An interactive JAVA microscope to visualize divergence and curl via zooming1 http://www.eas.asu.edu/~microscope
Shannon Holland2 Center for Innovation in Engineering Education Arizona State University
[email protected]
Matthias Kawski3 Department of Mathematics Arizona State University
[email protected]
Abstract. We introduce a sophisticated JAVA applet that allows the user to zoom in on any vector eld in the plane (de ned in terms of standard analytic expressions) to see its derivatives. We provide a wide array of di erent magnifying lenses that allow the user to focus on various parts of the approximate linearization. This applet is a keystone piece of a radically new approach to di erential vector calculus that consistently develops notions of derivatives in terms of local (almost) linearity. This approach also suggests to much closer link courses in vector calculus, di erential equations, and linear algebra. The use of the microscope inescapably enforces the understanding of derivatives (including curl and divergence) as objects quantifying local properties. This is most dramatic when used for the most important elds in physics and mathematics, the inverse square central force elds (gravity, electrostatics) and the magnetic eld about an in nitely long straight wire with constant current, or in mathematics the eld f (z) = z1 (the derivative of the complex logarithm) which is universally used for counting. The applet is freely available on the WWW [2] and runs under any standard JAVA-enabled WWW-browser, i.e. it is absolutely platform independent.
1 Introduction The ability to magnify a portion of the graph of a function by repeatedly pressing a ZOOM-button on a graphing calculator has radically changed the way students are introduced to the di erential calculus of a function of a single variable. This approach much better illustrates not only that taking a limit is a process, but also that the concept of di erentiability is intimately intertwined with arbitrarily good local approximability by linear objects. The second author raised the question: "If zooming is so compelling in the calculus of a single variable, why then is it not used throughout all di erential calculus?" [4, 5, 6, 7, 8, 10]. He is now developing an entire course in vector calculus that develops This utility was rst presented at the Tenth International Conference on Technology in Collegiate Mathematics in Chicago, November 1997 1 This work was partially supported by the NSF-sponsored Foundation Coalition through Cooperative agreement EEC 92-21460 2 This work was partially supported by the NSF-sponsored Foundation Coalition through Cooperative agreement EEC 92-21460 and by the ACEPT through NSF DUE 94-53610 1
1
vector di erential calculus via linear approximations, and that climaxes in very accessible presentations of the integral theorems in vector calculus. In the past, the technical obstacle standing in the way of such an approach was that it was practically impossible to visualize vector elds in real time { e.g. using chalk and black board, or pencil and paper. This inability forced a predominantly algebraic treatment (with very few pictures) which in turn formed a major obstacle for all but those students best trained in algebra. Consequently, today vector calculus is understood only by a very small part of the populace. The key concepts of di erential vector calculus, in nitesimal expansion rates (divergence) and in nitesimal rotations (curl) are of major importance from turbulent ows to elctro magnetic phenomena. We claim, that if not dogmatically insisting on a purely algebraic symbolic introduction, these concepts can be presented in very attractive ways to many people that may lack the preparation (and time) to go through traditional courses in this subject. Only in the last decade have computer hardware and software become available that allow one to instantaneously visualize a vector eld. However, until now, no utility comparable to the zooming button on the graphing calculators has been available that allowed one to instantaneously, and interactively, visualize the derivatives of vector elds. In previous presentations [4, 5, 6, 7, 8, 10] and in the upcoming book \Zooming and limits: From sequences to Stokes theorem" [11] we have traced this lack of a facility to a common incomplete understanding of the di erent modes of zooming: E.g. zooming for continuity (and Riemann-integrability) is structurally completely di erent from zooming for di erentiability (and for the behaviours about zeros of the vector eld). We here present the missing utility, and expect that it will have consequences reaching far beyond our current imagination. The curl really is a concept in three dimensions { and it is actually possible to see it via zooming in three dimensions, compare [4, 5, 6, 7, 8, 10]. However, for an introductory treatment the planar analogue is much more appropriate, and also requires much smaller computing resources. Throughout this article we abuse the term curl in the obvious way: For a vector eld F~ (x; y) = M (x; y)~{+ N (x; y)~| in the plane we @M ~ refer to the scalar quantity @N @x ? @y as the curl of F . Technically the term \in nitesimal rotation" is preferrable, but it is not commonly used. As there is no danger in this article to confuse the two-dimensional and the threedimensional setting, and the emphasis is on visualizing the vectorial derivatives, and to provide access to major concepts of vector calculus to a less technically focused audience we eventually decided to use the term curl which is widely recognized, and which best encapsulates the features that we are helping visualize.
