Foundation Coalition Computer Visualization Rethink Vector Calculus 3rd Int Conf Tech in Math...

Foundation Coalition

Computer Visualization Rethink Vector Calculus 3rd Int Conf Tech in Math Teaching. Koblenz: October 1997

Matthias Kawski, AZ State Univ. http://math.la.asu.edu/~kawski

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How CAS and Visualization lead to a complete rethinking of an intro to vector calculus

Matthias Kawski Department of Mathematics

Arizona State [email protected]

http://math.la.asu.edu/~kawski

Lots of MAPLE worksheets (in all degrees of rawness), plus plenty of other class-materials: Daily instructions, tests, extended projects

“VISUAL CALCULUS” (to come soon, MAPLE, JAVA, VRML)“VISUAL CALCULUS” (to come soon, MAPLE, JAVA, VRML)This work was partially supported by the NSF through Cooperative Agreement EEC-92-21460 (Foundation Coalition) and the grant DUE 94-53610 (ACEPT)




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You zoom in calculus I for derivatives / slopes

--Why then don’t you zoom in calculus III

for curl, div, and Stokes’ theorem ?

How CAS and Visualization lead to a complete rethinking of an intro to vector calculus

• Zooming• Uniform differentiability• Linear Vector Fields• Derivatives of Nonlinear Vector Fields• Animating curl and divergence• Stokes’ Theorem via linearizations• Controllability versus conservative fields / potentials

review: distinguish different kinds of zooming

side-track, regarding rigor etc.




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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.




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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???




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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 containinfo about and




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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.

1.97 1.98 1.99 2.00 2.01 2.02 2.031.03 2.1909 2.2304 2.2701 2.3100 2.3501 2.3904 2.43091.02 2.4409 2.4804 2.5201 2.5600 2.6001 2.6404 2.68091.01 2.6709 2.7104 2.7501 2.7900 2.8301 2.8704 2.91091.00 2.8809 2.9204 2.9601 3.0000 3.0401 3.0804 3.12090.99 3.0709 3.1104 3.1501 3.1900 3.2301 3.2704 3.31090.98 3.2409 3.2804 3.3201 3.3600 3.4001 3.4404 3.48090.97 3.3909 3.4304 3.4701 3.5100 3.5501 3.5904 3.6309

-3 -2 -1 0 1 2 3 1.7 1.8 1.9 2.0 2.1 2.2 2.33 0 -5 -8 -9 -8 -5 0 1.3 1.20 1.55 1.92 2.31 2.72 3.15 3.602 5 0 -3 -4 -3 0 5 1.2 1.45 1.80 2.17 2.56 2.97 3.40 3.851 8 3 0 -1 0 3 8 1.1 1.68 2.03 2.40 2.79 3.20 3.63 4.080 9 4 1 0 1 4 9 1.0 1.89 2.24 2.61 3.00 3.41 3.84 4.29-1 8 3 0 -1 0 3 8 0.9 2.08 2.43 2.80 3.19 3.60 4.03 4.48-2 5 0 -3 -4 -3 0 5 0.8 2.25 2.60 2.97 3.36 3.77 4.20 4.65-3 0 -5 -8 -9 -8 -5 0 0.7 2.40 2.75 3.12 3.51 3.92 4.35 4.80




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Zooming on contour diagrams

Easier than 3D. -- Important: recognize contour diagrams of planes!!




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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 ……..




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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!

Without the blue curve this isthe pictorial foundation forthe convergence of Euler’s andrelated methods for numericallyintegrating diff. equations




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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 vfsIt is extremely simple, just consistently apply rules all the way to vfs




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The usual boxes for continuity

This is EXACTLY the characterization of continuity at a point, butwithout these symbols. CAUTION: All usual fallacies of confusion oforder of quantifiers still apply -- but are now closer to common sense!




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Zooming of 0th kind in calculus IContinuity 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




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Convergence of R-sumsvia zooming of zeroth kind (continuity)

The zooming of 0th kind picture demonstrate that the limit exists! -- The first partfor the proof in advanced calculus: (Uniform) continuity => integrability. Key idea: “Further subdivisions will not change the sum” => Cauchy sequence.

The zooming of 0th kind picture demonstrate that the limit exists! -- The first partfor the proof in advanced calculus: (Uniform) continuity => integrability. Key idea: “Further subdivisions will not change the sum” => Cauchy sequence.

Common pictures demonstrate how areais exhausted in limit.

Common pictures demonstrate how areais exhausted in limit.




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Zooming of the 2nd kind, calculus IZooming 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 calculator?Answer: The first derivative test!

