Integrating Mathematica in the 
Undergraduate Science Curriculum:
Teaching Computer Literacy with  Mathematica

Fredrick  Olness
Southern Methodist University / Fermilab
http://www.physics.smu.edu/~olness

By the year 2000, American colleges and universities will be lean and mean, service oriented and science minded, multicultural and increasingly diverse---if they intend to survive their fiscal agony.
				      ("Campus of the Future," TIME, April 13, 1992.)

Introduction

If our educational institutions are to remain competitive, it is essential that our students be prepared to enter a society that is increasingly dependent on advanced technology. Therefore, students must be exposed to advanced computer tools in our science and mathematics courses--even in introductory courses--to prepare them for a world where computers play an important role.
	In the 10 years since Mathematica was introduced, it has already had a significant impact on the way science is taught and how research is performed. This presentation will highlight experiences during the last five years integrating Mathematica into the physics curriculum at SMU.
	In 1993 we added Mathematica as a component of our physics curriculum.  The students were given an initial tutorial at the beginning of the semester and then assigned problems from their text which were particularly well suited to the capabilities of Mathematica. For the lower-level courses, the user-friendly interface and the intuitive graphics capabilities of Mathematica encouraged and tempted the student to experiment. With intermediate level courses, the students were able to delve deeper into the capabilities of the program and solve more complex problems.
	
	In the following, I contrast traditional solutions with modern solutions using Mathematica.

A picture (plot) is worth 1000 words

x(t) = (Subscript[x, 0])+ (Subscript[v, 0]) t + (1/2)a (t^2)           v(t) = (Subscript[v, 0]) +  a  t          (v^2) -   (Subscript[v, 0]^2) =  
              2 a    Δx

Realistic Problems Become Elementary

\(TraditionalForm
    \`\(F\& ⇀ \)\  = \ 
      \(m\ \ \(x(t)\ \)\&¨ = \ 
        \(-\ m\)\ g\ \  - \ b\ \(\(x\& . \)(t)\)\)\)

Multi-Dimensional Vector Fields Become Intuitive

\(TraditionalForm
    \`∇\(·E\&⇀\)\  = \ 
      \(ρ\ \ \ \ \ \ \ \ \ \ \ \ ∇\(\[Cross]E\&⇀\)\  = 
        \(0\ \ \ \ \ \ \ \ \ \ \ \ \ \ 
            ∇\(·B\&⇀\)\  = 
          \(0\ \ \ \ \ \ \ \ \ \ \ \ ∇\(\[Cross]B\&⇀\)\  = 
            \ \(j\&⇀\ \ \ \ \ \ \ \ \ \ \ \ \ \ 
                E\&⇀\  = \ \(-\ ∇V\)\)\)\)\)\)

Graphical Proof of Identities:  Obscure => Obvious

\(TraditionalForm
    \`e\^\(i\ z\) = 
      ∑\+\(ℓ = 0\)\%∞ i\^\(\ ℓ\)\ 
          \((2\ ℓ + 1)\)\ \ \(\(J\_ℓ\)(r)\)\ \ 
          \(\(P\_ℓ\)(cos(θ))\)\)

“There are only two physics problems that can be solved exactly: the Kepler problem of an orbiting mass, and the simple harmonic oscillator. Everything else is perturbation theory.”

New Insight Into Practical Problems and Applications

\(TraditionalForm
    \`\(x(t)\  = \ 
      \([E\^\(t\ \((ω - γ)\)\)\ 
              \((v\_0 + \ \(x\_0\)(γ + ω)\ )\) - 
            \(E\^\(\(-t\)\ \((ω - γ)\)\)\)(
              v\_0 + \ \(x\_0\)(γ - ω))]\)/ω\ \)\)

Boundary Value Problems:  Is my answer correct?    Look at the picture!

V =  
      -(1/π) 
        (I (Subscript[V, 0]) 
            (Log[1 - 
                    (E^(-((I π x)/a) - ((π y)/b)))] - 
                Log[1 + 
                    (E^(-((I π x)/a) - ((π y)/b)))] - 
                Log[1 - (E^(((I π x)/a) - ((π y)/b)))] + 
                Log[1 + (E^(((I π x)/a) - ((π y)/b)))]))

Visualizing Complex Systems: The Hydrogen Atom

Ψ=(Subscript[R, n, ℓ]) (Subscript[Y, ℓ, m])(θ, ϕ)                         0=(k^2) (r^2) R(r) - ℓ R(r) - 
                (ℓ^2) R(r) + 2 r (R^′)(r) + 
                (r^2) (((R^′))^′)(r)
				(Subscript[R, n, ℓ]) = {
                ((2 (Z^(3/2)))/((Subscript[a, 0]^(3/2)) 
                    (E^(((r Z)/(Subscript[a, 0])))))), {
                  (((Z^(3/2)) (2 - ((r Z)/(Subscript[a, 0]))))/(2 (Sqrt[2]) 
                      (Subscript[a, 0]^(3/2)) 
                      (E^(((r Z)/(2 (Subscript[a, 0]))))))), 
                  ((r (Z^(5/2)))/(2 (Sqrt[6]) (Subscript[a, 0]^(5/2)) 
                      (E^(((r Z)/(2 (Subscript[a, 0])))))))})

Conclusions:

Mathematica has proven an efficient and powerful tool for teaching,  has been successfully incorporated into the general physics curriculum, and  is also being adopted by others for use in their courses. The Mathematica program gave the SMU students access to a powerful yet easy-to-use program to assist with their homework and help them visualize mathematical/physical concepts.  The student comments are favorable; the curriculum is being refined, and will continue to be used in the future.

References:

Given the space limitations, I have presented only a few examples. Many of these examples were taken from the text Mathematica for Physics. Much of this material, as well as sample homework assignments from various courses, is available on the web at:   www.physics.smu.edu/~olness. 

There is also a notebook of collected materials on display at this conference.

Student Comments: