CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Computer Algebra Systems in Vector Calculus: 1 A radically new approach based on visualization Matthias Kawski Department of Mathematics Arizona State University Tempe, AZ 85287 [email protected] 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) ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 2 Vector Calculus via Linearizations You zoom in calculus I for derivatives /
slopes -Why then dont you zoom in calculus III for curl, div, andreview: Stokes theorem ? distinguish different kinds of zooming Zooming Uniform differentiability side-track, regarding rigor etc. Linear Vector Fields Derivatives of Nonlinear Vector Fields Animating curl and divergence Stokes Theorem via linearizations Inverse questions and applications (controllability versus conservative) ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 The pre-calculator days 3 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. ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition
CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 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??? ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Calculators have ZOOM button! 5 Tickmarks contain info about and 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!) ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus
4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 6 Zooming in on numerical tables 3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 0 -5 -8 -9 -8 -5 5 0 -3 -4 -3 0 8 3 0 -1 0 3 9 4 1 0 1 4 8 3 0 -1 0 3 5 0 -3 -4 -3 0 0 -5 -8 -9 -8 -5 3 0 5 8 9 8 5 0 1.3 1.2 1.1 1.0 0.9 0.8
0.7 1.7 1.20 1.45 1.68 1.89 2.08 2.25 2.40 1.8 1.55 1.80 2.03 2.24 2.43 2.60 2.75 1.03 1.02 1.01 1.00 0.99 0.98 0.97 1.97 2.1909 2.4409 2.6709 2.8809 3.0709 3.2409 3.3909
1.98 2.2304 2.4804 2.7104 2.9204 3.1104 3.2804 3.4304 1.9 1.92 2.17 2.40 2.61 2.80 2.97 3.12 1.99 2.2701 2.5201 2.7501 2.9601 3.1501 3.3201 3.4701 2.0 2.31 2.56 2.79 3.00 3.19 3.36 3.51 2.1
2.72 2.97 3.20 3.41 3.60 3.77 3.92 2.00 2.3100 2.5600 2.7900 3.0000 3.1900 3.3600 3.5100 2.2 3.15 3.40 3.63 3.84 4.03 4.20 4.35 2.01 2.3501 2.6001 2.8301 3.0401 3.2301 3.4001 3.5501 2.3 3.60
3.85 4.08 4.29 4.48 4.65 4.80 2.02 2.3904 2.6404 2.8704 3.0804 3.2704 3.4404 3.5904 2.03 2.4309 2.6809 2.9109 3.1209 3.3109 3.4809 3.6309 This applies to all: single variable, multi-variable and vector calculus. Foundation In thisMatthias presentation only, graphical approach and analysis. ACEPT Kawski, AZ Stateemphasize Univ. http://math.la.asu.edu/~kawski
Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Zooming on contour diagram Easier than 3D. -- Important: recognize contour diagrams of planes!! ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition 7 CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 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 .. ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 9 Zooming for line-INTEGRALS of vfs
Without the blue curve this is the pictorial foundation for the convergence of Eulers 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! ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 10 Two kinds of zooming ItItisisextremely extremelysimple, simple,just justconsistently consistentlyapply applyrules rulesall allthe theway wayto tovfs vfs 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 domain and range Picture for differentiability (local linearity) Need to ignore (subtract) constant part -- picture can not show total magnitude!!! ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 The usual boxes for continuity 11 This is EXACTLY the 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! ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97
Zooming of 0 kind in calculus I th Continuity Continuityvia viazooming: zooming: Zoom Zoominindomain domainonly: only:Tickmarks Tickmarksshow show>0. >0. Fixed Fixedvertical verticalwindow windowsize sizecontrolled controlledby by ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition 12 CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Convergence of R-sums via zooming of zeroth kind (continuity)
13 Common Commonpictures pictures demonstrate demonstratehow howarea area isisexhausted exhaustedininlimit. limit. th The Thezooming zoomingof of00thkind kindpicture picturedemonstrate demonstratethat thatthe thelimit limitexists! exists!----The Thefirst firstpart part for forthe theproof proofininadvanced advancedcalculus: calculus:(Uniform) (Uniform)continuity continuity=> => integrability.
