Bridging the Gap between Mathematics and Physics

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Bridging the Gap between Mathematics and Physics Tevian Dray & Corinne A. Manogue

Tevian Dray & Corinne A. Manogue

Bridging the Gap between Mathematics and Physics

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Math vs. Physics Functions Dot Product Dictionary

Mathematics vs. Physics



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What are 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)?



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What are Functions?


A: T (r , θ) = kr 2



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What are Functions?


A: T (r , θ) = kr 2 B: T (r , θ) = k(r 2 + θ2 )



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What are 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)? r

A: T (r , θ) =

kr 2

B: T (r , θ) =

k(r 2


θ

+

y x

θ2 )


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What are Functions?

T

MATH = f (x, y ) = k(x 2 + y 2 )

T

= g (r , θ) = kr 2



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What are Functions?

T

MATH = f (x, y ) = k(x 2 + y 2 )

T

= g (r , θ) = kr 2

T

PHYSICS = T (x, y ) = k(x 2 + y 2 )

T

= T (r , θ) = kr 2



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What are Functions?

T

MATH = f (x, y ) = k(x 2 + y 2 )

T

= g (r , θ) = kr 2

T

PHYSICS = T (x, y ) = k(x 2 + y 2 )

T

= T (r , θ) = kr 2

Two disciplines separated by a common language...



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Projection: ~u · ~v = |~u||~v| cos θ ~u · ~v = ux vx + uy vy

θ



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θ

Law of Cosines: (~u − ~v) · (~u − ~v) = ~u · ~u + ~v · ~v − 2~u · ~v |~u − ~v|2 = |~u|2 + |~v|2 − 2|~u||~v| cos θ



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θ

Law of Cosines: (~u − ~v) · (~u − ~v) = ~u · ~u + ~v · ~v − 2~u · ~v |~u − ~v|2 = |~u|2 + |~v|2 − 2|~u||~v| cos θ

Addition Formulas: ~u = cos α ˆı + sin α ˆ ~v = cos β ˆı + sin β ˆ ~u · ~v = cos(α − β) Tevian Dray & Corinne A. Manogue


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Kerry Browne (Ph.D. 2002) Tevian Dray & Corinne A. Manogue


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Bridge Project Vector Differentials

The Vector Calculus Bridge Project

http://www.math.oregonstate.edu/bridge http://physics.oregonstate.edu/BridgeBook



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The Vector Calculus Bridge Project Differentials (Use what you know!) Multiple representations Symmetry (adapted bases, coordinates) Geometry (vectors, div, grad, curl)




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The Vector Calculus Bridge Project Differentials (Use what you know!) Multiple representations Symmetry (adapted bases, coordinates) Geometry (vectors, div, grad, curl) Small group activities Instructor’s guide (in preparation) http://www.math.oregonstate.edu/bridge http://physics.oregonstate.edu/BridgeBook



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Vector Differentials

dy ˆ

d~r dx ˆı

d~r = dx ˆı + dy ˆ



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dy ˆ

d~r dx ˆı

d~r = dx ˆı + dy ˆ


ˆ r dφ φ

d~r dr ˆr

ˆ d~r = dr ˆr + r dφ φ


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dy ˆ

d~r

ˆ r dφ φ

dx ˆı d~r = dx ˆı + dy ˆ

d~r dr ˆr

ˆ d~r = dr ˆr + r dφ φ

ds = |d~r| ~ = d~r1 × d~r2 dA dA = |d~r1 × d~r2 | dV

= (d~r1 × d~r2 ) · d~r3



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SUMMARY




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SUMMARY

I took this class a year ago, and I still remember all of it...




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