foundation coalition vector calculus via linearizations, 5th conf. teaching math, baltimore june...

Foundation Coalition

Vector Calculus via Linearizations, 5th Conf. Teaching Math, Baltimore June 1996

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

1

Vector Calculus via Linearizations

Matthias Kawski Department of Mathematics

Center for Innovation in Engineering EducationArizona State University

Tempe, AZ [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




2

You zoom in calculus I for derivatives / slopes--

Why then don’t you zoom in calculus III for curl, div, and Stokes’ theorem ?

Vector Calculus via Linearizations

• Zooming• Uniform differentiability• Linear Vector Fields• Derivatives of Nonlinear Vector Fields• Stokes’ Theorem

long motivation

side-track, regarding rigor etc.

This work was partially supported by the NSF through Cooperative Agreement EEC-92-21460 (Foundation Coalition) and the grant DUE 94-53610 (ACEPT)




3

The pre-calculator days

The textbook shows a static picture, the teacher thinks of the process,the students think limits mean factoring/canceling rational expressions (and anyhow are convinced tangent lines can only touch at one point)




4

Multi-media ???

With multi-media we can animate the process -- now the “process-idea”of a limit comes across-- but, just adapting new technology to old pictures




5

Calculators have ZOOM button!

New technologies provide new avenues: Each student zooms at a different point, leaves final result on screen, all get up, and …………..WHAT A MEMORABLE EXPERIENCE!(rigorous, and capturing the most important and idea of all!)




6

Zooming in multivariable calculus

Zoom in on a surface -- is the Earth round or flat ???




7

Zooming in on numerical tables

This applies to all: single variable, multi-variable and vector calculus.In this presentation only, emphasize graphical approach and analysis.

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




8

Zooming on contour diagram

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




9

Gradient field: Zooming out of normals!

Pedagogically correct order: Zoom in on contour diagram until linear, assign one normal vector to each magnified picture, then ZOOM OUT , put all small pictures together to BUILD a varying gradient field ……..




10

Naïve zooming on vector field

What we got?? Boring?? Not at all -- this is the key for INTEGRATION!

Be patient! Color will be utilized very soon, too.

Be patient! Color will be utilized very soon, too.




11

Zooming for line-INTEGRALS of vfs

Zooming for INTEGRATION?? -- derivative of curve, integral of field!YES, there are TWO kinds of zooming needed in introductory calculus!




12

Two kinds of zooming

Zooming of the first kind• Magnify domain only• Keep range fixed• Picture for continuity

(local constancy)• Existence of limits of

Riemann sums (integrals)

Zooming of the second 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




13

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!




14

Zooming of 1st 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




15Convergence of R-sumsvia zooming of first kind (continuity)

The zooming of 1st kind picture demonstrate that the limit exists! -- The first partfor the prrof in advanced calculus: (uniform) continuity --> integrability. Key idea: “Further subdivisions will not change the sum” => Cauchy sequence.

The zooming of 1st kind picture demonstrate that the limit exists! -- The first partfor the prrof in advanced calculus: (uniform) continuity --> integrability. Key idea: “Further subdivisions will not change the sum” => Cauchy sequence.

Common pictures demosntarte how areais exhausted in limit.

Common pictures demosntarte how areais exhausted in limit.




16

Zooming of the 2nd kind, calculus I

This is the usual calculator exercise -- this is remembered for whole life!




17

Zooming of the 2nd kind, calculus I

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




18

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!




19

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




20

Zooming of 1st kind in multivar.calc.

Surfaces become flat, contours disappear, tables become constant? Boring? Not at all! Only this allows us to proceed w/ Riemann integrals!




21

for unif. continuity in multivar. calc.

Graphs sandwiched in cages -- exactly as in calc I.Uniformity: Terrific animations of moving cages, fixed width.

21




22Convergence of R-sums in multivar.calc.via zooming of 1st kind (continuity)

Almost the little-oh proof, with uniform-cont. hypothesis also almost the complete proof. -- Remember THIS picture for advanced calc.!




23

Zooming of 2nd kind in multivar.calc.

If surface becomes flat after magnification, call it differentiable at point.Partial derivatives (cross-sections become straight).Gradients (contour diagrams become equidistant parallel straight lines)




24 for unif. differentiability in multivar.calc.