Why can’t one answer this by standard zooming on a calculator?Answer: The first derivative test!

Also: Zooming out of “n-th” kinde.g. to find power of polynomial,establish nonpol character of exp.




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Zooming of the 1st kind, calculus I

Slightly more advanced, characterization of differentiability at point.Useful for error-estimates in approximations, mental picture for proofs.




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Uniform continuity, pictoriallyA 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!




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Uniform differentiability, pictoriallyA short side-excursion, re rigor in proof of Stokes’ thm.

With the hypothesis of uniform differentiability much less trouble withorder 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




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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!




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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.

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Zooming of 1st kind in multivar.calc.

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)




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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

Still need lots of workfinding good examplesgood parameter values




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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”!!)




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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).




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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 homogeneityare much easier to spot thanadditivity

[y/4,(2*abs(x)-x)/9]




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What is the analogue of “slope” for vector fields?First recall: “linear” and slope in precalc

Consider divided differences,

rise over run

Linear <=> ratio is CONSTANT,INDEPENDENT of thechoice of points (xk,yk )

y y

x x2 1

2 1

y

x




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Constant ratios for linear fields

Work with polygonal paths in linear fields, each student has a differentbasepoint, a different shape, each student calculates the flux/circulation line integral w/o calculus (midpoint/trapezoidal sums!!), (and e.g. viamachine 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.




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L Tds L x y y i x L x x y j y

L x y y i x L x x y j y

c b x y

R

( , ) ( , )

( , ) ( , )

..( )... ( )

0 0 0 0

0 0 0 0

only using linearity

Algebraic formulas: tr(L), (L-LT)/2

(x0,y0)

(x0,y0 -y)

(x0,y0+y)

(x0+x,y0)(x0-x, y0)

for L(x,y) = (ax+by,cx+dy), using only midpoint rule (exact!) and linearity for e.g. circulation integral over rectangle

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 firstDevelop understanding where (a+d), (c-b) etc come from in limit free setting first




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Telescoping sumsRecall: For linear functions, the fundamental theorem is exact without limits, it is just a telescoping sum!

Want: Stokes’ theorem for linear fields FIRST!

F b F a

F x F x

F x F x

x xx

F x dx

k k

k k

k k

a

b

( ) ( )

( ( ) ( ))

( ) ( )

( )

1

1

1




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Telescoping sums for linear Greens’ thm.This extends formulas from line-integrals over rectangles / trianglesfirst to general polygonal curves (no limits yet!), then to smooth curves.

L Nds

L Nds

trL A

trL A

trL A

C

Ck

k k

kk

k

The picture new TELESCOPING SUMS matters (cancellations!)

Caution, when arguing withtriangulations of smooth surfaces




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Nonlinear vector fields, zoom 1st 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/curlto be the trace/skew symmetric part of the linear field we see after zooming.

The originalvector field,colored by rot

Same vector fieldafter subtractingconstant part (fromthe point for zooming)




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Check for understanding (important)

Zooming of the 1st kind on a linear object returns the same object!

After zooming of first kind!

originalv-fieldis linear

subtractconstantpart at p




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Student exercise: Limit

Fix a nonlin field, a few base points,a set of contours,different studentsset up & evaluateline integrals overtheir contour at theirpoint, and let thecontour shrink.

Report all results totransparency in thefront. Scale by area,SEE convergence.

Instead of ZOOMING,this perspective lets the contours shrink to a point.

Do not forget to alsodraw these contoursafter magnification!




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Integrals & continuity

Derivatives

Show all

Symm part only

Anti-symm part

dxdy-magnification factor

xy-magnification factor

1 10 100 1000 infinity

An interactive JAVA microscope to zoom for derivatives of vector fieldsrealized by Shannon Holland, ASU. http://www.eas.asu.edu/~asufc/microscopeFinal version to be presented at the ICTCM, Chicago in November 1997




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Rigor in the defn: DifferentiabilityRecall: Usual definitions of differentiability rely much on joint continuity of partial derivatives of component functions. This isnot 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 isnot geometric, and troublesome: diff’able not same as “partials exist”

Better: Do it like in graduate school -- the zooming picture is right!Better: Do it like in graduate school -- the zooming picture is right!

A function/map/operator F between linear spaces X and Z is uniformlydifferentiable on a set K if for every p in K there exists a linear mapL = Lp such that for every > 0 there exists a > 0 (indep.of p) such that| F(q) - F(p) - Lp(q-p) | < | q - p | (or analogous pointwise definition).

