Vector Calculus via Linearizations

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Vector Calculus via Linearizations
Matthias Kawski Department of Mathematics Center
for Innovation in Engineering Education Arizona
State University Tempe, AZ 85287 kawski_at_math.la.as
u.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
VISUAL CALCULUS (to come soon, MAPLE, JAVA,
VRML)
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Vector Calculus via Linearizations

• You zoom in calculus I for derivatives / slopes
• --
• Why then dont you zoom in calculus III
• for curl, div, and Stokes theorem ?

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

long motivation
side-track, regarding rigor etc.
This work was partially supported by the NSF
through Cooperative Agreement EEC-92-21460
(Foundation Coalition) and the grant DUE 94-53610
(ACEPT)
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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!
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!)
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Zooming in multivariable calculus

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

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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.
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Zooming on contour diagram
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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Naïve zooming on vector field
Be patient! Color will be utilized very soon,
too.
What we got?? Boring?? Not at all -- this is the
key for INTEGRATION! This picture is key to
convergence of Eulers method for integrating DEs
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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!
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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 domainand range
• Picture for differentiability(local linearity)
• Need to ignore (subtract) constant part --
  picture can not show total magnitude!!!

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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!
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Zooming of 0th kind in calculus I
Continuity via zooming Zoom in domain only
Tickmarks show dgt0. Fixed vertical window size
controlled by egt0
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Convergence of R-sumsvia zooming of zeroth kind
(continuity)
Common pictures demosntarte how area is
exhausted in limit.
The zooming of 0th kind picture demonstrate that
the limit exists! -- The first part for the proof
in advanced calculus (Uniform) continuity gt
integrability. Key idea Further subdivisions
will not change the sum gt Cauchy sequence.
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Zooming of the 1st kind, calculus I
This is the usual calculator exercise -- this is
remembered for whole life!
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Zooming of the 2nd kind, calculus I
Zooming at quadratic ratios (in range /domain)
exhibits local quadratic-ness near
nondegenerate extrema. Even more impressive for
surfaces!
Also Zooming out of n-th kind e.g. to find
power of polynomial, establish nonpol charater of
exp.
Pure meanness Instead of find the min-value,
ask for find the x-coordinate (to 12 decimal
places) of the min.
Why cant one answer this by standard zooming
on a calcuator? Answer The first derivative test!
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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.
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Uniform continuity, pictorially
A short side-excursion, re rigor in proof of
Stokes thm.
Demonstration Slide tubings of various radii
over bent-wire!
Many have argued that uniform continuity belongs
into freshmen calc. Practically all proofs
require it, who cares about continuity at a
point? Now we have the graphical tools -- it is
so natural, LET US DO IT!!
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Compare e.g. books by Keith Stroyan
A short side-excursion, re rigor in proof of
Stokes thm.
Uniform differentiability, pictorially
Demonstration Slide cones of various opening
angles over bent-wire!
With the hypothesis of uniform differentiability
much less trouble with order of quantifiers in
any proof of any fundamental/Stokes
theorem. Naïve proof ideas easily go thru, no
need for awkward MeanValueThm
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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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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.
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Convergence of R-sums in multivar.calc.via
zooming of 0th kind (continuity)
Almost the little-oh proof, with uniform-cont.
hypothesis also almost the complete e-d proof.
-- Remember THIS picture for advanced calc.!
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Zooming of 1st kind in multivar.calc.
If surface becomes planar (lienar) after
magnification, call it differentiable at
point. Partial derivatives (cross-sections become
straight -- compare T.Dick calculators) Gradient
s (contour diagrams become equidistant parallel
straight lines)
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e-d for unif. differentiability in multivar.calc.
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Animation Slide this cone (with tilting center
plane around) (uniformity)
Advanced calc Where are e and d ?
Still need lots of work finding good
examples good parameter values
Graphs sandwiched between truncated cones -- as
in calc I. New Analogous pictures for contour
diagrams (and gradients)
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Zooming of 0th kind in vector calc
Key application Convergence of R-sums for line
integrals After zooming work(precalc) (CONSTANT
force) dot (displacement) Further magnification
will not change sum at all (unif. cont./C.S.)
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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!!)
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Zooming of 1st kind in vector calc.
Now it is all obvious!! -- What will we get???
The original vector field, colored by div
Same vector field after subtracting constant part
(from the point for zooming)
Practically linear
Prep pictures for pointwise addition
(subtraction) of vfs recommended
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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? ---gt 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?
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Linearity ???
Definition A map/function/operator L X -gt Y is
linear if L(cP)c L(p) and
L(pq)L(p)L(q) for all ..
Can your students show where to find
L(p),L(pq). in the picture?
y/4,(2abs(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, ...
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Analogue(s) of slope
Want to later geometrically define divergence as
limit of flux-integral divided by enclosed
volume, curl/rotation as limit of circulation
integral divided by enclosed volume
What about the linear case? This is the PERFECT
SETTING to develop these concepts LIMIT-FREE --
in complete analogy with the development of the
slope of a straight line BEFORE calculus! Note,
line-integrals of linear fields over polygonal
paths do not require any integrals, ---
midpoint/trapezoidal SUMS are exact! -- in
complete analogy with area under a line in
PRECALCULUS!
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Recall linear and slope in precalc
Consider divided differences, rise over
run Linear ltgt ratio is CONSTANT, INDEPENDENT
of the choice of points (xk,yk )
Dy
Dx
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Rarely enough Linear in multi-var. calc.