integrability. Key Keyidea: idea:Further Furthersubdivisions subdivisionswill willnot notchange changethe thesum sum => =>Cauchy Cauchysequence. sequence. ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 14 Zooming of the 2 kind, calculus I nd Zooming Zoomingat atquadratic quadraticratios ratios(in (inrange range /domain) /domain)exhibits
exhibitslocal localquadratic-ness quadratic-ness near nearnondegenerate nondegenerateextrema. extrema. Even Evenmore moreimpressive impressivefor forsurfaces! surfaces! Also: Zooming out of n-th kind e.g. to find power of polynomial, establish nonpol character of exp. Pure Pure meanness: meanness: Instead Insteadof offind findthe themin-value, min-value, ask askfor forfind findthe thex-coordinate x-coordinate(to (to 12 12decimal decimalplaces) places)of
ofthe themin. min. Why Whycant cantone oneanswer answerthis thisby by standard standardzooming zoomingon onaacalculator? calculator? Answer: Answer:The Thefirst firstderivative derivativetest! test! ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Zooming of the 1 kind, calculus I st 15 Slightly more advanced, characterization of differentiability at point. Useful for error-estimates in approximations, mental picture for proofs.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 16 A short side-excursion, re rigor in proof of Stokes theorem Uniform continuity, pictorially 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!! ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 A short side-excursion, re rigor in proof of Stokes thm. Compare e.g. books by Keith Stroyan 17 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. Nave proof ideas easily go thru, no need for awkward MeanValueThm ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 18 Zooming of 0 kind in multivar.calc. th Surfaces become flat, contours disappear, tables become constant? Boring? Not at all! Only this allows us to proceed w/ Riemann integrals! ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 19 19 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.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Zooming of 1st kind in multivar.calc. 20 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) ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 21 for unif. differentiability in multivar.calc. Advanced Advancedcalc: calc: Where Whereare areand and Animation: Animation:Slide
Slide this thiscone cone(with (withtilting tilting center centerplane planearound) around) (uniformity) (uniformity) 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) ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 charact. for continuity in vector calc. 22 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!!) ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Linear vector fields ??? 23 Usually we see them only in the DE course (if at all, even there). Who Whoknows knowshow howto totell tellwhether whetheraapictured picturedvector vectorfield fieldisislinear? linear? ---> --->What Whatdo dolinear linearvector vectorfields fieldslook looklike? like?Do Dowe wecare?
care? ((Do ((Dostudents studentsneed needaabetter betterunderstanding understandingof oflinearity linearityanywhere?)) 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? ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 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, ... ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 25 What is the analogue of slope for vector fields? First recall: linear and slope in precalc Consider divided differences, rise over run y2 y1 x2 x1 y x Linear <=> ratio is CONSTANT, INDEPENDENT of the choice of points (xk,yk ) ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97
Constant ratios for linear fields 26 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. ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 27 Algebraic formulas: tr(L), (L-L )/2 T Develop Developunderstanding understandingwhere where(a+d), (a+d),(c-b) (c-b)etc etccome comefrom fromin inlimit limitfree freesetting settingfirst first
RL Tds L( x0 , y0 y) i x L( x0 x, y0 ) j y L( x0 , y0 y ) i x L( x0 x , y0 ) j y ..(only using linearity)... (c b) xy (x0,y0+y) (x0-x, y0) (x0,y0) (x0+x,y0) for L(x,y) = (ax+by,cx+dy), using only midpoint rule (exact!) and linearity for e.g. circulation integral over rectangle (x0,y0 -y) Coordinate-free GEOMETRIC arguments w/ triangles, simplices in 3D are even nicer ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97
Want: Stokes theorem for linear fields FIRST! Telescoping sums Recall: For linear functions, the fundamental theorem is exact without limits, it is just a telescoping sum! F (b) F (a ) ( F ( xk 1 ) F ( xk )) F ( xk 1 ) F ( xk ) x xk 1 xk b F ( x )dx a ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition 28 CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 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. CL Nds
