Interactive Visualization

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Interactive Visualization

• Matthias Kawski
• Department of Mathematics
• Arizona State University
• Tempe, Arizona U.S.A.

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Thanks for generous support by

• Department of Mathematics
• Center for Research in Education of Science,
• Mathematics, Engineering, and Technology
• Arizona State University
• INTEL Corporation through grant 98-34
• National Science Foundation through the grants
• DUE 97-52453 Vector Calculus via Linearization
• Visualization and
  Modern Applications
• DMS 00-xxxxx Algebra and Geometry of Nonlinear
  Control Systems
• EEC 98-02942 Engineering Foundation Coalition

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Change of talk For complex analysis,
differential geometry, and many others, see
AMS-ScandinavianCongress talkhttp//math.la.asu.
edu/kawski
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A short-course on curl divergenceusing
interactive visualization

• Goals Learn new points of view for a
  classical core topic.Experience visual language
  as powerful organizing principle
  (compare to traditional symbolic/algebraic
  only approach).
• Doing math experiment, make observations,
  conjecture, further test, formulate theorem,
  prove ? definition,.
• Coherence Very few fundamental concepts Build
  rich rooted concept images . . . .
  . . . . and remember them
  for life
  (as opposed to memorize formula for next
  exam only)
• Make connections, avoid fragmentation of
  knowledge
• Enjoy the beauty, have fun, become mesmerized .
  . .

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Vision

• We are at the beginning of a new era in which an
• interactive visual language not only complements,
  
• but often supersedes the traditional, almost
  exclusively
• algebraic-symbolic language which for generations
  
• has often been confused with mathematics itself,
• (and which may be largely responsible for the
  isolation,
• poor public perception, and extremely difficult
  re-entry
• into mathematics due to the imposed vertical
  structure).

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Changing environment

• New opportunities!foremost information
  technology
• New needs, expectations demands for higher
  efficiency/productivity
• Case in point Attitude towards black boxes,
  graphical interfaces and a visual language,
• not just graphing calculators and CAS
• numerical integration of any dynamical system
• e.g. record a macro (EXCEL, Visual
  Basic/C/Java)
• Op-amps (PSPICE, SIMULINK)
• We do not have a choice if we want to keep our
  jobs.

System of differential equations in modern
visual language
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What is our mission? Goal? Objective?

• Keep math alive -- raise next generation of
  mathematicians(React to changing
  demands/needs/environs, but dont betray our
  tradition)
• Applications service to other disciplines/society
  (what are willing to compromise, and what will
  we not compromise?)
• Math as a twin of philosophy, search for
  truthlearn to argue, prove beyond any doubt...
• Math as a science Experiment and discover...

Which of these (and others) require x and y
symbols? When? Which may be (possibly better?)
served via interactive
graphical/visual languages? When?
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Case study The curl divergence
The central object of study in vector calculus. A
horrible formula that few students
remember beyond the next exam.
Traditionally almost exclusive use of algebraic
symbols

• little insight (one-sided, or fragmented,
  concept image)
• major hurdle for re-entry students
• invitation to further study higher math?

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Curl divergence ? derivatives?
?
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Curl Coherence or fragmentation?
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Compartmentalization / Fragmentation !
Complex Analysis
Linear Algebra
Differential Equations
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Coherence DE ? VC ? LA
The visual languageprovides the glue
thatconnects differentaspectsof the
samemathematicalobjects!
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It all started w/ a simple question

• If zooming is so effective for introducing
  derivatives in calculus I . . . .
• why then dont we use zooming in calc III
• for curl, divergence, Stokes theorem ?

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Secant lines ? Zooming ?

• Some of us grew up w/ secant lines and all the
  well-documented misconceptions of tangent lines

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Secant lines ? Zooming ?

• Some of us grew up w/ secant lines and all the
  well-documented misconceptions of tangent
  lines
• Today students zoom on graphing calculators

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Secant lines ? Zooming ?