2
2 Features of the microscope
2.1 General description
Our utility plots a user-speci ed vector eld in the plane (represented as usual by arrows) in a user-speci ed window referred to as \the background". The user may select a magnifying lens from a wide selection of di erent lenses, described below, and drag this lens (using the mouse) across the background. Inside the lens the results of zooming (in the various modes) are displayed almost instantaneously. Depending on the choice of the lens, these illustrate the (almost) constant drift near the point of zooming, or the local variation from a constant drift. The vector elds displayed inside the lens are calculated numerically via standard divided di erences. As the magni cation factor is increased, the latter rapidly converge to the components of the local linearization. As alternatives to the complete linearization we provide for plotting only special geometrically meaningful components that converge e.g. to the skew-symmetric part or the symmetric part of the linearization. Due to the nite resolution, and the very limited number of pixels available to display any individual arrow, already very modest, nite magni cations yield the image of the limit, which is correct pixel-by-pixel! We plan to add features that allow the user to specify (draw) polygonal volume elements, whose time-evolution under the ow of the vector eld, or under the drift plus components of the linear part are displayed in order to visualize the integrals of the rotation and the divergence, both of the in nitesimal and the approximate microscopic version.
2.2 Data input and control parameters
We judged it of critical importance that the user can explore any vector eld of her/his choice, rather than be restricted to a nite collection (or nitely parameterized family) of vector elds. We thus included a powerful fast parser [12], that allows the user to type in essentially any vector eld that can be expressed in terms of standard analytic functions. This is complemented with a pull-down menu of commonly used elds, e.g. the normalized gravitational (electro-static) eld about a point mass (point charge), and the normalized magnetic eld about an in nitely long straight wire through which there
ows a constant current. | ~ (x; y) = x~2| ? y~2{ G~ (x; y) = xx~2{ ++ y~ H (1) 2 y x +y These are notoriously dicult to plot due to their singularities at the origin together with the built-in feature of most plotting packages that auto-scale every eld. The parser handles this very nicely as it allows one to use cut-o functions, which encapsulate the physically meaningful nite radius " of the mass, charged body, or of the wire (inside which the elds vanish). We have preprogrammed truncated elds, using the \box"-function provided by the parser, which is closely related to the Heaviside function u(t) = 1 if t 0 3
and u(t) = 0 if t < 0 (using a typical choice of radius ") | ~ 0 (x; y) = u(x2 + y2 ? "2) x~2| ? y~2{ (2) H G~ 0 (x; y) = u(x2 + y2 ? "2) xx~2{ ++ y~ 2 y x +y These are available from a pull-down menu. This menu provides several more vector elds that exhibit interesting features. We caution the user to avoid polynomial formulas like those commonly found in traditional calculus textbooks: These not only are mostly unphysical and rather irrelevant for practical applications, but they also exhibit very undesirable growth behaviours (from the standpoint of visualization). Much more useful are oscillatory and, even globally bounded rationally functions as components. These produce images that are more familiar from the study of dynamical systems, in particular, typical ows from uid dynamics. The user also can control standard features such as the location and size of the window, the grid on which the arrows are drawn etc. Moreover, the user may vary the magni cation factor for the various kinds of zoom lenses described in the following section. Changing the magni cation factors allows one to slowly experience the convergence to a linear eld.
