Matthias Kawski Interactive Visualization 5 th atcm Chiang

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Interactive Visualization Matthias Kawski Department of Mathematics Arizona State University Tempe, Arizona U. S. A. http: //math. la. asu. edu/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 http: //math. la. asu. edu/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 A short-course on curl & divergence using 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Changing environment • New opportunities! foremost: information technology System of differential equations in modern “visual” language… • 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…. http: //math. la. asu. edu/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 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 truth learn 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Curl & divergence derivatives? ? http: //math. la. asu. edu/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Curl: Coherence or fragmentation? http: //math. la. asu. edu/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Compartmentalization / Fragmentation ! Complex Analysis Linear Algebra Differential Equations http: //math. la. asu. edu/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Coherence: DE VC LA The visual language provides the glue that connects different “aspects” of the same mathematical objects! http: //math. la. asu. edu/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 JAVA - Vector field analyzer • Start the program http: //math. la. asu. edu/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Zooming for continuity • Magnify the domain continuity, R-integrability http: //math. la. asu. edu/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Zooming for continuity/derivatives • Magnify the domain continuity, R-integrability • Magnify domain & range at equal rates differentiability http: //math. la. asu. edu/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Zoom for derivative of vector field 1. Subtract the “drift”: (DF) (x , y ) = F( x , y ) - F( x 0 , y 0 )

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Zoom for derivative of vector field 1. Subtract the “drift”: (DF) (x , y ) = F( x , y ) - F( x 0 , y 0 ) 2. Zoom at equal rates in domain and range

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Zoom for derivative of vector field 1. Subtract the “drift”: (DF) (x , y ) = F( x , y ) - F( x 0 , y 0 ) 2. Zoom at equal rates in domain and range

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Zoom for derivative of vector field 1. Subtract the “drift”: (DF) (x , y ) = F( x , y ) - F( x 0 , y 0 ) 2. Zoom at equal rates in domain and range

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Zoom for derivative of vector field 1. Subtract the “drift”: (DF) (x , y ) = F( x , y ) - F( x 0 , y 0 ) 2. Zoom at equal rates in domain and range Observe rapid convergence to the derivative (DF)(x, y)

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Derivative of a vector field ? ? ? Differentiability means. . . ? ? ? What kind of object is the derivative (of a vector field)?

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Derivative of a vector field ? ? ? Differentiability means. . approximability by a linear object. and “that linear object” is the derivative at that point …

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

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

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

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Linearity A key concept in sophomore curriculum – “superposition” Definition: A map/function/operator L: X Y is linear if L( c. P ) = c L(p) and L( p + q ) = L(p) + L(q) for all …. .

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 Decompose linear field L(x, y) = (ax+by) i + (cx+dy) j into symmetric Recall: Decompose scalar function into even and odd parts. and skew symmetric parts

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 JAVA - Vector field analyzer • Return to the program http: //math. la. asu. edu/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 A short-course on curl & divergence using interactive visualization o o o o 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 A short-course on curl & divergence using interactive visualization o o o o 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 A short-course on curl & divergence using interactive visualization o Learn new points of view for a classical core topic. o Experience visual language as powerful organizing principle o o o (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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 A short-course on curl & divergence using interactive visualization o Learn new points of view for a classical core topic. o Experience visual language as powerful organizing principle (compare to traditional symbolic/algebraic –only approach). o Doing math: experiment, make observations, conjecture, o o 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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 A short-course on curl & divergence using interactive visualization o Learn new points of view for a classical core topic. o Experience visual language as powerful organizing principle (compare to traditional symbolic/algebraic –only approach). o Doing math: experiment, make observations, conjecture, further test, formulate theorem, prove definition, …. Coherence: Very few fundamental concepts o o Build rich “rooted” concept images. . . . and remember them for life o o (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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 A short-course on curl & divergence using interactive visualization o Learn new points of view for a classical core topic. o Experience visual language as powerful organizing principle (compare to traditional symbolic/algebraic –only approach). o Doing math: experiment, make observations, conjecture, further test, formulate theorem, prove definition, …. Coherence: Very few fundamental concepts o Build rich “rooted” concept images. . o. . and remember them for life o o (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/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 A short-course on curl & divergence using interactive visualization o Learn new points of view for a classical core topic. o Experience visual language as powerful organizing principle (compare to traditional symbolic/algebraic –only approach). o Doing math: experiment, make observations, conjecture, further test, formulate theorem, prove definition, …. Coherence: Very few fundamental concepts o Build rich “rooted” concept images. . o. . and remember them for life (as opposed to: memorize formula for next exam only Make connections, avoid fragmentation of knowledge o o Enjoy the beauty, have fun, become mesmerized. . . http: //math. la. asu. edu/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 A short-course on curl & divergence using interactive visualization o Learn new points of view for a classical core topic. o Experience visual language as powerful organizing principle (compare to traditional symbolic/algebraic –only approach). o Doing math: experiment, make observations, conjecture, further test, formulate theorem, prove definition, …. Coherence: Very few fundamental concepts o Build rich “rooted” concept images. . o. . and remember them for life (as opposed to: memorize formula for next exam only Make connections, avoid fragmentation of knowledge o Enjoy the beauty, have fun, become mesmerized. . . o http: //math. la. asu. edu/~kawski@asu. edu

Matthias Kawski Interactive Visualization 5 th atcm Chiang Mai, Thailand. December 2000 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 • Power. Point presentations from most past conferences • Also on-line: All publications, all classes (Writing. Proofs, Business. Calc, Calc I, III, ODEs, Lin. Alg, Vect. Calc, PDEs, Engin. Math, Complex, Diff. Geom, Adv. Math. Via. Tech, …), and extensive MAPLE, MATLAB depositories. . http: //math. la. asu. edu/~kawski@asu. edu