THE VECTOR CALCULUS BRIDGE PROJECT

Transcription

1 THE VECTOR CALCULUS BRIDGE PROJECT Tevian Dray & Corinne Manogue Oregon State University I: Mathematics Physics II: The Bridge Project III: A Radical View of Calculus

2 Support Mathematical Association of America Professional Enhancement Program Oregon State University Department of Mathematics Department of Physics National Science Foundation DUE DUE DUE DUE

3 Support Grinnell College Noyce Visiting Professorship Mount Holyoke College Hutchcroft Fund Oregon Collaborative for Excellence in the Preparation of Teachers

4 Mathematics Physics Mathematics Physics

5 The Grand Canyon Mathematics Physics

6 QUESTION Suppose the temperature on a rectangular slab of metal is given by T (x, y) = k(x 2 + y 2 ) where k is a constant. What is T (r, θ)?

7 QUESTIONS Suppose the temperature on a rectangular slab of metal is given by T (x, y) = k(x 2 + y 2 ) where k is a constant. What is T (r, θ)? Answers A: T (r, θ) = kr 2 B: T (r, θ) = k(r 2 + θ 2 )

8 MATH T = f(x, y) = k(x 2 + y 2 ) T = g(r, θ) = kr 2 PHYSICS T = T (x, y) = k(x 2 + y 2 ) T = T (r, θ) = kr 2

9 MATH T = f(x, y) = k(x 2 + y 2 ) T = g(r, θ) = kr 2 PHYSICS T = T (x, y) = k(x 2 + y 2 ) T = T (r, θ) = kr 2 Differential Geometry! T (x, y) T (x, y) 1 T (r, θ) T (r, θ) 1 (x, y) S T (x, y) T R 2 R

10 EXAMPLE (Stewart 4th edition 15.6: 12, 13, 14) Find the directional derivative of the function at the given point in the direction of v. (a) f(x, y) = x/y, (6, 2), v = 1, 3 (b) g(s, t) = s 2 e t, (2, 0), v = î + ĵ (c) g(r, θ) = e r sin θ, (0, π/3), v = 3 î 2 ĵ

11 EXAMPLE (Stewart 4th edition 15.6: 12, 13, 14) Find the directional derivative of the function at the given point in the direction of v. (a) f(x, y) = x/y, (6, 2), v = 1, 3 (b) g(s, t) = s 2 e t, (2, 0), v = î + ĵ (c) g(r, θ) = e r sin θ, (0, π/3), v = 3 î 2 ĵ Context matters!

12 THE BRIDGE PROJECT Small group activities Instructor s guide (in preparation) CWU, LBCC, MHC, OSU, UPS, UWEC Workshops, Minicourses (PREP, MAA,...)

13 THE BRIDGE PROJECT Small group activities Instructor s guide (in preparation) CWU, LBCC, MHC, OSU, UPS, UWEC Workshops, Minicourses (PREP, MAA,...) Differentials (Use what you know!) Multiple representations Symmetry (adapted bases, coordinates) Geometry (vectors, div, grad, curl)

14 PEOPLE Barbara Edwards Tom Dick, Christine Escher, Dianne Hart, John Lee, Hal Parks Johnner Barrett, Corina Contstantinescu, Sam Cook, Nicole Webb Stuart Boersma, Giuliana Davidoff, Alan Durfee, Marc Goulet, Martin Jackson, Greg Quenell, Alex Smith, Judy de Szoeke David Griffiths, Harriet Pollatsek Bill McCallum, Jim Stewart

15 PUBLICATIONS 1. Tevian Dray and Corinne A. Manogue, The Vector Calculus Gap: Mathematics Physics, PRIMUS 9, (1999). 2. Tevian Dray and Corinne A. Manogue, Electromagnetic Conic Sections, Am. J. Phys. 70, (2002). 3. Tevian Dray & Corinne A. Manogue, Spherical Coordinates, College Math. J. 34, (2003). 4. Tevian Dray & Corinne A. Manogue, The Murder Mystery Method for Determining Whether a Vector Field is Conservative, College Math. J. 34, (2003). 5. Tevian Dray and Corinne A. Manogue, Using Differentials to Bridge the Vector Calculus Gap, College Math. J. 34, (2003). 6. Tevian Dray & Corinne A. Manogue, The Geometry of the Dot and Cross Products, College Math. J. (submitted).

16 2005: WORKSHOPS Corvallis, OR, 6/25 29/05 Joint Mathematics Meetings, Atlanta, GA, 1/6 8/ : MathFest, Providence, RI, 8/12 14/04 Corvallis, OR, 8/7 11/04 PNWMAA Meeting, Anchorage, AK, 6/24/04 PREP Workshop, South Hadley, MA, 6/18 22/04 AAPT Meeting, Miami, FL, 1/24/ : Corvallis, OR, 8/13 17/03 MathFest, Boulder, CO, 7/31 8/1/03

17 MATHEMATICIANS LINE INTEGRALS Start with Theory F d r = F ˆT ds = F ( r(t)) r (t) r (t) r (t) dt = F ( r(t)) r (t) dt =... = P dx + Q dy + R dz Do examples starting from next-to-last line Need parameterization r = r(t)

18 PHYSICISTS LINE INTEGRALS Theory Chop up curve into little pieces d r. Add up components of F parallel to curve (times length of d r) Do examples directly from F d r Need d r along curve

19 MATHEMATICS F (x, y) = y î + x ĵ x 2 + y 2 r = x î + y ĵ F d r = = π 2 0 π 2 0 x = 2 cos θ y = 2 sin θ F (x(θ), y(θ)) r (x(θ), y(θ)) dθ 1 ( sin θ î + cos θ ĵ) 2( sin θ î + cos θ ĵ) dθ 2 =... = π 2

20 PHYSICS F = ˆθ r d r = r dθ ˆθ I: F = const; F d r = F d r = 1 2 (2 π 2 ) II: Do the dot product π 2 F d r = 0 ˆθ 2 2 dθ ˆθ = π 2 0 dθ = π 2

21 PHYSICS F = ˆθ r d r = r dθ ˆθ I: F = const; F d r = F d r = 1 2 (2 π 2 ) II: Do the dot product π 2 F d r = 0 ˆθ 2 2 dθ ˆθ = π 2 0 dθ = π 2 Symmetry matters!

22 CORIANDER I: In the small town of Coriander, the library can be found by starting at the center of the town square, walking 25 meters north, turning 90 to the right, and walking a further 60 meters. II: It turns out that magnetic north in Coriander is approximately 14 degrees east of true north. If you use a compass to find the library (!), the above directions will fail. Instead, you must walk 39 meters in the direction of magnetic north, turn 90 to the right, and walk a further 52 meters. y Y x 52 X

23 VECTOR DIFFERENTIALS dy ĵ d r dx î r dθ ˆθ d r dr ˆr d r = dx î + dy ĵ d r = dr ˆr + r dθ ˆθ ds = d r d S = d r 1 d r 2 ds = d r 1 d r 2 dv = (d r 1 d r 2 ) d r 3 df = f d r

24 A RADICAL VIEW OF CALCULUS The central idea in calculus is not the limit. The central idea of derivatives is not slope. The central idea of integrals is not area. The central idea of curves and surfaces is not parameterization. The central representation of a function is not its graph.

25 A RADICAL VIEW OF CALCULUS The central idea in calculus is the differential. The central idea of derivatives is rate of change. The central idea of integrals is total amount. The central idea of curves and surfaces is use what you know. The central representation of a function is data attached to the domain.