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Vector Calculus via Linearizations Matthias Kawski Department of Mathematics Center for Innovation in Engineering Education Arizona State University Tempe, AZ 85287 firstname.lastname@example.org 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)
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 • Stokes’ Theorem long motivation side-track, regarding rigor etc. This work was partially supported by the NSF through Cooperative Agreement EEC-92-21460 (Foundation Coalition) and the grant DUE 94-53610 (ACEPT)
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.
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???
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!)
Zooming in multivariable calculus Zoom in on a surface -- is the Earth round or flat ???
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.
Zooming on contour diagram Easier than 3D. -- Important: recognize contour diagrams of planes!!
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 ……..
Naïve zooming on vector field Be patient! Color will be utilized very soon, too. 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
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!
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
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!
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
Convergence of R-sumsvia zooming of zeroth kind (continuity) Common pictures demosntarte 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.
Zooming of the 1st kind, calculus I This is the usual calculator exercise -- this is remembered for whole life!
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 charater 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 calcuator? Answer: The first derivative test!
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.
Uniform continuity, pictorially A short side-excursion, re rigor in proof of Stokes’ thm. Demonstration: Slide tubings 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!!
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
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!
21 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.
Convergence of R-sums in multivar.calc.via zooming of 0th kind (continuity) Almost the little-oh proof, with uniform-cont. hypothesis also almost the complete e-d proof. -- Remember THIS picture for advanced calc.!
Zooming of 1st 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)
24 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)
Zooming of 0th 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.)
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”!!)
Zooming of 1st kind in vector calc. Now it is all obvious!! -- What will we get??? The original vector field, colored by div Same vector field after subtracting constant part (from the point for zooming) Practically linear Prep: pictures for pointwise addition (subtraction) of vfs recommended
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?
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, ...
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!
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
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.
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!)
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.
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
More formulas in linear setting 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 firstby 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! E.g. Translation-invariance in linear fields, additivity in integrand, line integrals of constant fields over closed curves vanish (constant fields) -- pictorial arguments for
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!
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!)
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.
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!
Student exercise: Limit Instead of ZOOMING, this perspective lets thecontours 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.
After zooming of 1st 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!
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.
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:
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!
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!
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. 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….?
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!
Prep: axis of rotation in 3d Preliminary: Review that each scalar function may be written as a sum of even and odd part. 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.