Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
How CAS and Visualization lead to a
1
complete rethinking of an intro to vector calculus
Matthias Kawski
Department of Mathematics
Arizona State University
kawski@asu.edu
http://math.la.asu.edu/~kawski
Lots of MAPLE worksheets (in all degrees of rawness), plus plenty
of other class-materials: Daily instructions, tests, extended projects
This
work was partially
supported by(to
thecome
NSF through
Cooperative
Agreement
“VISUAL
CALCULUS”
soon, MAPLE,
JAVA,
VRML)
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
How CAS and Visualization lead to a
2
complete rethinking of an intro to vector calculus
You zoom in calculus I for derivatives / slopes
-Why then don’t you zoom in calculus III
for curl, div, and Stokes’ theorem ?
•
•
•
•
•
•
•
review: 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
Controllability versus conservative fields / potentials
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
5
Calculators have ZOOM button!
Tickmarks contain
info about e and d
New technologies provide new avenues:
Each student zooms at a different point, leaves final result on screen,
all get up, and …………..WHAT A MEMORABLE EXPERIENCE!
(rigorous, and capturing the most important and idea of all!)
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
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.
In this presentation only, emphasize graphical approach and analysis.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
7
Zooming on contour diagrams
Easier than 3D. -- Important: recognize contour diagrams of planes!!
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
9
Zooming for line-INTEGRALS of vfs
Without the blue curve this is
the pictorial foundation for
the convergence of Euler’s and
related methods for numerically
integrating diff. 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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
10
Two kinds of zooming
It is extremely simple, just consistently apply rules all the way to 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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
11
The usual e-d boxes for continuity
This is EXACTLY the e-d characterization of continuity at a point, but
without these symbols. CAUTION: All usual fallacies of confusion of
order of quantifiers still apply -- but are now closer to common sense!
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
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
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
12
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
Convergence of R-sums
via zooming of zeroth kind (continuity)
13
Common pictures
demonstrate how area
is exhausted in limit.
The zooming of 0th kind picture demonstrate that the limit exists! -- The first part
for the proof in advanced calculus: (Uniform) continuity => integrability.
Key idea: “Further subdivisions will not change the sum” => Cauchy sequence.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
14
Zooming of the 2nd kind, calculus I
Zooming at quadratic ratios (in range
/domain) exhibits “local quadratic-ness”
near nondegenerate extrema.
Even more impressive for surfaces!
Also: Zooming out of “n-th” kind
e.g. to find power of polynomial,
establish nonpol character of exp.
Pure meanness:
Instead of “find the min-value”,
ask for “find the x-coordinate (to
12 decimal places) of the min”.
Why can’t one answer this by
standard zooming on a calculator?
Answer: The first derivative test!
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
15
Zooming of the 1st kind, calculus I
Slightly more advanced, e-d characterization of differentiability at point.
Useful for error-estimates in approximations, mental picture for proofs.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
16
A short side-excursion, re rigor in proof of Stokes’ thm.
Uniform continuity, pictorially
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!!
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
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.
Naïve proof ideas easily go thru, no need for awkward MeanValueThm
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
18
Zooming of 0th kind in multivar.calc.
Surfaces become flat, contours disappear, tables become constant?
Boring? Not at all! Only this allows us to proceed w/ Riemann integrals!
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
19
19
e-d for unif. continuity in multivar. calc.
Graphs sandwiched in cages -- exactly as in calc I. Uniformity:
Terrific JAVA-VRML animations of moving cages, fixed size.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
21
e-d for unif. differentiability in multivar.calc.
Advanced calc:
Where are e and d ?
Animation: Slide
this cone (with tilting
center plane around)
(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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
22
e-d charact. for continuity in vector calc.
Warning: These are uncharted waters -- we are completely unfamiliar
with these pictures. Usual = continuity only via components functions;
Danger: each of these is rather tricky Fk(x,y,z) JOINTLY(?) continuous.
Analogous animations for uniform continuity, differentiability, unif.differentiability.
Common problem: Independent scaling of domain / range ??? (“Tangent spaces”!!)
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
23
Linear vector fields ???
Usually we see them only in the DE course (if at all, even there).
Who knows how to tell whether a pictured vector field is linear?
---> What do linear vector fields look like? Do we care?
((Do students need a better understanding of linearity anywhere?))
What are the curl and the divergence of linear vector fields?
Can we see them? How do we define these as analogues of slope?
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
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
Dy
Dx
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
26
Constant ratios for linear fields
Work with polygonal paths in linear fields, each student has a different
basepoint, a different shape, each student calculates the flux/circulation
line integral w/o calculus (midpoint/trapezoidal sums!!), (and e.g. via
machine for circles etc, symbolically or numerically), then report
findings to overhead in front --> easy suggestion to normalize by area
--> what a surprise, independence of shape and location! just like slope.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
27
Algebraic formulas: tr(L), (L-LT)/2
Develop understanding where (a+d), (c-b) etc come from in limit free setting first
 