• Some of us grew up w/ secant lines and all the
  well-documented misconceptions of tangent
  lines
• Today students zoom on graphing calculators

17
Secant lines ? Zooming ?

• Some of us grew up w/ secant lines and all the
  well-documented misconceptions of tangent
  lines
• Today students zoom on graphing calculators

18
Secant lines ? Zooming ?

• Some of us grew up w/ secant lines and all the
  well-documented misconceptions of tangent
  lines
• Today students zoom on graphing calculators

19
Secant lines ? Zooming ?

• Some of us grew up w/ secant lines and all the
  well-documented misconceptions of tangent
  lines
• Today students zoom on graphing calculators

• Better math. interactive, definition,
  applicability, even e and d

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JAVA - Vector field analyzer

• Start the program

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Zooming for continuity

• Magnify the domain ? continuity, R-integrability

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Zooming for continuity/derivatives

• Magnify the domain ? continuity, R-integrability
• Magnify domain range at equal rates ?
  differentiability

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Zoom for derivative of vector field

• Subtract the drift
• (DF) (x , y ) F( x , y ) - F( x0 , y0 )

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Zoom for derivative of vector field

• Subtract the drift
• (DF) (x , y ) F( x , y ) - F( x0 , y0
  )
• 2. Zoom at equal rates in domain and range

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Zoom for derivative of vector field

• Subtract the drift
• (DF) (x , y ) F( x , y ) - F( x0 , y0
  )
• 2. Zoom at equal rates in domain and range

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Zoom for derivative of vector field

• Subtract the drift
• (DF) (x , y ) F( x , y ) - F( x0 , y0
  )
• 2. Zoom at equal rates in domain and range

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Zoom for derivative of vector field

• Subtract the drift
• (DF) (x , y ) F( x , y ) - F( x0 , y0
  )
• 2. Zoom at equal rates in domain and range

Observe rapid convergence to the derivative
(DF)(x,y)
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Derivative of a vector field ???
Differentiability means . . . . .???
What kind of object is the
derivative
(of a vector field)?
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Derivative of a vector field ???
Differentiability means . . . . . .
. . . . approximability by a linear object.
and that linear object is the derivative
at that point
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Derivative of a vector field ???
Differentiability means . . . . . .
. . . . approximability by a linear object.
What kind of object is that L, the
derivative? (today here stay
w/ a calculus level viewpoint)
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Did you do your precalculus
before proceeding to calculus??
Differentiability means . . . . . .
. . . . approximability by a linear
object. Calculus I Before tangent lines and
derivatives, study lines and slopes for a
year.Calculus III Before tangent planes and
gradients, study planes and normal
vectors. Vector Calculus Before curl and
divergence, did you study linear vector
fields?Complex Analysis Before Cauchy Riemann
equns, multiply by complex number) Grad.school
Before convex analysis, study linear functional
analysis for a year. ) T.Needham
amplitwist
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Linear vector fields ???
Do you recognize a linear vector field when you
see one?Why differentiate a vector field? What
is the goal, purpose?
Differentiability means approximability by a
linear object. Calculus I Before tangent lines
and derivatives, study lines and slopes for a
year.Calculus III Before tangent planes and
gradients, study planes and normal
vectors. Vector Calculus Before curl and
divergence, did you study linear vector
fields?Complex Analysis Before Cauchy Riemann
equns, multiply by complex number) Grad.school
Before convex analysis, study linear functional
analysis for a year. ) T.Needham
amplitwist
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Linearity
A key concept in sophomore curriculum
superposition
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 ..
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Decompose linear field
L(x,y) (axby) i (cxdy) j
Recall Decompose scalar function into even and
odd parts.
into symmetric and skew symmetric parts
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JAVA - Vector field analyzer

• Return to the program

36
A short-course on curl divergenceusing
interactive visualization

• Learn new points of view for a classical core
  topic.
• Experience visual language as powerful organizing
  principle (compare to
  traditional symbolic/algebraic only approach).
• Doing math experiment, make observations,
  conjecture, further test, formulate theorem,
  prove ? definition,.
• Coherence Very few fundamental concepts
• Build rich rooted concept images . . . .
  . . . . and remember
  them for life
  (as opposed to memorize formula for
  next exam only
• Make connections, avoid fragmentation of
  knowledge
• Enjoy the beauty, have fun, become mesmerized . .
  .

37
A short-course on curl divergenceusing
interactive visualization
?

• Learn new points of view for a classical core
  topic.
• Experience visual language as powerful organizing
  principle (compare to
  traditional symbolic/algebraic only approach).
• Doing math experiment, make observations,
  conjecture, further test, formulate theorem,
  prove ? definition,.
• Coherence Very few fundamental concepts
• Build rich rooted concept images . . . .
  . . . . and remember
  them for life
  (as opposed to memorize formula for
  next exam only
• Make connections, avoid fragmentation of
  knowledge
• Enjoy the beauty, have fun, become mesmerized . .
  .

