1. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 1 Vector Calculus via Linearizations Matthias Kawski Department of Mathematics Center for Innovation in Engineering Education Arizona State University Tempe, AZ 85287 kawski@math.la.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 “VISUAL CALCULUS” (to come soon, MAPLE, JAVA, VRML)

2. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 2 This work was partially supported by the NSF through Cooperative Agreement EEC-92-21460 (Foundation Coalition) and the grant DUE 94-53610 (ACEPT) You zoom in calculus I for derivatives / slopes -- Why then don’t you zoom in calculus III for curl, div, and Stokes’ theorem ? Vector Calculus via Linearizations Zooming Uniform differentiability Linear Vector Fields Derivatives of Nonlinear Vector Fields Stokes’ Theorem long motivation side-track, regarding rigor etc.

3. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 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. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 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. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 5 Calculators have ZOOM button! 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!) Tickmarks contain info about  and 

6. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 6 Zooming in multivariable calculus Zoom in on a surface -- is the Earth round or flat ???

7. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 7 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.

8. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 8 Zooming on contour diagram Easier than 3D. -- Important: recognize contour diagrams of planes!!

9. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 9 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 ……..

10. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 10 Naïve zooming on vector field What we got?? Boring?? Not at all -- this is the key for INTEGRATION! This picture is key to convergence of Euler’s method for integrating DE’s Be patient! Color will be utilized very soon, too.

11. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 11 Zooming for line-INTEGRALS of vfs Zooming for INTEGRATION?? -- derivative of curve, integral of field! YES, there are TWO kinds of zooming needed in introductory calculus!

12. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 12 Two kinds of zooming 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!!! It is extremely simple, just consistently apply rules all the way to vfs

13. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 13 The usual  boxes for continuity 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!

14. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 14 Zooming of 0 th kind in calculus I Continuity via zooming: Zoom in domain only: Tickmarks show  >0. Fixed vertical window size controlled by  Continuity via zooming: Zoom in domain only: Tickmarks show  >0. Fixed vertical window size controlled by 

15. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 15 Convergence of R-sums via zooming of zeroth kind (continuity) The zooming of 0 th 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. The zooming of 0 th 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. Common pictures demosntarte how area is exhausted in limit. Common pictures demosntarte how area is exhausted in limit.

16. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 16 Zooming of the 1 st kind, calculus I This is the usual calculator exercise -- this is remembered for whole life!

17. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 17 Zooming of the 2 nd kind, calculus I Zooming at quadratic ratios (in range /domain) exhibits “local quadratic-ness” near nondegenerate extrema. Even more impressive for surfaces! Zooming at quadratic ratios (in range /domain) exhibits “local quadratic-ness” near nondegenerate extrema. Even more impressive for surfaces! Pure meanness: Instead of “find the min-value”, ask for “find the x-coordinate (to 12 decimal places) of the min”. 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 calcuator? Answer: The first derivative test! Why can’t one answer this by standard zooming on a calcuator? Answer: The first derivative test! Also: Zooming out of “n-th” kind e.g. to find power of polynomial, establish nonpol charater of exp.

18. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 18 Zooming of the 1 st kind, calculus I Slightly more advanced,  characterization of differentiability at point. Useful for error-estimates in approximations, mental picture for proofs.

19. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 19 Uniform continuity, pictorially A short side-excursion, re rigor in proof of Stokes’ thm. 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!! Demonstration : Slide tubings of various radii over bent-wire!

20. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 20 Uniform differentiability, pictorially A short side-excursion, re rigor in proof of Stokes’ thm. 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 Demonstration : Slide cones of various opening angles over bent-wire! Compare e.g. books by Keith Stroyan

21. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 21 Zooming of 0 th 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!

22. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 22  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. 21

23. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 23 Convergence of R-sums in multivar.calc. via zooming of 0 th kind (continuity) Almost the little-oh proof, with uniform-cont. hypothesis also almost the complete  proof. -- Remember THIS picture for advanced calc.!

24. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 24 Zooming of 1 st kind in multivar.calc. If surface becomes planar (lienar) 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)

25. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 25  for unif. differentiability in multivar.calc. Graphs sandwiched between truncated cones -- as in calc I. New: Analogous pictures for contour diagrams (and gradients) Animation: Slide this cone (with tilting center plane around) (uniformity) Animation: Slide this cone (with tilting center plane around) (uniformity) Advanced calc: Where are  and  Advanced calc: Where are  and  24 Still need lots of work finding good examples good parameter values

26. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 26 Zooming of 0 th kind in vector calc Key application: Convergence of R-sums for line integrals After zooming: work=(precalc) (CONSTANT force) dot (displacement) Further magnification will not change sum at all (unif. cont./C.S.)

27. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 27  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 F k (x,y,z) JOINTLY(?) continuous. Analogous animations for uniform continuity, differentiability, unif.differentiability. Common problem: Independent scaling of domain / range ??? (“Tangent spaces”!!)

28. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 28 Zooming of 1 st kind in vector calc. Now it is all obvious!! -- What will we get??? Prep: pictures for pointwise addition (subtraction) of vfs recommended The original vector field, colored by div Same vector field after subtracting constant part (from the point for zooming) Practically linear

29. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 29 Linear vector fields ??? 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?)) 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? Usually we see them only in the DE course (if at all, even there).

30. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 30 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? 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,... Odd-ness and homogeneity are much easier to spot than additivity Odd-ness and homogeneity are much easier to spot than additivity [y/4,(2*abs(x)-x)/9]

31. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 31 Analogue(s) of “slope” Want to later geometrically define divergence as limit of flux-integral divided by enclosed volume, curl/rotation as limit of circulation integral divided by enclosed volume What about the linear case? This is the PERFECT SETTING to develop these concepts LIMIT-FREE -- in complete analogy with the development of the slope of a straight line BEFORE calculus! Note, line-integrals of linear fields over polygonal paths do not require any integrals, --- midpoint/trapezoidal SUMS are exact! -- in complete analogy with area under a line in PRECALCULUS!

32. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 32 Recall: “linear” and slope in precalc Consider divided differences, rise over run Linear ratio is CONSTANT, INDEPENDENT of the choice of points (x k,y k ) yy xx

33. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 33 Rarely enough: “Linear” in multi-var. calc. Using tables of function values, or contour diagrams, consider appropriate “divided differences” --> partial deriv.’s, gradient,... In each fixed direction, ratios are constant, independent of choice of points, in particular independent w.r.t. parallel translation.

34. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 34 As usual, first develop pictorial notion of circulation and divergence. BEFORE calculations For NONlinear fields pictorially the local character of divergence and rotation is obvious -- for LINEAR vfs local and global are the same. (Students looking at magnetic field about wire always falsely agree that it is rotational!)

35. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 35 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.

36. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 36 Algebraic formulas: tr(L), (L-L T )/2 (x 0,y 0 ) (x 0,y 0 -  y) (x 0,y 0 +  y) (x 0 +  x,y 0 )(x 0 -  x, y 0 ) for L(x,y) = (ax+by,cx+dy), using only midpoint rule (exact!) and linearity for e.g. circulation integral over rectangle Coordinate-free GEOMETRIC arguments w/ triangles, simplices in 3D are even nicer Develop understanding where (a+d), (c-b) etc come from in limit free setting first

37. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 37 More formulas in linear setting E.g. Translation-invariance in linear fields, additivity in integrand, line integrals of constant fields over closed curves vanish (constant fields) -- pictorial arguments for Develop analogous formulas for flux integrals in 2d and 3d, again relying only on the midpoint rule for straight edges or flat parallelograms. In order to later get general formulas via triangulation's (?!), replace rectangle first by right triangles (trivial!), then by general triangles --> compare next slide on telescoping sums, developing the arguments like “fluxes over interior surfaces cancel”. Warning: To make sense out of div, rot, curl, need to have a notion of angle (inner product…), i.e. cannot get formulas in purely affine setting. Purely geometric (coordinate-free) proof in triangles are very neat & instructive!

38. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 38 Telescoping sums Recall: For linear functions, the fundamental theorem is exact without limits, it is just a telescoping sum! Want: Stokes’ theorem for linear fields FIRST!

39. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 39 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. The picture new TELESCOPING SUMS matters (cancellations!) Caution, when arguing with triangulations of smooth surfaces

40. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 40 Nonlinear vector fields, zoom 1 st kind 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. The original vector field, colored by rot Same vector field after subtracting constant part (from the point for zooming)

41. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 41 Check for understanding (important) Zooming of the 1 st kind on a linear object returns the same object! After zooming of first kind! original v-field is linear subtract constant part at p

42. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 42 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. Report all results to transparency in the front. Scale by area, SEE convergence. Instead of ZOOMING, this perspective lets thecontours shrink to a point. Do not forget to also draw these contours after magnification!

43. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 43 After zooming of 1 st kind Subtract constant part, and zoom: A familiar picture occurs: As the field appears to be closer to linear the ratios integral divided by area become independent of choice of contour, the limits appear to make sense! Subtract constant part, and zoom: A familiar picture occurs: As the field appears to be closer to linear the ratios integral divided by area become independent of choice of contour, the limits appear to make sense!

44. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 44 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” 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 = L p such that for every  > 0 there exists a  > 0 (indep.of p) such that | F(q) - F(p) - L p (q-p) | <  | q - p | (or analogous pointwise definition). 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 = L p such that for every  > 0 there exists a  > 0 (indep.of p) such that | F(q) - F(p) - L p (q-p) | <  | q - p | (or analogous pointwise definition). Advantage of uniform: Never any problems when working with little-oh: F(q) = F(p) + L p (q-p) +o( | q - p | ) -- all the way to proof of Stokes’ thm.

45. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 45 Divergence, rotation, curl For a differentiable field define (where contour shrinks to the point p, circumference -->0 ) 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): Use your judgment worrying about independence of the contour here…. Use your judgment worrying about independence of the contour here…. Consequence:

46. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 46 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….). 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! 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!

47. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 47 About little-oh’s & uniform differentiability By hypothesis, for every p there exist a linear field L p such that for every  > 0 there is a  > 0 (independent of p (!)) such that | F(q) - F(p) - L p (q - p) | <  | q - p | for all q such that | q - p | < . By hypothesis, for every p there exist a linear field L p such that for every  > 0 there is a  > 0 (independent of p (!)) such that | F(q) - F(p) - L p (q - p) | <  | q - p | for all q such that | q - p | < . Key: Stay away from pathological, arbitrary large surfaces bounding arbitrary small volumes, 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! Except for small number (lower order)of outside regions, hypothesize a regular subdivision, i.e. without pathological relations between diameter, circumference/surface area, volume! The errors in the two approximate equalities in the nonlinear telescoping sum:

48. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 48 From 2d to 3d Key: DO IT SLOWLY. Develop the concepts in a planar setting - so you can see them! In planar setting develop the notions of line-integrals, linear fields, trace(divergence), rotation, approximation by linear fields, and integral theorems. After full mastery go to the hard-to-see 3d-case. In planar setting develop the notions of line-integrals, linear fields, trace(divergence), rotation, approximation by linear fields, and integral theorems. After full mastery go to the hard-to-see 3d-case. SPECIAL: The direction of the curl in 3d -- compare next slide! I personally have not yet made up my mind about surface integrals -- I talked to Keith Stroyan, and sympathize with actually playing with Schwarz’ surface (beautiful animations of triangulations --> lighting/shading tilting……) I do not like to start with parameterized surfaces, but instead parameterizable ones….? I personally have not yet made up my mind about surface integrals -- I talked to Keith Stroyan, and sympathize with actually playing with Schwarz’ surface (beautiful animations of triangulations --> lighting/shading tilting……) I do not like to start with parameterized surfaces, but instead parameterizable ones….?

49. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 49 Schwarz’ surface Pictorially the trouble is obvious. SHADING! Simple fun limit for proof Not at all unreasonable in 1 st multi-var calculus Entertaining. Warning about limitations of intuitive arguments, … yet it is easy to fix!

50. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 50 Prep: axis of rotation in 3d Decompose linear, planar vectorfields 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. Decompose linear, planar vectorfields 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. Preliminary: Review that each scalar function may be written as a sum of even and odd part. Preliminary: Review that each scalar function may be written as a sum of even and odd part.

51. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 51 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? Requires prior development of decomposition symmetric/antisymmetric in planar case. Addresses additivity of rotation (angular velocity vectors) -- who believes that? 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 1 st kind), jiggle a little, discover order, rotate until look down a tube, each student different axis 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 1 st 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: “Visual calculus”, or WWW-site. usual nonsense 3d-fieldjiggle -- wait, there IS order!It is a rigid rotation!

52. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 52 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. 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 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) 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, grad=> irrotational (w/ limits)

53. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 53 Animate curl & div, integrate DE (drift) Color by rot: red=left turn green=rite turn divergence controls growth

54. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 54 Spinning corks in linear, rotating field Period indep.of radius compare harmonic oscillator - pend clock Always same side of the moon facing the Earth -- one rotation per full revolution.

55. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 55 Spinning corks in magnetic field Irrotational (black). Angular velocity drops sharply w/ increasing radius.

56. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 56 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 fro position, calculates curl, integrates angular momentum equations, and creates animation using rotation matrices. Colored faces crucial!

57. Foundation Coalition Vector Calculus via Linearizations, 9th Int Conf Tech Coll Math, RenoNV, Nov 1996 Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski 57 Stokes’ theorem & magnetic field 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,... Do your students have a mental picture of the objects in the equn?