2.3 Many di erent zoom lenses
As discussed in detail in previous presentations [5, 6, 7, 8, 10] there is a need for di erent modes of zooming. We realize these via di erent lenses from which the user may select any one at a time. The most familiar one magni es only the domain, eventually yielding a constant eld. This is "the picture" for understanding the convergence of Euler's method (and related methods for the numerical integration of di erential equations) and for the convergence of Riemann sums de ning line integrals. Derivatives in turn always involve taking a di erence. Naturally one plots the local variation of the vector eld from its value at the reference point ~F (x; y) = F~ (x0 + x; y0 + y) ? F~ (x0; y0) (3) which vanishes at the center of the lens. This di erence makes perfect sense in a Euclidean setting, while in a general setting on a manifold the minus sign requires a more sophisticated interpretation, compare e.g. [9]. Using magni cation factors for the arrows (see below) that are commensurate with the magni cations of the domain (lens versus background) this vector eld ~F is readily ~ of the original eld F~ . observed to converge to the local linearization DF ! a b To facilitate the discussion we abuse notation and identify a matrix A = c d with the linear vector eld LA (x; y) = (ax + by)~{ + (cx + dy)~|. Any matrix may be decomposed in many di erent ways. In this setting aimed at di erential vector calculus we consider A = Askew + Adiag + Asymm;0 (4) which respectively are the skew symmetric part, a multiple of the identity, and the tracezero-symmetric part: T b?c ! A ? A 0 Askew = 2 = c?b 02 (5) 2
4
a+d
Adiag = trace(A) I22 = Asymm;0
T = A+A ?A
2
diag
2
(6)
a+d
0
=
!
0 2
a?d b+c b+2 c d?2 a 2
2
!
(7)
The most common decomposition is A = Askew + Asymm where Asymm = Adiag + Asymm;0. However, both for geometric reasons (see section 3), and for the sake of being able to visualize each part separately we decided to include lenses that correspond to any one of the components above. The relation to vector calculus is straightforward: The curl and rotation correspond to the skew symmetric part of the Jacobian (DF ), the divergence (rate of expansion of volume) is the trace of (DF ), while the symmetric part includes both information about rate of change of volume as well as principal directions of expansion and compression. The latter are studied e.g. in chaotic dynamical systems. While such decompositions are straightforward for linear elds, they make no immediate sense for the still nonlinear elds seen after any nite amount of magni cation. Let us here just state that various lenses are provided corresponding to these di erent components. Their implementation as well as the geometric rationale will be discussed in the section 3.
2.4 Integrating curl and divergence
To round out this utility we are planning to add features that allow one to visualize the integrated ow, and the integrals of the components discussed in the previous subsection, in particular the integrals of the curl and the divergence. More speci cally, many software packages allow the user to click anywhere in the graphics window to initialize a di erential equation, and the program integrates the associated initial value problem and draws the integral curve through this point. This is not enough to illustrate the e ects of the derivatives visualized so far. Indeed, one needs to consider small volume elements and track them along the ow. Our utility shall allow the user to specify such initial volume elements e.g. by interactively drawing circles or polygons in the graphics window. The time evolution of the volume, shape, orientation, and the location of the vertices, edges or general boundaries will then be displayed according to one's speci cation. E.g. integrating only the drift and the skew symmetric part illustrates the rotation (spinning). Integrating only the drift and the diagonal part illustrates the volume changes. Integrating only the drift and the symmetric part illustrates the volume changes together with rst order distortions as they have become familiar from studying chaotic systems. Noninteractive versions of selected such animations have been implemented in MAPLE by the second author [5, 6, 7, 8, 10], and have been widely used in teaching (see the second author's website http://math.la.asu.edu/ ~ kawski for worksheets for in-class use). We anticipate that this utility will be able to help visualize even ne points like the di erence between integrating the linearization (the e ects on idealized, in nitesimal small volume elements) and integrating the di erential euqation for several initial conditions that are close to each other (the e ects on microscopic or macroscopic objects). For 5
example, linear ows map line segments to line segments and thus transform polygons into polygons, and ellipses into ellipses, while general nonlinear ows may dramatically distort the shapes. Critical issues under development are a well-designed user-interface, selection of practical, yet suciently accurate integration schemes, and e.g. adaptive schemes to select suitable number of points along the boundaries in order to follow the transformation of the edges under the various ows. At this time we may point the reader to an impressive JAVA applet by David Austin [1] that implements some of these features, but is restricted to a small selection of prede ned elds (it lacks a parser). Even with its limitations this applet provides amazing insights, e.g. see how squares evolve under the ow of the (irrotational and divergence-free) \magnetic eld" H~ (x; y).
3 Technical and geometric background 3.1 Four scales
Most programs that plot vector elds (represented by arrows) automatically scale the arrow so that the longest arrow in the plotting window does not overlap neighboring grid points. We do not see any way to avoid this, but since the microscope may be used to zoom in near zeros of the vector eld, it is important that near these points the eld (inside the lens) is correctly drawn as almost vanishing (arrows that are not scaled up), or, alternatively, that the user at least has access to the information lost in the autoscaling. Indeed, we consider it worthwhile that the user re ect on the four di erent scales used, and that this be explicitly discussed in class: In the background window one should distinguish the SCALE 1 used in the domain (the gridpoints) from the SCALE 2 used for the arrows . This SCALE 2 is determined for any xed plotting window (before anything is drawn) by evaluating the vector eld at all grid points and then normalizing it: 1 (arrow drawn at (xi ; yj )) = F~ (xi; yj ) (8) ~ maxi;j kF (xi ; yj )k1 Like most programs that plot vector elds, ours will not show any di erence between plots of positive multiples of a xed eld { however, the scaling factor is shown maxi;j kF~ (xi; yj )k1 on a side bar. The lens has its own SCALE 3 for the part of the domain that is shown. (This is NOT the tangent space, as the magni cation remains nite.) The SCALE 3 is related to SCALE 1 by a user-selected magni cation or zooming factor. The lens also has its own scale SCALE 4 for the arrows. When zooming for continuity (local almost constancy, to understand Riemann integrals), then the default is that the SCALE 4 for the arrows in the lens is the same as the SCALE 2 for the arrows in the background. If one zooms for the derivative (or any of its components) then the default is that the magni cation factor (SCALE 2)/(SCALE 4) for the arrow lengths changes in a lock step fashion with the magni cation factor (SCALE 1)/(SCALE 3) as determined by the user. The rationale is that in this case the lens shows an almost linear eld, i.e. almost 6
homogeneous eld, and one needs to zoom at equal rates in domain and range to observe convergence. The initial choice for the SCALE 4 is somewhat tricky: While it is straightforward to choose an appropriate scale at any xed point of zooming, the scales used at di erent points (when dragging the lens) should be comparable (e.g. to see the di erences in magnitude as the lens is placed near a zero of the vector eld). Technically, the information needed is the C 1-norm
@F2 (p) ; @F1 (p) ; @F2 (p) (9) 1 kF~ jK k1;1 = sup k(DF )(p)k1 = sup max @F ( p ) ; @x @x @x @x p2K p2K of the vector eld F~ restricted to the plotting window K . A reasonable estimate for this (
)
norm is obtained by numerically evaluating of the sup-norm at about 2 dozen pseudorandomly selected points pk = (xx; yk ), k = 1; : : : N :
F~ (xk + x; yk ) ? F~ (pk ) kF~ jK k1;1 1max max kN y 8 <
:
~ ~ ; F (xk ; yk +yy) ? F (pk ) 1
1
9 = ;
(10) The increments x and y are chosen small relative to the SCALE 1 of the background window. To ensure that the images shown are reproducible, we use a set of points that is xed in screen coordinates. However, if one changes the background window K , one should expect that the lengths of the arrows shown inside the lens are scaled di erently. For vector elds that have a large variation or whose derivatives have singularities inside the window, these estimates typically are severe underestimates. Consequently, as one moves the lens close to such singularities or points where k(DF )k is large one will observe wildly varying and extremely long arrows { while surprising in the rst moment we found that such a display vividly illustrates the singularity, and we now consider this display very desirable { it really \gets the message across". This behavior is most commonly encountered when zooming on the gravitational eld G~ or the magnetic eld H~ : If the eld is not truncated this wild behavior is encountered near the singularity at the origin. If truncated, then this behavior is observed when moving the lens to the circle at which the eld is truncated. For the cases where one is interested only in the directions of the eld inside the lens, we provide a switch to \Show direction only". This allows SCALE 4 to vary from point to point, autoscaling the eld at each position of the lens so that (at any point of zooming) the arrows shown have maximum length without overlapping.
