Equipping students to connect multivariable calculus with the...

Equipping students to connect multivariablecalculus with the physical world

David RoundyOregon State University

DUE-0837829National Science Foundation

DUE-1141330DUE-1323800

Computational Lab and Electrostatics Energy and Entropy Conclusion

Challenges transitioning from multivariable calculus

multidimensional integrals

in math... integral is area under a curve

in physics... integration is chopping and adding

in math... functions have at most two arguments, the functionis the third dimension

in physics... fields exist in three dimensions

in math... students integrate specified functions

in physics... setting up an integral is the primary challenge

partial derivatives

in math... students differentiate explicit functions and variables

in physics... we differentiate physically measurable quantities

in math... “everything else” is held constant

in physics... it matters what is held constant

2 / 31










partial derivatives




in physics... it matters what is held constant2 / 31


Addressing challenges from multivariable calculus

multidimensional integrals (computational lab)

I integration as chopping and adding:students perform summation of integral explicitly

I fields exist in three dimensions:students visualize fields in three dimensions

I setting up an integral is the primary challenge:once students have written down an integral, coding it up isrelatively easy (by the end of the course)

partial derivatives (Energy and Entropy Paradigm)

I taking derivatives of physically measurable quantities:students measure partial derivatives experimentally

I it matters what is held constant:students measure and discuss how to fix a given quantity

3 / 31


Computational lab parallel to “traditional” courses

A computational lab for traditional courses

I reinforce learning in traditional courses

I save time by not having to introduce the physics

I students have a very busy schedule: just one credit

All work is done in the lab

I Today all physicists need to program

I Struggling students make little progress outside of class

I These are the students who need a computational course4 / 31


Python and matplotlib

Reasoning

I free software, readily available to students

I ease of use and power comparable to Matlab

I used by professional scientists

I tutorials and help readily available on web

5 / 31


Teaching programming to physics students

No templates needed

I students write their programs from scratch

I they google for help

Pair programming

I students work in pairs: a driver and a navigator

I swap roles every 30 minutes or so

I “show and tell” when projects are done

6 / 31


Electrostatics

The first two Paradigms cover electrostatics and magnetostatics.

Learning goals shared with the Paradigms

I how to compute distances

I taking advantage of symmetry

I curvilinear coordinates

I visualizing in 3D with slices and lines through V (~r)

I integration as chopping and adding (how to set up integrals)

Learning goals specific to computing

I plotting

I writing functions and using loops

7 / 31


Electrostatics (student work)Day 1: 4 point charges

4 3 2 1 0 1 2 3 4Distance (m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0Po

tent

ial (

V)

I how to compute distances I how to plot8 / 31


Electrostatics (student work)Day 2: 4 point charges

3 2 1 0 1 2 3X (m)

2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

2.0

Y (m

)

15000000000.000 1500

0000

000.

000

v(X,Y,.1)

I visualizing slices of a three-dimensional field9 / 31


Electrostatics (student work)Day 3: Square of charge

10 5 0 5 1010

5

0

5

10

4000.000 4000.000

4000.000 4000.000

6000.000

8000.000

10000.00012000.000

Simplest default with labels

I chopping and adding I googling for help 10 / 31


Electrostatics (student work)Day 4: Square of charge (with varying density on left)

0 1 2 3 4 5z

0.0

0.5

1.0

1.5

2.0

2.5V

Sig = f, Potential @x=0, y=0

0 1 2 3 4 5z

01234567

V

Sig = 1, Potential @x=0, y=0

6 4 2 0 2 4 6x

6

4

2

0

2

4

6

y

Sig = f, Potential @z=.01

321

012345

6 4 2 0 2 4 6x

6

4

2

0

2

4

6

y

Sig = 1, Potential @z=.01

01234567

I visualizing fields in multiple dimensions11 / 31


Computational conclusion

I all physicists need to write computer programs

I use physics that is relevant to students

I computational lab with pair programming works well

I integration as chopping and adding

I visualization of three-dimensional fields

I setting up an integral is sufficient to find solution(numerically)

12 / 31


Challenges in Energy and Entropy

I no thermo in our lower-division sequence

I students have never seen a differential in a math course

I variables p, V , T and S are unfamiliar

I partial derivative notation is new:(∂T∂V

)p

or ∂T∂V

∣∣p

I “everything else is held constant”I partial derivative manipulations are also new

I cyclic chain ruleI Clairaut’s theoremI ordinary chain ruleI inverse of a partial derivative

I Thompson, Manogue, Roundy, Mountcastle. PERC 2011 Proceedings. (2012).

I Kustusch, Roundy, Dray, and Manogue. PERC 2012 Proceedings. (2013).

I Kustusch, Roundy, Dray, and Manogue. submitted to Phys. Rev. Special Topics – PER.