RL  Tds = L( x0 , y0 - Dy)  i Dx  L( x0  Dx, y0 )  j Dy




- L( x0 , y0  Dy )  i Dx - L( x0 - Dx , y0 )  j Dy
=..(only using linearity)... = (c - b) DxDy
(x0,y0+Dy)
(x0-Dx, y0)
(x0,y0)
(x0+Dx,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 -Dy)
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
28
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 ( x k 1 ) - F ( x k )
=
 Dx
x k 1 - x k
b
=  F ( x )dx
a
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
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  DAk

Caution, when arguing with
= trL   k DAk
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
30
Nonlinear vector fields, zoom 1st kind
The original
vector field,
colored by rot
Same vector field
after subtracting
constant part (from
the point for zooming)
If after zooming of the first kind we obtain a linear field, we declare the
original field differentiable at this point, and define the divergence/rotation/curl
to be the trace/skew symmetric part of the linear field we see after zooming.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
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.
32
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
33
An interactive JAVA microscope to zoom for derivatives of vector fields
realized by Shannon Holland, ASU.
http://www.eas.asu.edu/~asufc/microscope
Final version to be presented at the ICTCM, Chicago in November 1997
Integrals & continuity
Derivatives
Show all
Symm part only
Anti-symm part
xy-magnification factor
1
10
100
1000
infinity
dxdy-magnification factor
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
34
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.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
35
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
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 your judgment worrying about
independence of the contour here….
1
Consequence: | div ( F - L)| 
 e (diam)  (circumference)  4e
(area )
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
36
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.
 F  Nds =
 
=   F  Nds

  trF ( p )  DA )
   div ( F )dA
C
k Ck
k
k
k
k
Rk
=  div ( F )dA
R
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!
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
37
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:
| 
 

( F - Lp )  NdS |  e  diam(Vk )  area ( Sk )
| 
 
div ( F - Lp )dV |  e  vol (Vk )
Sk
Vk
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!
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
38
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
39
Decompositions
Preliminary: Review that each scalar function
may be written as a sum of even and odd part.
Decompose linear, planar vector fields
into sum of symm. & skew-symm. part
(geometrically -- hard?, angles!!,
algebraically = link to linear algebra).
(Good place to review the additivity of
((line))integral
drift + symmetric+antisymmetric.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
40
“CURL”: An axis of rotation in 3d
Requires prior development of decomposition symmetric/antisymmetric in planar case.
Addresses additivity of rotation (angular velocity vectors) -- who believes that?
usual nonsense 3d-field
jiggle -- wait, there IS order!
It is a rigid rotation!
Don’t expect to see much if plotting vector field in 3d w/o special (bundle-) structure,
however, plot ANY skew-symmetric linear field (skew-part after zooming 1st kind),
jiggle a little, discover order, rotate until look down a tube, each student different axis
For more MAPLE files (curl in coords etc) see book: “Zooming and limits, ...”, or WWW-site.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
Proposed class outline
41
Assuming multi-variable calculus treatment as in Harvard Consortium Calculus,
with strong emphasis on Rule of Three, contour diagrams, Riemann sums, zooming.
• What is a vector field: Pictures. Applications. Gradfields <-->ODEs.
• Constant vector fields. Work in precalculus setting!.
Nonlinear vfs. (Continuity). Line integrals via zooming of 0th kind.
Conservative <=>circulation integrals vanish <=> gradient fields.
• Linear vector fields. Trace and skew-symmetric-part via line-ints.
Telescoping sum (fluxes over interior surfaces cancel etc….),
grad<=>all circ.int.vanish<=>irrotational (in linear case, no limits)
OPPOSITE: nonintegrable (not exact) <==> “controllable”
• Nonlinear fields: Zoom, differentiability, divergence, rotation, curl.
Stokes’ theorem in all versions via little-oh modification of
arguments in linear settings. Magnetic/gravitat. fields revisited.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
42
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
43
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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
44
Tumbling “soccer balls” in 3D-field
Need to see the animation!
At this time:
User supplies vector field and init
cond’s or uses default example.
MAPLE integrates DEs for position,
calculates curl, integrates angular
momentum equations, and creates animation
using rotation matrices. Colored faces crucial!
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
45
Stokes’ theorem & magnetic field
 
 
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 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, ...
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
46
Animation of the re-orientation
da - F1(q1, q2) dq1 - F2(q1, q2) dq2 = 0
q2
T = 3 Dt
q1
T = 7 Dt
T=0
Three linked rigid bodies. Total angular momentum Great application of Green’s
always zero = conserved. Yet by moving through a theorem. Fun animations. Good
loop is shape-space (q1q2-space) the attitude a may projects. Link to recent research.
be changed! (Satellite w/ antenna, falling cat …)
CAS required for algebra!
.......
T = 9 Dt
T = 11 Dt
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
a
T = 25 Dt
Foundation
Coalition
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
The graph of the rotation of F(q1,q2)
Selecting a suitable loop in shape space
that results in “maximal” attitudinal change
47
Da =   =  d
C
Da = 
C
R
 
 
F  dr =    FdA
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 Green’s 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
Computer Visualization => Rethink Vector Calculus
3rd Int Conf Tech in Math Teaching. Koblenz: October 1997
48
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
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 -- don’t just pop it on the screen.
ACEPT Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski
Foundation
Coalition