3.2 Geometric background
This microscope is designed for vector elds in the plane. It takes as input the components in Cartesian coordinates. The grid is adapted to Cartesian coordinates. The arrows are normalized (auto-scaled) using the sup-norm again relative to standard Cartesian coordinates. (The use of a square lens is an artifact due to the utilities readily available in the JAVA development kit; we would have preferred a round hour glass, but the programming e ort would have been signi cantly larger). 7
However, derivatives are geometric, i.e., they are intrinsically coordinate free. Moreover, the divergence and the curl are geometric objects, too, even though most students only encounter them in terms of sums and di erences of partial derivatives of component functions. The central theme in the program to develop an introductory vector calculus based on linearity is that a derivative is a linear object that locally approximates the given object { and linearity is a geometric notion, independent of the choice of basis. We have thus strived to design the utility, and the formulas used in the program, to re ect as much as possible the geometric nature of the objects under investigation. But there are limits to this, e.g. what are the symmetric and skew symmetric parts of a general vector eld on a manifold? The more sophisticated interpretations of the curl and the divergence in terms of tangent and co-tangent vector elds, and in terms of exterior di erential forms are beyond the scope of this utility. Yet the current implementation is geometrically sound. The main problem arises from the objective to have lenses which show (ideally after in nite zooming) only the skew-symmetric part, the diagonal part, or the symmetric part of the derivative of the vector eld at the point of zooming. When the magni cation factor is practically in nite (i.e. where the eld shown inside the lens agrees pixel-by-pixel with the true limit) it is obvious (simple linear algebra) what the lens shall show. However, for nite magni cation factors, the numerical di erentiation yields clearly visible deviations from linear elds. Our objective is to split these nonlinear elds in a geometrically sound fashion, such that as the magni cation factor increases the images shown inside the respective lenses converge to the usual components of the linearization (diagonal, symmetric, skew-symmetric). The central idea behind our construction is to utilize a map built on the projection L : gl(2) 7! so(2) ft I22 : t 2 Rg de ned by : A 7! J ?1AJ where J is the matrix
J = 10 ?01
!
(11)
Here J may be either seen as a rotation J 2 SO(2) or as the in nitesimal generator J 2 so(2), since exp( 2 J ) = J . The skew symmetric part Askew and the diagonal part Adiag both commute with J considered as an element of so(2), and consequently they commute with any rotation matrix exp(tJ ) for any t 2 R. On the other hand any trace-zero-symmetric matrix B , and thus also Asymm;0, does not commute with J , but instead (B ) = ?B . In our understanding the actions of SO(2) and so(2) on the domain and the range of
any vector eld in the plane are a sound foundation on which to base a decomposition of the nonlinear elds obtained as numerical estimates for the derivative and shown inside the lens. Indeed, for a xed position of the lens, centered at p = (x0 ; y0), let (~F ) = (~pF )(x; y) denote the vector eld (~F )(x; y) = F~ (x0 + x; y0 + y) ? F~ (x0; y0)
(12)
which vanishes at (x; y) = (0; 0). The following table lists (with the usual abuse of notation) the components of the nonlinear vector eld (~F ) shown by the various lenses (up to the scaling as discussed in 8
the previous subsection). Here %~ denotes the unit radial vector eld %~(u; v) = pu21+v2 (u~{ + v~|), %~? the unit tangential vector eld %~?(u; v) = pu21+v2 (?v~{ + u~|), and < ; > is the standard Euclidean inner product = uu0 + vv0. Name of lens
Formula for calculation in general case
all
~F
Linear case !
radial tangential
~F rad =< ~F; %~ > %~ ~F tang =< ~F; %~? > %~? = ~F ? ~F rad
not linear not linear !
commuting anti-commuting multiple of identity skew symmetric symmetric
a b c d
~F comm = 14 ?J ?1~FJ + 2~F ? J~FJ ?1 ~F anti = 14 J ?1~FJ + 2~F + J~FJ ?1 = ~F ? ~F comm
~F diag =< ~F comm; %~ > %~
~F skew =< ~F comm; %~? > %~? = ~F
a+d c?b b?2 c a+2 d 2 2 b + 0 2c b+c 0 2
a+d 2
? ~F diag
~F sym = ~F diag + ~F anti = ~F ? ~F skew
0 0
b?c 2
a
b+c 2
0
a+d 2 c?b
!