k L Nds Ck k trL Ak Caution, when arguing with trL k Ak triangulations of smooth surfaces trL A The picture new TELESCOPING SUMS matters (cancellations!) ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 30 Nonlinear vector fields, zoom 1 kind st 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.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 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! ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 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? ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Student exercise: Limit 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. 33 Instead of ZOOMING, this perspective lets the contours shrink to a point. Do not forget to also
draw these contours after magnification! Report all results to transparency in the front. Scale by area, SEE convergence. ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 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! ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition
CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 35 Rigor in the defn: Differentiability Recall: Recall:Usual Usualdefinitions definitionsof ofdifferentiability differentiabilityrely relymuch muchon onjoint joint continuity continuityof ofpartial partialderivatives derivativesof ofcomponent componentfunctions. functions.This Thisisis not notgeometric, geometric,and andtroublesome: troublesome:diffable diffablenot notsame sameas aspartials partialsexist exist
Better: Better:Do Doititlike likein ingraduate graduateschool school----the thezooming zoomingpicture pictureisisright! right! AAfunction/map/operator function/map/operatorFFbetween betweenlinear linearspaces spacesXXand andZZisis uniformly uniformly differentiable differentiableon onaaset setKKififfor forevery everyppin inKKthere thereexists existsaalinear linearmap map LL==LLppsuch suchthat thatfor forevery every> >00there thereexists
existsaa> >00(indep.of (indep.ofp)p)such such that that Advantage of--uniform: whenpointwise working with little-oh: | |F(q) LLpp(q-p) <<| |qqany --pp| |problems (or definition). F(q)--F(p) F(p) (q-p)| |Never (oranalogous analogous pointwise definition). F(q) = F(p) + Lp (q-p) +o( | q - p | ) -- all the way to proof of Stokes thm. ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Divergence, rotation, curl 36
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 L Nds L Nds C C (and showed independence tr ( L) ( area ) Nds of everything): C 2 2 ( x y ) / 4 For a differentiable field F Nds define (where contour div ( F )( p) lim C shrinks to the point p,
(area ) circumference -->0 ) ( x 2 y 2 ) / 4 Use Useyour yourjudgment judgmentworrying worryingabout about independence independenceofofthe thecontour contourhere. here. 1
Consequence: | div ( F L)| (diam) ( circumference) 4 ( area ) ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 37 Proof of Stokes theorem, nonlinear In Incomplete completeanalogy analogyto tothe theproof proof of ofthe thefundamental fundamentaltheorem theoremin in calc calcI:I:telescoping telescopingsums sums++limits limits (+uniform (+uniformdifferentiability, differentiability,or
or MVTh, MVTh,or orhandwaving.). 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. F Nds F Nds trF ( p ) A ) div ( F )dA C k Ck k k k k Rk div ( F )dA R ItItstraightforward straightforwardto toput putin
inlittle-ohs, little-ohs,use useuniform uniformdiff., diff.,and andcheck checkthat that the theorders ordersof oferrors errorsand andnumber numberof ofterms termsin insum sumbehave behaveas asexpected! expected! ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 38 About little-ohs & uniform differentiability By Byhypothesis, hypothesis,for forevery
everyppthere thereexist existaalinear linearfield fieldLLppsuch suchthat that for forevery every>>00there thereisisaa>>00(independent (independentof ofpp(!)) (!))such suchthat that | |F(q) F(q)--F(p) F(p)--LLpp(q (q--p) p)| |<<| |qq--pp| |for forall allqqsuch suchthat that| |qq--pp| |<<.. The errors in the two approximate equalities in the nonlinear telescoping sum: | ( F Lp ) NdS | diam(Vk ) area ( S k ) Sk | div ( F Lp )dV | vol (Vk ) Vk
Key: Key:Stay Stayaway awayfrom from pathological, pathological,arbitrary arbitrary large largesurfaces surfacesbounding bounding arbitrary arbitrarysmall smallvolumes, volumes, Except Exceptfor forsmall smallnumber number(lower (lowerorder)of order)ofoutside outsideregions, regions,hypothesize hypothesizeaaregular regularsubdivision, subdivision, i.e. i.e.without withoutpathological pathologicalrelations relationsbetween betweendiameter, diameter,circumference/surface circumference/surfacearea, area,volume!
volume! ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 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! ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 40 Decompositions Preliminary: Preliminary:Review Reviewthat thateach eachscalar
scalarfunction function may may be bewritten writtenas asaasum sumof ofeven evenand andodd oddpart. part. Decompose Decomposelinear, linear,planar planarvector vectorfields fields into intosum sumof ofsymm. symm.& &skew-symm. skew-symm.part part (geometrically (geometrically ----hard?, hard?,angles!!, angles!!, algebraically algebraically==link linkto tolinear linearalgebra).