• Using tables of function values, or contour
  diagrams, consider appropriate divided
  differences --gt partial deriv.s, gradient, ...
• In each fixed direction, ratios are constant,
  independent of choice of points, in particular
  independent w.r.t. parallel translation.

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As usual, first develop pictorial notion of
circulation and divergence. BEFORE calculations
For NONlinear fields pictorially the local
character of divergence and rotation is obvious
-- for LINEAR vfs local and global are the
same. (Students looking at magnetic field about
wire always falsely agree that it is rotational!)
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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 --gt easy suggestion to normalize by
area--gt what a surprise, independence of shape
and location! just like slope.
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Algebraic formulas tr(L), (L-LT)/2
Develop understanding where (ad), (c-b) etc come
from in limit free setting first
(x0,y0Dy)
for L(x,y) (axby,cxdy), using only midpoint
rule (exact!) and linearity for e.g. circulation
integral over rectangle
(x0,y0)
(x0Dx,y0)
(x0-Dx, y0)
(x0,y0 -Dy)
Coordinate-free GEOMETRIC arguments w/ triangles,
simplices in 3D are even nicer
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More formulas in linear setting
Develop analogous formulas for flux integrals in
2d and 3d, again relying only on the midpoint
rule for straight edges or flat
parallelograms. In order to later get general
formulas via triangulation's (?!), replace
rectangle firstby right triangles (trivial!),
then by general triangles --gt compare next slide
on telescoping sums, developing the arguments
like fluxes over interior surfaces
cancel. Warning To make sense out of div, rot,
curl, need to have a notion of angle (inner
product), i.e. cannot get formulas in purely
affine setting. Purely geometric
(coordinate-free) proof in triangles are very
neat instructive!
E.g. Translation-invariance in linear fields,
additivity in integrand, line integrals of
constant fields over closed curves vanish
(constant fields) -- pictorial arguments for
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Telescoping sums
Want Stokes theorem for linear fields FIRST!
Recall For linear functions, the fundamental
theorem is exact without limits, it is just a
telescoping sum!
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Telescoping sums for linear Greens thm.
This extends formulas from line-integrals over
rectangles / triangles first to general polygonal
curves (no limits yet!), then to smooth curves.
Caution, when arguing with triangulations of
smooth surfaces
The picture new TELESCOPING SUMS matters
(cancellations!)
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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.
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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!
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Student exercise Limit
Instead of ZOOMING, this perspective lets
thecontours shrink to a point. Do not forget
to also draw these contours after magnification!
Fix a nonlin field, a few base points,a set of
contours, different students set up
evaluate line integrals over their contour at
their point, and let the contour shrink. Report
all results to transparency in the front. Scale
by area, SEE convergence.
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After zooming of 1st kind
Subtract constant part, and zoom A familiar
picture occurs As the field appears to be closer
to linear the ratios integral divided by
area become independent of choice of contour, the
limits appear to make sense!
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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 diffable 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 gt 0 there exists a d gt 0
(indep.of p) such that F(q) - F(p) - Lp(q-p)
lt 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.
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Divergence, rotation, curl
Intuitively define the divergence of F at p to be
the trace of L, where L is the linear field to
which the zooming at p converges (!!).
For a linear field we defined (and showed
independence of everything)
For a differentiable field define (where
