The Geometry of Calculus

Transcription

1 Introduction Department of Mathematics Oregon State University

2 Introduction Acknowledgments Places: People: Stuart Boersma, Johnner Barrett, Sam Cook, Aaron Wangberg Corinne Manogue

3 Introduction Mathematics vs. Physics Physics Functions Physics is about things. Physicists can t change the problem. Mathematicians do algebra. Physicists do geometry.

4 What are Functions? Introduction Physics Functions 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)? A: T(r,θ) = kr 2 B: T(r,θ) = k(r 2 +θ 2 ) r θ x y

5 What are Functions? Introduction Physics Functions 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 Two disciplines separated by a common language... physical quantities functions

6 Trig Introduction Algebra Geometry Infinitesimals Ý rdθ Ü Θ r x 2 +y 2 = r 2 = x dx +y dy = 0 ( ) ds 2 = r 2 dθ 2 = dx 2 +dy 2 = dx 2 1+ x2 y 2 = r 2 dx2 y 2, dθ 2 = dx2 y 2 = dy2 x 2 = = d sinθ = cosθdθ dy = x dθ dx = y dθ d cosθ = sinθdθ

7 Introduction Proof Without Words Algebra Geometry Infinitesimals x dy rdθ dy = (r dθ)cosθ d(r sinθ) = r cosθdθ d(sinθ) = cosθdθ r Θ (with Aaron Wangberg), College Math J. 44, (2013).

8 Infinitesimals Introduction Algebra Geometry Infinitesimals But so great is the average person s fear of the infinite that to this day calculus all over the world is being taught as a study of limit processes instead of what it really is: infinitesimal analysis. (Rudy Rucker)... many mathematicians think in terms of infinitesimal quantities: apparently, however, real mathematicians would never allow themselves to write down such thinking, at least not in front of the children. (Bill McCallum)

9 Introduction Derivatives Example d(u +cv) = du +c dv d(uv) = udv +v du d (u n ) = nu n 1 du d (e u ) = e u du d(sinu) = cosudu d(cosu) = sinudu d(lnu) = 1 u du

10 Introduction Derivatives (& Integrals) Derivatives Example Derivatives: Chain rule: d d sinu sinu = = cosudu = cosu du du du d d sinu sinu = = cosudu = cosu du dx dx dx dx Inverse functions: q = lnu = e q = u = d(e q ) = e q dq = du = dq = du e q = du u Integrals: 2x cos(x 2 )dx = cosudu = d(sinu) = sinu = sin(x 2 )

11 Coherence Introduction Derivatives Example Instead of: chain rule related rates implicit differentiation derivatives of inverse functions difficulties of interpretation (units!) One coherent idea: Zap equations with d & Corinne A. Manogue, Putting Back into Calculus, College Math. J. 41, (2010).

12 Introduction Derivatives Example Example: Tangent line to a circle Find the slope of the tangent line to the circle of radius 5 centered at the origin at the point ( 3, 4). x 2 +y 2 = 25 = 2x dx +2y dy = 0 = dy dx = x y = 3 4

13 Introduction Bridge Project Vector Gradient The Bridge Project (Use what you know!) Multiple representations Symmetry (adapted bases, coordinates) Geometry (vectors, div, grad, curl) Small group activities Instructor s guide Online text ( DUE , DUE , DUE

14 Introduction Bridge Project Vector Gradient The Bridge Project Bridge Project homepage hits in 2009

15 Vector Introduction Bridge Project Vector Gradient dy ŷ d r dx ˆx r dφ ˆφ d r drˆr d r = dx ˆx+dy ŷ d r = drˆr+r dφ ˆφ ds = d r d A = d r 1 d r 2 da = d r 1 d r 2 dv = (d r 1 d r 2 ) d r 3

16 Introduction The Geometry of Gradient Bridge Project Vector Gradient df = f d r df small change in f d r small step level curve = f = constant = df = 0 = f d r = 0 = f level curve The gradient points uphill...

17 Introduction Bridge Project Vector Gradient Raising Calculus to the Surface (Aaron Wangberg) (Each surface is dry-erasable, as are the matching contour maps.)

18 The Hill Introduction Bridge Project Vector Gradient Suppose you are standing on a hill. You have a topographic map, which uses rectangular coordinates (x,y) measured in miles. Your global positioning system says your present location is at one of the points shown. Your guidebook tells you that the height h of the hill in feet above sea level is given by h = a bx 2 cy y 0-5 where a = 5000 ft, b = 30 ft mi 2, and c = 10 ft mi x

19 Conclusions A Radical View of Calculus Thick Derivatives 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 physical quantities is not functions. Calculus is about infinitesimal reasoning. Derivatives are ratios of small quantities. The central idea of integrals is chop, multiply, add. The central idea of curves and surfaces is use what you know. The central representation of physical quantities is relationships (equations) between them.

20 Conclusions A Radical View of Calculus Thick Derivatives Partial Derivatives Machine horizontal pulleys Developed for junior-level thermodynamics course Two positions, x i, two string tensions (masses), F i. Find x F. Idea: Measure x, F; divide. Mathematicians: That s not a derivative! Paradigms in Physics Project DUE , DUE spring the system flags thumb nuts vertical pulleys weights

21 Conclusions A Radical View of Calculus Thick Derivatives Graphical Verbal Symbolic Numerical Physical Rate of Difference Ratio of Slope Measurement Change Quotient Changes Ratio average rate of change f(x+ x) f(x) x y 2 y 1 x 2 x 1 numerically Limit instantaneous... lim x 0...with x small Function... at any point/time f (x) =... depends on x tedious repetition An extended framework for the concept of the derivative.

22 Conclusions A Radical View of Calculus Thick Derivatives No entry for symbolic differentiation!! Captures roundoff error, measurement error, quantum mechanics... thick derivatives David Roundy et al., RUME Proceedings 2015, MAA, pp ( Thick Derivatives, AMS Blog: On Teaching and Learning Mathematics (May 31, 2016). (

23 Conclusions A Radical View of Calculus Thick Derivatives PUT DIFFERENTIALS BACK into CALCULUS!