!
!
2
0
b+c
!
2
d
This is a very large number of di erent lenses { and in many cases one may only want to look at a few of these. However, once the code was written for the essential ~ Fskew , the others were almost obtained as corollaries, and it is conceivable that for certain applications/investigations one may want to look at any one of these. Probably the most standard decomposition of a matrix is into its symmetric and skew-symmetric parts
A = Asymm + Askew = A +2 A + A ?2 A T
T
(13)
The skew symmetric part essentially is the same as the rotation (or curl) in our setting. The symmetric part combines information about rate of change of volume (area) and some information about the change of shape. On the other hand, more geometric decompositions use the projections (id + )=2 and (id ? )=2 onto the parts that commute with rotations and those that anticommute with a rotation by 2 . The rst of these two components may be further decomposed
A = Acomm + Aanti = (Askew + Adiag ) + Aanti
(14)
where the multiple of the identity (encapsulating the divergence) may be either considered alone or grouped together with the skew symmetric part (rotations), or with the tracezero-symmetric part Aanti . This latter component is the least often considered part; it roughly encapsulates shears. 9
4 Typical explorations and exercises
4.1 Understanding the limitations of a plot
Due to the nite number of pixels only a very limited number of distinct arrows can be drawn and distinguished at any grid point. The main trade-o is between a ne or coarse grid, and the number of di erent arrows that can be distinguishes at any grid point. To get a feeling for the coarse grid, plot a typical periodic eld (e.g. F~ (x; y) = cos x sin y~{ + (sin x ? cos y)~| on various windows K = [?a; a] [?a; a]. When a increases slowly, say from a = 1 to a = 10 one nds the expected images. Yet with larger values of a, say around a = 50 many users are surprised by the aliasing e ects due to the inadequate sampling frequency (the grid is very coarse when compared to the natural frequencies of the vector eld). To make this a fun game, one may want to search for values for a that are in resonance with the number of pixels, the grid size, and the frequencies of the vector eld. Such dramatically undersampled images yield another surprise when zooming for the drift, slowly dragging the lens, and seeing rapid oscillations inside the lens while the background appears very calm (due to the undersampling). Indeed, zooming of the zero-th kind simply means increasing the sampling frequency (inside the lens)! To get a feeling for the severe limitations due to the nite number of pixels, each user should plot a number of polynomial vector elds as well as the usual singular elds G~ and H~ . For the latter two, the user should compare the images for even and odd grid sizes (each window centered about the singularity): The vector F~ (p) with the largest magnitude (one-norm) among all p 2 K is used to scale all arrows in the window. Thus if variations in magnitude are large, many arrows may be displayed as points only. (This is one reason why one often considers only direction elds when studying di erential equation.) Some playing with such unsatisfactory examples will lead most users to rather work with globally bounded vector elds, whose components typically are globally bounded rational functions of two variables, or formed as sums and products of trigonometric expressions, as well as the truncations of the singular elds (multiplied with a cuto function ). To understand the autoscaling we suggest that any rst time user plot any reasonable vector eld, and then compare its image with those of positive multiples of the same eld. They should be identical images.
4.2 Integrating vector elds
Whether for line integrals (de ned through limits of Riemann sums) or for solutions of di erential equations (Euler's method, as well as existence and uniqueness results) an essential property is the continuity of the vector eld This means that sucient magni cation (zooming of the rst kind) yields a practically constant vector eld. While generally relatively unexciting, even this kind of zooming can yield surprises, e.g. when the eld is undersampled (as described in the previous subsection). An interesting exploration is to start with the magni cation factor 1, and while holding the lens xed, to slowly increase the magni cation factor, so as to experience the process of taking the limit. A further valuable exercise is to drag the lens to nd zeros (equilibria) of the vector 10
eld. Here one may turn the switch to \show direction only" and try to place the lens exactly at the zero to determine the kind of equilibrium point of the associated di erential equation. Equilibria are the only points in continuous vector elds where even after any magni cation one observes very rapid changes of the eld shown inside the lens as one moves the lens (but only when using the option \show direction only").