Graphs sandwiched between truncated cones -- as in calc I.New: Analogous pictures for contour diagrams (and gradients)

Animation: Slide this cone (with tilting center plane around)(uniformity)

Animation: Slide this cone (with tilting center plane around)(uniformity)

Advanced calc:Where are and

Advanced calc:Where are and

24

Still need lots of workfinding good examplesgood parameter values




25Zooming of 1st kind in vector calc.

Key application: Convergence of R-sums for line integralsAfter zooming: work=(precalc) (CONSTANT force) dot (displacement)Further magnification will not change sum at all (unif. cont./C.S.)




26

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: Scaling domain / range independently ??? (“Tangent spaces”!!)




27

Zooming of 2nd kind in vector calc.Now it is all obvious!! -- What will we get???

Prep: pictures for pointwise addition (subtraction) of vfs recommended

The originalvector field,colored by div

Same vector fieldafter subtractingconstant part (fromthe point for zooming)

Practically linearPractically linear




28

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, there).




29

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

Odd-ness and homogeneityare much easier to spot thanadditivity

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




30

Analogue(s) of “slope”Want to later geometrically define divergence as limit of flux-integral divided by enclosed volume, curl/rotation as limit of circulation integral divided by enclosed volume

What about the linear case?

This is the PERFECT SETTING to develop these concepts LIMIT-FREE -- in complete analogy with the development of the slope of a straight line in BEFORE calculus!

Note, line-integrals of linear fields over polygonal paths do notrequire any integrals, midpoint/trapezoidal sums are exact! --again in complete analogy with area under a line is PRECALCULUS!




31

Recall: “linear” and slope in precalc

Consider divided differences,

rise over run

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

y y

x x2 1

2 1

y

x




32

Rarely enough: “Linear” in multi-var. calc.

Using tables of function values, or contour diagrams, consider appropriate “divided differences” --> partial deriv.’s, gradient, ...

In each fixed direction, ratios are constant, independent of choice of points, in particular independent w.r.t. parallel translation.

-3 -2 -1 0 1 2 33 -9 -6 -3 0 3 6 92 -6 -4 -2 0 2 4 61 -3 -2 -1 0 1 2 30 0 0 0 0 0 0 0-1 3 2 1 0 -1 -2 -3-2 6 4 2 0 -2 -4 -6-3 9 6 3 0 -3 -6 -9




33As usual, first develop pictorial notion of

circulation and divergence. BEFORE calculations

For linear fields there can be no misunderstanding about local character of divergence or rotation- for linear vfs local and global are the same.(Students looking at magnetic field about wire always falsely agree that it is rotational!)




34

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.




35

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 a 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




36

More formulas in linear setting

E.g. Translation-invariance in linear fields, additivity in integrand, line integrals of constant fields over closed curves vanish (constant fields) -- pictorial arguments for

L Tds L p L Tds L p Tds L Tds L Tds

p C C C C C

( ( ) ) ( ) 0

Develop analogous formulas for flux integrals in 2d and 3d, again relying only on themidpoint rule for straight edges or flat parallelograms.

In order to later get general formulas via triangulation's (?!), replace rectangle firstby right triangles (trivial!), then by general triangles --> compare slide ontelescoping sums, developing the arguments like “fluxes over interior surfaces cancel”.

Warning: To make sense out of div, rot, curl, need to have notion of angle (inner product, dual space, linear pairing,…), i.e. cannot get formulas in purely affine setting. Purely geometric (coordinate-free) proof in triangles are very neat & instructive!




37

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




38

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




39

Nonlinear vector fields, zoom 2nd kind

If after zooming of the second 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)




40

Check for understanding (important)

Zooming of the 2nd kind on a linear object returns the same object!

After zooming of second kind!

originalv-fieldis linear

subtractconstantpart at p




41

Student exercise: LimitFix 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 thecontours shrink to a point.

Do not forget to alsodraw these contoursafter magnification!




42

After zooming of 2nd kind

Subtract constant part, and zoom:A familiar picture occurs: As thefield appears to be closer to linearthe ratios integral divided by areabecome independent of choice ofcontour,the limits appear to make sense!




