NSF DUE 97-52453, Matthias Kawski, Arizona State U.

Vector Calculus via Linearization: Visualization and Modern Applications

Abstract. This project is developing a radically new approach to vector calculus that is completely aligned with recent reform efforts in other college entry-level math, science, and engineering (MSE) courses. As part of this larger MSE curriculum, it brings more coherence and consistency to it, emphasizing the common foundation for much of these courses. Sophisticated computer visualization is vigorously exploited as it has opened a completely new avenue to vector calculus, allowing to reconnect the concepts of vectorial derivatives back to their foundation, local linearity. Students now can see the curl by zooming.Connections to concepts of linear algebra and differential equations perspectives are omnipresent. The curriculum is ideally suited for cooperative learning environments where students are guided to make many key discoveries themselves via often dramatic computer experiments. The new course is also broadening the range of applications (beyond the traditional EM and fluid dynamics) by drawing on modern geometric control which has ubiquitous applications, even connecting to recent research achievements that fit into this sophomore course. The curriculum is being implemented in the form of interactive texts, and is using modern electronic media and the WWW for rapid dissemination. Assessment of the enhanced learning experiences is an integral part of the project. This curriculum is having a high impact on both MSE majors and future math and science teachers by conveying a coherent and modern view of advanced mathematics.

“Via Linearization”

We asked: “If zooming for derivatives is so effective in calculus I, then why don’t we zoom in vector calculus for curl and divergence?”

Our curriculum demonstrates that zooming for derivatives of vector fields is not only possible, but yields even more powerful insights than zooming in single variable calculus.

To illustrate the traditional fragmentation and lack of coherence simply ask: “What do curl and divergence have to do with derivatives – i.e. where is the linear object?” Recall: any kind of differentiability means “approximability by linear objects”

Our approach brings back coherence, starting with a solid foundation of vector pre-calculus, that is the analysis of linear and constant vector fields before proceeding to differential and integral vector calculus.

A coherent, connected curriculum!

Compartmentalized, fragmented knowledge

Linear Algebra

Complex Analysis

Differential Equations

L Tds L x y y i x L x x y j y

L x y y i x L x x y j y

c b x y

R

( , ) ( , )

( , ) ( , )

..( )... ( )

0 0 0 0

0 0 0 0

only using linearity

Vector pre-calculus: Study linear fields first!& understand in depth where curl and div come from

(x0,y0)

(x0,y0 -y)

(x0,y0+y)

(x0+x,y0)(x0-x, y0)

for L(x,y) = (ax+by,cx+dy), using only midpoint rule (exact!) and linearity for e.g. circulation integral over rectangle --- no calculus yet!

Vector pre-calculus: Study linear fields first!& integrate over polygonal domains

L Nds

L Nds

trL A

trL A

trL A

C

Ck

k k

kk

k

No calculus yet – develop the fundamental concepts first in the pre-calculus setting! The picture of new TELESCOPING SUMS matters (cancellations!)

The JAVA vector field analyzer

• Freely available on the WWW• Built-in parser & predefined common examples• Different lenses to interactively zoom for derivatives, make connection with linear algebra!• Visualize nonlinear and (parts of) linearized flows make connection between vect calc and diff equns • Coming: A panel to experiment w/ line/flux integrals • Coming: Illustrated workbook (200 pp.)

Example: An irrotational, divergence free vector field

Example: The linearized flow illustrates the integrals of div and curl

Develop deeply rooted concept images

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

Modern applications: ControlThe traditional vector calculus course emphasized applications to fluid mechanics and electro-magnetics – we emphasize engineering control!

• The primarily descriptive physics point of view changes to the much more engaging engineering point of view that asks inverse questions: “What do we have to do so that ..... (something desirable occurs)?”

This kind of questioning fills the need for “story-lines” behind such inverse questions necessitated by the advent of computer algebra systemsthat trivialized the traditional (forward) exercises – example: “ask for a curve such that a line integral of a given differential form has a desired value”.

• Modern applications, such as re-orienting satellites, falling cats, divers and gymnasts appear more tangible than the traditional sole applications fromE & M, and they appear to be more gender neutral. Students are excited when as sophomores working on “projects” with links to NASA .....

Project: Re-orientation via internal shape changes

Traditional emphasis, physics point of view:Conservative ( = integrable ) vector fields,“closed loops lift to a potential surface”

Modern emphasis, engineering point of view: Controllable ( = nonintegrable ) vector fields, “design the closed loop in base so that the vertical gap of the lifted curve is as desired”.

CrdF

Traditional: Given F and C find (boring w/ computer algebra system)Modern: Given F and find C (intelligent, ubiquitous applications)

1

2

Modern technology efficiently handles messy calculations.

Theory matters!

It is the theory that matters!

Students are directly connected to current research, via onlinesearches. Especially references from National Research Council) expand in very subtle way the 3rd semester picture to the waytop researchers think today.

Other “products” under preparation

• Background “textbook”

• Library of MAPLE worksheets

• Workbook to accompany JAVA vector field analyzer

• “Falling cat” project booklet

• “Book of zooming”: From limits to Stokes theorem

• Assessment and evaluation instruments with comparison data from field-tests (w/ Sean Larsen)

Contact info / to learn moreAddress: Matthias Kawski

Department of Mathematics Arizona State University Tempe, AZ 85287-1804 U.S.A.

URL: http://math.la.asu.edu/~kawskie-mail: kawski@asu.eduPhone: (480) 965 0107

“Everything is on the WWW”, worksheets, class-materials, JAVA, MAPLE worksheets, publications, conference presentations (as *.ppt, offering the fastest way to get an overview of the project...)