4.3 The derivative of a vector eld
In order to make best use of the microscope it is mandatory that the user (student) deeply understands the concept of linearity in terms of
L is linear , (8p; q)(8c) L(p + q) = L(p) + L(q) ^ L(cp) = cL(p):
(15)
The user must be able to identify whether a picture of a vector eld represents a linear eld. (Note that homogeneity, or the lack of it, is much easier to detect than additivity, or the lack of it.) As an important check for understanding, every user should zoom for the derivative of a linear eld. At any rate of magni cation the lens shows a copy of the original eld! (linear objects \look the same on any scale".) The rst exploration is to start with the magni cation factor 1, and while holding the lens xed, to slowly increase the magni cation factor, so as to experience the process of taking the limit, seeing how the eld shown in the lens converges to a linear eld. The rst place where the linearization plays an important is near zeros (equilibria) of the vector eld, and as long as the derivative of the vector eld is continuous, one does not even need to exactly center the lens at the zero, to clearly see the how the linearization characterizes the behaviors of the di erential equation near the equilibrium. After this, one may zoom in at various points where the derivative does not vanish, and see that no matter where one goes, one always nds a well de ned linear eld. Two very important elds for which to study the derivative are (the normalized truncation of) the gravitational/electro static eld G~ and (the normalized truncation of) the magnetic eld H~ , which are both irrotational and divergence free (at every point other than the singularity). However, since neither is constant, their derivatives cannot vanish. Most users have never seen their derivatives { they vividly show the irrotational character (which starkly contrasts the global picture of the magnetic eld H~ ), and with a little closer observation one also easily sees that the two eigenvalues are negatives of each other, i.e. the elds are divergence free (which starkly contrasts the global picture of the gravitational/electro static eld G~ ).
4.4 Looking at the components
Classical analytic geometry is the place where most have seen how to decompose linear mappings of the plane for the rst time, linear algebra is a more abstract setting. Our program of introducing vector calculus via linearizations under development by the second author devotes much e ort to the decompositions of linear vector elds before proceeding to the di erential calculus of nonlinear elds. Students so prepared nd it most natural to look at one component of the (almost) linear eld at a time. 11
For each component (each of the many additional lenses) one should again start with slowly increasing the magni cation factor from 1 to practical in nity, to observe the process how the component of the nonlinear eld converges to the usual component of the linear eld representing its derivative. The rst decomposition the student should try to understand is the decomposition into the radial and the tangential part. This is best done by repeatedly changing the lens type between the entire (approximation of the) derivative and these two components, rst at any one xed point, then while moving the lens. The observed mix of inward/outward ow or forward/backward rotation the suggest further decompositions, essentially into the average of the radial (or tangential) component and the remainder. Just as before, the user should again repeatedly switch between the three corresponding lens types, to get an intuitive feeling of how these decompositions work (and how they relate to the algebraic formulae). The skew symmetric part clearly corresponds to the net torque, responsible for a rotating or spinning motion of any volume element. The diagonal part evidently encapsulates the net rate of change of volume (area). In the planar case the lenses clearly show that each of these can be represented by a scalar quantity, the trace (divergence) and the two dimensional analogue of the curl. The e ects of the trace-zero-symmetric part is best described by "shears". A more familiar picture arises from combining this part with the diagonal part to form the symmetric part. This part yields a important surprise, that directly illustrates the important theorem of linear algebra that every self-adjoint linear operator (matrix) can be orthogonally diagonalized. Indeed, in addition to showing the rates of change in volume this view also shows the principal directions of the squashing and pulling. In all nondegenerate cases one inescapably observes two perpendicular principal directions, agreeing with the eigen vectors of the symmetric part.
