Interactive Visualization

Interactive Visualization Matthias Kawski Department of Mathematics Arizona State University Tempe, Arizona U.S.A. http://math.la.asu.edu/~kawskikawski@asu.edu

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 http://math.la.asu.edu/~kawskikawski@asu.edu

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

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 . . . http://math.la.asu.edu/~kawskikawski@asu.edu

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). http://math.la.asu.edu/~kawskikawski@asu.edu

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… http://math.la.asu.edu/~kawskikawski@asu.edu

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? http://math.la.asu.edu/~kawskikawski@asu.edu

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? http://math.la.asu.edu/~kawskikawski@asu.edu

Curl & divergence  derivatives? ? http://math.la.asu.edu/~kawskikawski@asu.edu

Curl: Coherence or fragmentation? http://math.la.asu.edu/~kawskikawski@asu.edu

Compartmentalization /Fragmentation ! Complex Analysis Linear Algebra Differential Equations http://math.la.asu.edu/~kawskikawski@asu.edu

Coherence: DE  VC  LA The visual languageprovides the glue thatconnects different“aspects”of the samemathematicalobjects! http://math.la.asu.edu/~kawskikawski@asu.edu

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 ?” http://math.la.asu.edu/~kawskikawski@asu.edu

Secant lines  Zooming ? • Some of us grew up w/ secant lines and all the well-documented misconceptions of tangent lines http://math.la.asu.edu/~kawskikawski@asu.edu

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 http://math.la.asu.edu/~kawskikawski@asu.edu

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 http://math.la.asu.edu/~kawskikawski@asu.edu

JAVA - Vector field analyzer • Start the program http://math.la.asu.edu/~kawskikawski@asu.edu

Zooming for continuity • Magnify the domain  continuity, R-integrability http://math.la.asu.edu/~kawskikawski@asu.edu

Zooming for continuity/derivatives • Magnify the domain  continuity, R-integrability • Magnify domain & range at equal rates differentiability http://math.la.asu.edu/~kawskikawski@asu.edu

Zoom for derivative of vector field • Subtract the “drift”: • (DF) (x , y ) = F( x , y ) - F( x0 , y0 ) http://math.la.asu.edu/~kawskikawski@asu.edu

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 http://math.la.asu.edu/~kawskikawski@asu.edu

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) http://math.la.asu.edu/~kawskikawski@asu.edu

Derivative of a vector field ??? Differentiability means . . . . .??? What kind of object is the derivative(of a vector field)? http://math.la.asu.edu/~kawskikawski@asu.edu

Derivative of a vector field ??? Differentiability means . . . . . . . . . . approximability by a linear object. and “that linear object” is the derivative at that point … http://math.la.asu.edu/~kawskikawski@asu.edu

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…) http://math.la.asu.edu/~kawskikawski@asu.edu

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” http://math.la.asu.edu/~kawskikawski@asu.edu

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” http://math.la.asu.edu/~kawskikawski@asu.edu

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 ….. http://math.la.asu.edu/~kawskikawski@asu.edu

Decompose linear field L(x,y) = (ax+by) i + (cx+dy) j Recall: Decompose scalar function into even and odd parts. into symmetric and skew symmetric parts http://math.la.asu.edu/~kawskikawski@asu.edu

JAVA - Vector field analyzer • Return to the program http://math.la.asu.edu/~kawskikawski@asu.edu

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 . . . http://math.la.asu.edu/~kawskikawski@asu.edu

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 . . . http://math.la.asu.edu/~kawskikawski@asu.edu

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 . . .  http://math.la.asu.edu/~kawskikawski@asu.edu

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 . . .   http://math.la.asu.edu/~kawskikawski@asu.edu

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 . . .    http://math.la.asu.edu/~kawskikawski@asu.edu

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 . . .     http://math.la.asu.edu/~kawskikawski@asu.edu

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 . . .      http://math.la.asu.edu/~kawskikawski@asu.edu

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 . . .       http://math.la.asu.edu/~kawskikawski@asu.edu

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@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 . . . . http://math.la.asu.edu/~kawskikawski@asu.edu

Interactive Visualization