interactive visualization

http://math.la.asu.edu/~kawski [email protected]

Matthias Kawski Interactive Visualization 5th atcm Chiang Mai, Thailand. December 2000

Interactive Visualization

Matthias KawskiDepartment of Mathematics

Arizona State UniversityTempe, Arizona U.S.A.

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 ApplicationsDMS 00-xxxxx Algebra and Geometry of Nonlinear Control Systems

EEC 98-02942 Engineering Foundation Coalition

Change of talk:For complex analysis, differential geometry, and many others, see AMS-ScandinavianCongress talkhttp://math.la.asu.edu/~kawski

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 . . .

VisionWe are at the beginning of a new era in which an interactive visual language not only complements,

but often supersedes the traditional, almost exclusivelyalgebraic-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).

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…

What is our mission? Goal? Objective?

• Keep math alive -- raise next generation of mathematicians(React to changing demands/needs/environ’s, but don’t 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?

Case study: The curl & divergenceThe 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?

Curl & divergence derivatives?

Curl: Coherence or fragmentation?

Compartmentalization / Fragmentation !

Linear Algebra

Complex Analysis

Differential Equations

Coherence: DE VC LA

The visual languageprovides the glue thatconnects different“aspects”of the samemathematicalobjects!

It all started w/ a simple question:

“If zooming is so effective for introducing derivatives in calculus I . . . .

why then don’t we use zooming in calc III for curl, divergence, & Stokes’ theorem ?”

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 and

JAVA - Vector field analyzer

• Start the program

Zooming for continuity

• Magnify the domain continuity, R-integrability

Zooming for continuity/derivatives

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

Zoom for derivative of vector field

1. Subtract the “drift”: (F) (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)

Derivative of a vector field ???

Differentiability means . . . . .???

What kind of object is the derivative (of a vector field)?

Differentiability means . . . . . . . . . . approximability by a linear object.

and “that linear object” is the derivative at that point …

What kind of object is that “L”, the derivative? (today & here stay w/ a calculus level viewpoint…)

Did you do your precalculus before proceeding to calculus??

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”

Linear vector fields ???

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”

Do you recognize a linear vector field when you see one?Why differentiate a vector field? What is the goal, purpose?

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 …..

Decompose linear field

Recall: Decompose scalar function into even and odd parts.

into symmetric and skew symmetric parts

L(x,y) = (ax+by) i + (cx+dy) j

JAVA - Vector field analyzer

• Return to the program

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 conceptsBuild rich “rooted” concept images . . . .

. . . . and remember them for life (as opposed to: memorize formula for next exam only

Make connections, avoid fragmentation of knowledgeEnjoy the beauty, have fun, become mesmerized . . .

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: [email protected]

• 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 . . . .

interactive visualization

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