5 Anticipated bene ts and conclusion This utility makes the advanced concepts of vectorial derivatives like divergence and curl accessible to a large population which may lack some of the formal training that is the usual prerequisite for a standard course in vector calculus. Some may use our utility to only develop a pictorial understanding e.g. relating the images of our microscope to the weather maps familiar from every day's news-cast. Yet the utility is very exible and can also provide for rich experiences at advanced levels and thus serve as a training ground to develop visualization skills that complement the acquisition of traditional formal algebraic skills to analyze vector elds. The most critical messages which one cannot escape are: Taking limits is a process that is intimately related to zooming. Di erentiability means arbitrarily good approximability by linear objects. \Irrotational" and \divergence free" are local, not global concepts. Students who have experienced this microscope will not think that the gravitational eld or the 12
magnetic eld have nontrivial divergence or nontrivial rotation when exposed to the usual, fallacy-inviting pictures.
This utility (as well as the program under development by the second author) demon-
strates the close links between vector calculus, di erential equations, and linear algebra, which under typical conditions is so easily lost on the students.
While we may not expect that everyone has the time to develop a very deep technical understanding of vector calculus, we believe that this utility makes it possible to communicate to the large audiences the core concepts of vector calculus (and hopefully win over a few who otherwise would not have considered formally studying more math and science). The aesthetic aspects of the utility are particularly important for this objective. Today, every calculus students experiences zooming in the study of derivatives of functions of a single variable. Out utility sends a strong message that the connection between zooming and derivatives (as experienced in single variable calculus) is not an isolated coincidence, but rather a manifestation of a general principle applicable to any kind of derivative. We may never see a complete integration of the traditional sophomore courses vector calculus, di erential equations, and linear algebra. Yet, this microscope visually demonstrates the connections, in a way that the usage of common (or not so common) algebraic symbols may never be able to do. At last, this microscope may nally end the common frustration experienced by many instructors who have a hard time to convey to students that the picture of the magnetic eld H~ illustrates an irrotational eld { a property critical for understanding electromagnetics.
Acknowledgment. The second author gratefully acknowledges very helpful discussions with Edwin Ihrig which much clari ed the geometrical background of the various lenses.
References [1] D. Austin, Flows of vector elds, http:/www.math.ubc.ca/javamath/Flow/flow.html [2] S. Holland and M. Kawski, A JAVA microscope to zoom for derivatives of vector elds, http://www.eas.asu.edu/~ microscope [3] Kaput, Simcalc, http://tango.mth.umassd.edu/ [4] M. Kawski, Vector Calculus via Linearizations, Joint Spring Meeting MAA - AZMATYC, Flagsta , 1996. [5] M. Kawski, Vector Calculus via Linearizations, 5th Conference on the Teaching of Mathematics, Baltimore, 1996. 13
[6] M. Kawski, Vector Calculus via Linearizations, 9th ICTCM (International Conference on Technology in Collegiate Mathematics), Reno, 1996. [7] M.Kawski, Conmputer Visualization and Vector Calculus Proc. 2nd Asian Techn. Conf. Math., Penang, Malaysia, 1997. [8] M. Kawski, Computer algebra systems and vector calculus: A radically new approach based on visualization 4th Int. Symp. on Derive and the TI-92, Kungsbacka, Sweden, 1997. [9] M. Kawski and H. Sussmann, Noncommutative power series and formal Lie-algebraic techniques in nonlinear control in: Operators, Systems and Linear Algebra, U. Helmke, D. Pratzel-Wolters, E. Zerz, eds., Teubner, 1997. [10] M. Kawski, How CAS and Visualization lead to a Complete Rethinking of an Introduction to Vector Calculus, Proc. 3rd Internat. Conf. on Techn. in Math. Teaching, Koblenz, Germany, 1997. [11] M. Kawski, Limits and zooming: From continuity to Stokes' theorem, (book under preparation). [12] D. Liu, Precise calculator, http://reality.sgi.com/employees/davidliu mti/java/calc/calc.html. [13] J. Marsden and A. Tromba, Vector Calculus, Freeman 1976. [14] David O'Tall, Making reserach in mathematics ducation relevant to research mathematicians, Joint Ann. Meetg. AMS and MAA, San Diego, 1997. [15] H. Schey, div, grad, curl, and all that, Norton, 1973. [16] E. Purcell Electricity and Magnetism, Berkeley Physics Course, McGraw-Hill, 1963. [17] M. Spivak, A Comprehensive Introduction to Di erential Geometry, Houston, 1970.
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