38
A short-course on curl divergenceusing
interactive visualization
?

• Learn new points of view for a classical core
  topic.
• Experience visual language as powerful organizing
  principle (compare to
  traditional symbolic/algebraic only approach).
• Doing math experiment, make observations,
  conjecture, further test, formulate theorem,
  prove ? definition,.
• Coherence Very few fundamental concepts
• Build rich rooted concept images . . . .
  . . . . and remember
  them for life
  (as opposed to memorize formula for
  next exam only
• Make connections, avoid fragmentation of
  knowledge
• Enjoy the beauty, have fun, become mesmerized . .
  .

?
39
A short-course on curl divergenceusing
interactive visualization
?

• Learn new points of view for a classical core
  topic.
• Experience visual language as powerful organizing
  principle (compare to
  traditional symbolic/algebraic only approach).
• Doing math experiment, make observations,
  conjecture, further test, formulate theorem,
  prove ? definition,.
• Coherence Very few fundamental concepts
• Build rich rooted concept images . . . .
  . . . . and remember
  them for life
  (as opposed to memorize formula for
  next exam only
• Make connections, avoid fragmentation of
  knowledge
• Enjoy the beauty, have fun, become mesmerized . .
  .

?
?
40
A short-course on curl divergenceusing
interactive visualization
?

• Learn new points of view for a classical core
  topic.
• Experience visual language as powerful organizing
  principle (compare to
  traditional symbolic/algebraic only approach).
• Doing math experiment, make observations,
  conjecture, further test, formulate theorem,
  prove ? definition,.
• Coherence Very few fundamental concepts
• Build rich rooted concept images . . . .
  . . . . and remember
  them for life
  (as opposed to memorize formula for
  next exam only
• Make connections, avoid fragmentation of
  knowledge
• Enjoy the beauty, have fun, become mesmerized . .
  .

?
?
?
41
A short-course on curl divergenceusing
interactive visualization
?

• Learn new points of view for a classical core
  topic.
• Experience visual language as powerful organizing
  principle (compare to
  traditional symbolic/algebraic only approach).
• Doing math experiment, make observations,
  conjecture, further test, formulate theorem,
  prove ? definition,.
• Coherence Very few fundamental concepts
• Build rich rooted concept images . . . .
  . . . . and remember
  them for life
  (as opposed to memorize formula for
  next exam only
• Make connections, avoid fragmentation of
  knowledge
• Enjoy the beauty, have fun, become mesmerized . .
  .

?
?
?
?
42
A short-course on curl divergenceusing
interactive visualization
?

• Learn new points of view for a classical core
  topic.
• Experience visual language as powerful organizing
  principle (compare to
  traditional symbolic/algebraic only approach).
• Doing math experiment, make observations,
  conjecture, further test, formulate theorem,
  prove ? definition,.
• Coherence Very few fundamental concepts
• Build rich rooted concept images . . . .
  . . . . and remember
  them for life
  (as opposed to memorize formula for
  next exam only
• Make connections, avoid fragmentation of
  knowledge
• Enjoy the beauty, have fun, become mesmerized . .
  .

?
?
?
?
?
43
A short-course on curl divergenceusing
interactive visualization
?

• Learn new points of view for a classical core
  topic.
• Experience visual language as powerful organizing
  principle (compare to
  traditional symbolic/algebraic only approach).
• Doing math experiment, make observations,
  conjecture, further test, formulate theorem,
  prove ? definition,.
• Coherence Very few fundamental concepts
• Build rich rooted concept images . . . .
  . . . . and remember
  them for life
  (as opposed to memorize formula for
  next exam only
• Make connections, avoid fragmentation of
  knowledge
• Enjoy the beauty, have fun, become mesmerized . .
  .

?
?
?
?
?
?
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Further information

• Almost all my work, and links to related
  sites,is available on-line
• http//math.la.asu.edu/kawski,
• else send e-mail kawski_at_asu.edu
• JAVA vector field analyzer (work on-line, or
  download all)JAVA 2 update, workbook, ..
  coming soon
• PowerPoint presentations from most past
  conferences
• Also on-line All publications, all classes
  (WritingProofs, BusinessCalc, Calc I,II,III,
  ODEs, LinAlg, VectCalc, PDEs, EnginMath, Complex,
  DiffGeom, AdvMathViaTech,), and extensive
  MAPLE, MATLAB depositories . . . .