contour shrinks to the point p, circumference
--gt0 )
Use your judgment worrying about independence of
the contour here.
Consequence
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Proof of Stokes theorem, nonlinear
In complete analogy to the proof of the
fundamental theorem in calc I telescoping sums
limits (uniform differentiability, or MVTh, or
handwaving.).
Here the hand-waving version The critical steps
use the linear result, and the observation
that on each small region the vector field is
practically linear.
It straightforward to put in little-ohs, 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-ohs uniform differentiability
By hypothesis, for every p there exist a linear
field Lp such that for every e gt 0 there is a d
gt 0 (independent of p (!)) such that F(q) -
F(p) - Lp(q - p) lt e q - p for all q such
that q - p lt d.
The errors in the two approximate equalities in
the nonlinear telescoping sum
Key Stay away from pathological,
arbitrary large surfaces bounding arbitrary small
volumes,
Except for small number (lower order)of outside
regions, hypothesize a regular subdivision, i.e.
without pathological relations between diameter,
circumference/surface area, volume!
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From 2d to 3d
Key DO IT SLOWLY. Develop the concepts in a
planar setting - so you can see them!
In planar setting develop the notions of
line-integrals, linear fields, trace(divergence),
rotation, approximation by linear fields,
and integral theorems. After full mastery go to
the hard-to-see 3d-case.
SPECIAL The direction of the curl in 3d --
compare next slide!
I personally have not yet made up my mind about
surface integrals -- I talked to Keith Stroyan,
and sympathize with actually playing with
Schwarz surface (beautiful animations of
triangulations --gt lighting/shadingltgttilting) I
do not like to start with parameterized
surfaces, but instead parameterizable ones.?
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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!
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Prep axis of rotation in 3d
Preliminary Review that each scalar function
may be written as a sum of even and odd part.
Decompose linear, planar vectorfields into sum
of symm. skew-symm. part (geometrically --
hard?, angles!!, algebraicallylink to linear
algebra). (Good place to review the additivity of
((line))integral drift symmetricantisymmetric
.
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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!
Dont 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 Visual calculus, or WWW-site.
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Proposed class outline
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 lt--gtODEs.
• Constant vector fields. Work in precalculus
  setting!.Nonlinear vfs. (Continuity). Line
  integrals via zooming of 0th kind.Conservative
  ltgtcirculation integrals vanish ltgt gradient
  fields.
• Linear vector fields. Trace and
  skew-symmetric-part via line-ints.Telescoping
  sum (fluxes over interior surfaces cancel
  etc.),gradltgtall circ.int.vanishltgtirrotational
  (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, gradgt irrotational (w/ limits)

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Animate curl div, integrate DE (drift)
Color by rot redleft turn greenrite
turn divergence controls growth
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Spinning corks in linear, rotating field
Period indep.of radius compare harmonic
oscillator - pend clock Always same side of the
moon facing the Earth -- one rotation per full
revolution.
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Spinning corks in magnetic field
Irrotational (black). 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
conds or uses default example. MAPLE
integrates DEs fro position, calculates curl,
integrates angular momentum equations, and
creates animation using rotation matrices.
Colored faces crucial!
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Stokes theorem magnetic field
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
Jacobs ladder). 3Dviews, jiggling necessary to
obtain understanding how curve sits relative to
wire.More impressive curve formed from torus
knots with arbitrary winding numbers, ...