A function/map/operator F between linear spaces X and Z is uniformlydifferentiable on a set K if for every p in K there exists a linear mapL = Lp such that for every > 0 there exists a > 0 (indep.of p) such that| F(q) - F(p) - Lp(q-p) | < | 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.




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| ( )|( )

( ) ( )div F Larea

diam circumference 1

4

Divergence, rotation, curl

For a differentiable fielddefine (where contourshrinks 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 independenceof everything):

tr LL Nds

Nds

L Nds

areaC

C

C( )( )

div F pF Nds

areaC( )( ) lim( )

Use your judgment worrying about independence of the contour here….

Use your judgment worrying about independence of the contour here….

Consequence:

( ) /x y2 2 4

( ) /x y2 2 4




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Proof of Stokes’ theorem, nonlinearIn complete analogy to the proofof the fundamental theorem incalc I: telescoping sums + limits(+uniform differentiability, orMVTh, or handwaving….).

In complete analogy to the proofof the fundamental theorem incalc I: telescoping sums + limits(+uniform differentiability, orMVTh, or handwaving….).

F Nds

F Nds

trF p A

div F dA

div F dA

C

Ck

k k k

Rk

R

k

k

( ) )

( )

( )

Here the hand-waving version:The critical steps use the linearresult, and the observation thaton each small region the vectorfield 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!




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About little-oh’s & uniform differentiability

| ( ) | ( )div F L dV vol VpV kk

| ( ) | ( ) ( ) F L NdS diam V area SpS k k

k

Key: Stay away from pathological, arbitrarylarge surfaces boundingarbitrary small volumes,

Key: Stay away from pathological, arbitrarylarge surfaces boundingarbitrary 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:

By hypothesis, for every p there exist a linear field Lp such that for every > 0 there is a > 0 (independent of p (!)) such that | F(q) - F(p) - Lp(q - p) | < | q - p | for all q such that | q - p | < .

By hypothesis, for every p there exist a linear field Lp such that for every > 0 there is a > 0 (independent of p (!)) such that | F(q) - F(p) - Lp(q - p) | < | q - p | for all q such that | q - p | < .




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Trouble w/ surface integrals: “Schwarz’ surface”

Pictorially the troubleis obvious. SHADING!

Simple fun limit for proof

Not at all unreasonablein 1st multi-var calculus

Entertaining. Warningabout limitations ofintuitive arguments, …yet it is easy to fix!




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Decompositions

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.

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.

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.




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“CURL”: An axis of rotation in 3dRequires 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 1st 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 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.

usual nonsense 3d-field jiggle -- wait, there IS order! It is a rigid rotation!




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Proposed class outlineAssuming 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) 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.




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Animate curl & div, integrate DE (drift)

Color by rot:red=left turngreen=rite turn

divergencecontrols growth




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Spinning corks in linear / magnetic field

Period indep.of radius compare harmonic oscillator - pendulum clockAlways same side of the moon faces the Earth -- one rotation per full revolution.

Irrotational (black = no color). Angular velocity drops sharply w/increasing radius.




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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 angularmomentum equations, and creates animationusing rotation matrices. Colored faces crucial!




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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, ...

F Tds F Tds

F NdS

C C

S

1 2

2 0

Do your students have a mental picture of the objects in the equn?




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Animation of the re-orientation

T = 0

T = 3 t T = 7 t

T = 25 tT = 9 t

T = 11 t

. . . . . . .

d - F1(1, 2) d1 - F2(1, 2) d2 = 0

Three linked rigid bodies. Total angular momentumalways zero = conserved. Yet by moving through a loop is shape-space (12-space) the attitude may be changed! (Satellite w/ antenna, falling cat …)

Great application of Green’stheorem. Fun animations. Goodprojects. Link to recent research.

CAS required for algebra!

12




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The graph of the rotation of F(1,2)

Selecting a suitable loop in shape space that results in “maximal” attitudinal change

Note the very sharp peaks and pits =-=> key to make this a great project. Randomlychosen curves lead to unpredictable attitude changes. Understanding of Green’s thm <==> systematic choice of loops in shape space to achieve desired attitude change

dRC

F dr FdA

RC




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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

Contrast this with conservative == integrable fields. There (as here) DYNAMICALLYGROW the potential surface using many lifted LOOPS -- don’t just pop it on the screen.

CRITICAL: Dynamically animate the loop and the lifted curve. Contrast with potential surface for conservative fields.

Foundation Coalition Computer Visualization Rethink Vector Calculus 3rd Int Conf Tech in Math...

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