43

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: differentiable or partials exist??

Recall: Usual definitions of differentiability rely much on joint continuity of partial derivatives of component functions. This isnot geometric, and troublesome: differentiable or 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.




44

| ( )|( )

( ) ( )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




45

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!




46

About little-oh’s & uniform differentiability

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

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

k

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 | < .

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:




47

From 2d to 3dKey: DO IT SLOWLY. Develop the concepts in a planar setting - so you can see them!Key: DO IT SLOWLY. Develop the concepts in a planar setting - so you can see them!

In planar setting develop the notions of line-integrals, linear fields,trace(divergence), rotation, approximation by linear fields, andintegral theorems. After full mastery go to the hard-to-see 3d-case.

In planar setting develop the notions of line-integrals, linear fields,trace(divergence), rotation, approximation by linear fields, andintegral theorems. After full mastery go to the hard-to-see 3d-case.

SPECIAL: The direction of the curl in 3d -- compare next slide!SPECIAL: The direction of the curl in 3d -- compare next slide!

I personally have not yet made up my mind about surface integrals -- I talked toKeith Stroyan, and sympathize with actually playing with Schwarz’ surface(beautiful animations of triangulations --> lighting/shading<=>tilting……)I do not like to start with parameterized surfaces, but instead parameterizable ones….?

I personally have not yet made up my mind about surface integrals -- I talked toKeith Stroyan, and sympathize with actually playing with Schwarz’ surface(beautiful animations of triangulations --> lighting/shading<=>tilting……)I do not like to start with parameterized surfaces, but instead parameterizable ones….?




48

Prep: axis of rotation in 3d

Then split linear, planar vectorfields intosymmetric and anti-symmetric part(geometrically -- hard?, angles!!, algebraically=link to linear algebra).(Good place to review the additivity of ((line))integral

drift + symmetric+antisymmetric.

Then split linear, planar vectorfields intosymmetric and anti-symmetric 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.




49

Axis of rotation in 3dRequires prior development of split symmetric/antisymmetric in planar case.Addresses additivity of rotation (angular velocity vectors) -- who believes that?

Requires prior development of split 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 2nd 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 2nd kind),jiggle a little, discover order, rotate until look down a tube, each student different axis

For more MAPLE files for projections etc. see the ICTCM 96, Reno, or my WWW-site.

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




50Proposed class scheduleAssuming 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 1st kind.Conservative <=>circulation integrals vanish <=> gradient fields.

• Linear vector fields. Trace and skew-symmetric-part via line-ints.Telescoping sum (fluxes over interior surfaces cancel etc….),grad<=>all circ.int.vanish<=>irrotational (in linear case, no limits)

• Nonlinear fields: Zoom, differentiability, divergence, rotation, curl.Stokes’ theorem in all versions via little-oh modification of arguments in linear settings.Magnetic/gravitat. fields revisited, grad=> irrotational (w/ limits)




51

Sneak previewThere are many more fantastic graphical views related to vector calculus.For a sneak-preview, check out the files on my WWW-pages.

For the full show come to the ICTCM 96 in November in Reno, NV.

E.g. coloring by divergence (red/blue) and by rotation (red/green).E.g. corks floating through a velocity field, all the while spinningaccording to the curl, and shrinking/growing (in 2d and in 3d)!E.g. animations of “homotoping” (morphing) a contour/surface in anirrotational/divergence free field, Gauss’ theorem, winding numbers.

and many more!!




52

Animate curl & div, integrate DE (drift)

Color by rot:red=left turngreen=rite turn

divergencecontrols growth




53

Spinning corks in linear, rotating field

Period indep.of radius.

Always same side ofthe moon facing theEarth -- one rotationper full revolution.




54

Spinning corks in magnetic field

Irrotational (black).

Angular velocity drops sharply w/increasing radius.




55

Corks spinning in general rotating field

This example:

Inside spin forward (red), outside spin backwards (green)

divergence alternating




56

Stokes’ theorem & magnetic field

Homotop the blue curve into the magenta circle WITHOUT TOUCHING THE WIRE(beautiful animation -- pink curve sliding down, sweeping out cyan surface……).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?

foundation coalition vector calculus via linearizations, 5th conf. teaching math, baltimore june...

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