13 / 31


Inverse of a partial derivative

Physics(∂p

∂V

)S

=1(

∂V∂p

)S

physical observables

p = p(V ,S)

p = p(V ,T )

V = V (p,S)

V = V (p,T )

Math

∂u

∂x6= 1

∂x∂u

functions

u(x , y)

v(x , y)

x(u, v)

y(u, v)(∂u

∂x

)y

6= 1(∂x∂u

)v

14 / 31


Mathematical interlude: a mechanical analogue for thermo

7 hours of thermodynamics math on a mechanical system

I integrating over a path to find work

I path independence to get potential energy

I small differences or tangent slope to find partial derivatives

I “holding everything else constant” is not possible

I partial derivative manipulations

I connection with experiment

I total differential for energy conservation

I Maxwell relations

I Legendre transformations

I Scherer, Kustusch, Manogue and Roundy, PERC 2013 Proceedings (in press).I Manogue and Roundy, PERC 2013 Proceedings (in press).I Roundy, Gupta, Wagner, Dray, Kustusch, van Zee, and Manogue. PERC 2013 Proceedings (in press).

15 / 31


The partial derivative machineThe system

xy

xF

yF

I one hidden elastic system

I two controllable and measurable coordinates x and y

I two controllable forces Fx and FyI one potential energy U, not directly measurable

16 / 31


Measuring partial derivatives(∂x

∂Fx

)y

vs

(∂x

∂Fx

)Fy

I students consistently believe these are the same

I they are taught to “hold everything else constant”

17 / 31


Approaches for finding a derivative

I make a small change, measure a small change, take a ratioI mistake: use a very small change and get noiseI mistake: use a large change and assume linear responseI try a different small change to ensure “small enough”

I measure many values, make a plotI mistake: fit to a straight lineI find tangent line (by eye)

I mistake: seek analytic form to differentiate (black box helps)18 / 31


A mechanical analogue for thermo: First Law

I energy conservation and path independence: differentials

I looks like thermodynamic identity: dU = Fxdx + Fydy

I students integrate work to find potential energy

I paths from state A to state B (like pV plots)

19 / 31


A mechanical analogue for thermo: Maxwell relations

dU = Fxdx + Fydy

I Maxwell relation(∂(∂U∂x

)y

∂y

)x

=

∂(∂U∂y

)x

∂x

y(

∂Fx∂y

)x

=

(∂Fy∂x

)y

I can verify this experimentally

20 / 31


A mechanical analogue for thermo: Legendre transforms

I Legendre transform: imagine one mass is inside the black boxI you cannot change one force Fy

I you cannot measure the value y

I add potential energy of mass causing Fy :

V ≡ U − Fyy

I like enthalpy and Helmholtz free energy

I gives more Maxwell relations

21 / 31


What is this derivative?

(∂p

∂V

)S

22 / 31


Name the experiment!

I give student groups specific derivatives

I students sketch an experiment to measure that derivative

I groups share their experiments with the class

23 / 31


Name the experiment!Adiabatic compressibility

(∂p∂V

)S

=

24 / 31


Name the experiment!General learning goals

I operational definitions of thermodynamic quantitiesI how to measureI how to fix

I what is held constant matters

I “canonical” thought experiments

I thermodynamic derivatives are physically measurable

25 / 31


Name the experiment!Specific learning goals

I adiabatic processes:(∂T∂V

)S

,(∂T∂p

)S

and(∂V∂p

)S

I changing temperature without heating:(∂T∂V

)S

and(∂T∂p

)S

I the First Law:(∂U∂T

)V

and(∂U∂p

)S

I heat capacity:(∂S∂T

)V

and(∂S∂T

)p

I heating without changing temperature:(∂S∂V

)T

and(∂S∂p

)T

I using Maxwell relations:(∂S∂V

)T

,(∂S∂p

)T

,(∂S∂p

)V

and(∂S∂V

)p

I turning derivatives upside down: many of the above

I D. Roundy, M. B. Kustusch, and C. A. Manogue. Am. J. Phys. (in press).

26 / 31


Measuring partial derivatives with experiment

Integrated labs enable tight coupling ofcoursework with experiments, includingteaching while data is being collected.

I ice-water calorimetry

I ice melting in water

I rubber band tension vs. L and T

Learning goals

I heat, heat capacity, latent heat

I work, free energy

I integrating experimental data

I measuring derivatives

I integrating to find ∆S

I Maxwell relationsI D. Roundy and M. Rogers. Am. J. Phys. (2013). 27 / 31


Thermal conclusion

I connect math with tangible reality

I partial derivative machine

I name the experiment

I perform actual experiments

28 / 31










partial derivatives




in physics... it matters what is held constant29 / 31


Addressing challenges from multivariable calculus

multidimensional integrals (computational lab)

I integration as chopping and adding:students perform summation of integral explicitly

I fields exist in three dimensions:students visualize fields in three dimensions

I setting up an integral is the primary challenge:once students have written down an integral, coding it up isrelatively easy (by the end of the course)

partial derivatives (Energy and Entropy Paradigm)

I taking derivatives of physically measurable quantities:students measure partial derivatives experimentally

I it matters what is held constant:students measure and discuss how to fix a given quantity

30 / 31


Acknowledgements

Paradigms team

I Corinne Manogue

I Tevian Dray

I Mary Bridget Kustusch

I Henri Jansen

I Janet Tate

I David McIntyre

Energy and Entropy collaborators

I John Thompson (U. Maine)

I Michael Rogers (Ithaca College)

Students

I Eric Krebs and Jeff Schulte

I Grant Sherer

31 / 31

Equipping students to connect multivariable calculus with the...

Documents

Transcript of Equipping students to connect multivariable calculus with the...