desired “phase-shift” 2