algebra). (Good (Goodplace placeto toreview reviewthe theadditivity additivityof of ((line))integral ((line))integral drift drift++symmetric+antisymmetric. symmetric+antisymmetric. ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 41 CURL: An axis of rotation in 3d Requires Requiresprior priordevelopment developmentof ofdecomposition decompositionsymmetric/antisymmetric symmetric/antisymmetricininplanar planarcase. case. Addresses Addressesadditivity
additivityof ofrotation rotation(angular (angularvelocity velocityvectors) vectors)----who whobelieves believesthat? that? usual nonsense 3d-field jiggle -- wait, there IS order! It is a rigid rotation! Dont Dontexpect expecttotosee seemuch muchififplotting plottingvector vectorfield fieldinin3d 3dw/o w/ospecial special(bundle-) (bundle-)structure, structure, st however, however,plot plotANY ANYskew-symmetric skew-symmetriclinear linearfield field(skew-part
(skew-partafter afterzooming zooming11 stkind), kind), jiggle jiggleaalittle, little,discover discoverorder, order,rotate rotateuntil untillook lookdown downaatube, tube,each eachstudent studentdifferent differentaxis axis For more MAPLE files (curl in coords etc) see book: Zooming and limits, ..., or WWW-site. ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Proposed class outline 42 Assuming Assumingmulti-variable multi-variablecalculus calculustreatment treatmentas
asininHarvard HarvardConsortium ConsortiumCalculus, Calculus, with withstrong strongemphasis emphasison onRule RuleofofThree, Three,contour contourdiagrams, diagrams,Riemann Riemannsums, sums,zooming. 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 0 th 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. ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 43
Animate curl & div, integrate DE (drift) Color by rot: red=left turn green=rite turn divergence controls growth ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 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. ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 45 Tumbling soccer balls in 3D-field Need to see the animation!
At this time: User supplies vector field and init conds or uses default example. MAPLE integrates DEs for position, calculates curl, integrates angular momentum equations, and creates animation using rotation matrices. Colored faces crucial! ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Stokes theorem & magnetic field 46 CF1 Tds CF2 Tds F NdS 2 0 S 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 Jacobs 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, ... ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Dont ask the old questions! 47 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] , ? ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Asking inverse questions in DE and VC48 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. ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Value of the inverse questions 49 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... ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97
50 Greens 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 Greens 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, with conservation of angular momentum constraint. Compare most recent literature on: Falling cats, gymnasts, and satellites 2 1 Choose 1(t), 2(t) from (0,0) to (0,0) such that is the desired phase-shift ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 Animation of the re-orientation
51 d - F1(1, 2) d1 - F2(1, 2) d2 = 0 T = 3 t 2 1 T = 7 t T=0 Three linked rigid bodies. Total angular momentum Great application of Greens always zero = conserved. Yet by moving through a theorem. Fun animations. Good loop is shape-space (12-space) the attitude may projects. Link to recent research. be changed! (Satellite w/ antenna, falling cat ) CAS required for algebra! ....... T = 9 t T = 11 t ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski T = 25 t Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 52 The differential form =F1(, 2)d1+ F2(1, 2)d2
Colored by d (positive = green, negative = red) F dr FdA C R A closer look at the interesting location == shape when and 2 both are almost equal to , i.e. the arms are folded inward. The most efficient loop (for for maximal phase-shift) encloses as much green and as little red as possible. ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 The graph of the rotation of F(1,2) Selecting a suitable loop in shape space that results in maximal attitudinal change 53 d C R
F dr FdA C R Note the very sharp peaks and pits =-=> key to make this a great project. Randomly chosen curves lead to unpredictable attitude changes. Understanding of Greens thm <==> systematic choice of loops in shape space to achieve desired attitude change ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition CAS in Vector Calculus 4th Intl. Symp. DERIVE & TI-92, Kungsbacka Sweden, Aug 97 The loop in the base-shape-space and the lifted curve in the total space Observe the nonzero holonomy -- the lifted curve does not close 54 CRITICAL: Dynamically animate the loop and the lifted curve. Contrast with potential surface for conservative fields. Contrast this with conservative == integrable fields. There (as here) DYNAMICALLY GROW the potential surface using many lifted LOOPS -- dont just pop it on the screen. ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski Foundation Coalition