Bridging the Vector Calculus Gap - Oregon State University

Tevian Dray & Corinne Manogue
Oregon State University
Mount Holyoke College
Grinnell College
http://www.physics.orst.edu/bridge

Support
• National Science Foundation
– DUE-9653250, 0231194
– DUE-0088901, 0231032
• Oregon Collaborative for
Excellence in the Preparation of
Teachers
• Oregon State University
• Grinnell College
– Noyce Visiting Professorship
• Mount Holyoke College
– Hutchcroft Fund

Mathematics
Physics
Mathematics
Physics

The Grand Canyon
Mathematics
Physics

A Question for the Audience
2 2
Txy (,) = kx ( + y)
Trθ (, ) =

Main Differences
• Physics is about things.
• Physicists can’t change the problem.

Physics is about things
• What sort of a beast is it
• Physics is independent of coordinates.
– Vectors as arrows
– Geometry of dot and cross products.
• Graphs are about the relationships of physical
things.
• Fundamental physics is highly symmetric.

What Are Vectors
Mathematics:
• Triples of numbers
v = v , v , v
1 2 3
Physics:
• Arrows in space
v = v ı+ v j+
v k
1 2 3ˆ

Planes
n
( )
r r 0
0
r −r ⋅ n = 0
⇒r⋅ n = r ⋅ n =
0
const.

Plane Wave Activity
Students connect points
with equal value of
k ⋅ r
What is
cos k r
( ⋅ −ωt)

Physics is about things
• What sort of a beast is it
• Physics is independent of coordinates.
• Graphs are about the relationships of physical
things.
• Fundamental physics is highly symmetric.
– Spheres and cylinders vs. parabolas.
– Interesting physics problems can involve trivial
math.
– rˆ, ˆ θ , ˆ φ.

Physicists can’t change the problem.
• Physics involves the creative synthesis of multiple ideas.
• Physics problems may not be well-defined math problems.
– No preferred coordinates or independent variables.
– No parameterization.
– Unknowns don’t have names.
– Getting to a well-defined math problem is part of the problem.
– If you can’t add units, it’s a poor physics problem.
• Physics problems don’t fit templates.

An Example
• Typical of EARLY upper-division work for
physics majors and many engineers.
• Solution requires:
– many mathematical strategies,
– many geometrical and visualization strategies,
– only one physics concept.
• Demonstrates different use of language.

Potential Due to Charged Disk
z
What is the electrostatic
potential at a point, on
axis, above a uniformly
charged disk

One Physics Concept
• Coulomb’s Law:
V
=
4
1
πε
0
q
r

Superposition
• Superposition for solutions of linear
differential equations:
V
=
4
1
πε
0
→ V( r)
=
q
r
4
1 σ ( r′ ) da′
∫
r − r′
πε
0

Chopping and Adding
r′ + z
2 2
z
Integrals involve
chopping up a part of
space and adding up a
physical quantity on
each piece.
r′

Computational Skill
• Can the students set-up and do the integral
1 σ ( r′ ) da′
V( r)
= ∫
4πε0
r − r′
2π
R
σ dr′ r′ dθ
′
= ∫∫
4πε
2 2
0 0 0 r′ + z
2πσ
= + −
4πε
0
( )
2 2
R z z

Limits (Far Away)
2πσ
Vr ( ) = R+ z −z
4πε
0
0
( )
2 2
2
2πσ
⎛ R ⎞
= z 1+ −z
2
4πε
⎜
0
z ⎟
⎝
⎠
2
2πσ
⎛ ⎛ 1 R ⎞
= ⎜ z 1
2
4πε
⎜ + + …
0
2 z
⎟−
⎝ ⎝
⎠
2
1 πR
σ
≈
4πε
z
⎞
z⎟
⎠

Physicists can’t change the problem.
• Physics involves the creative synthesis of multiple
ideas.
• Physics problems may not be well-defined math
problems.
• Physics problems don’t fit templates.
• Physics involves the interplay of multiple
representations.
– Dot product.
– Words, graphs, symbols

Relating Multiple
Representations
1. Flux is the total amount of electric field through a given area.
2.
Φ =
E
∫ ⋅da
E⋅
da
E
3.
da

Eigenvectors Activity
• Draw the initial vectors below on a single graph
• Operate on the initial vectors with your group's
matrix and graph the transformed vectors

Eigenvectors Activity
• Note any differences between the initial and
transformed vectors. Are there any vectors
which are left unchanged by your
transformation
• Sketch your transformed vectors on the
chalkboard.

Simple Curriculum Additions
• Equations describe the relationship of physical
quantities to other physical quantities.
• Emphasize superposition of solutions to linear
differential equations.
• Integrals involve chopping a part of space and
adding up a physical quantity on each piece.
• “Limits” require one to say what dimensionless
quantity is small.

Goals
Mathematics:
• Visualization
• Arbitrary surfaces
• 2-D is primary example
• Immediate sophistication
• Domain is important
• End point flexible
Physics:
• Visualization
• Simple geometries only
• 3-D is primary example
• Many years later
• Surface is important
• End point fixed
– Divergence theorem
– Stokes’ theorem

Opportunities
Beta testers are being sought to test materials:
• Vector Calculus (lower-division math)
http://www.physics.oregonstate.edu/bridge
MathFest (8/12-14/04)
Summer Workshops (6/18-22/04 MA) (8/7-11/04 OR)
• Thermodynamics and Quantum Mechanics
http://www.physics.oregonstate.edu/paradigms
Summer Workshop (6/25-28/04)
AAPT Workshop (1/25/04)