Mathematica Lab • Identities & Mini Research

From exercises to experiments

This page treats Mathematica as a research assistant. The aim is to help NAMSSN UI students turn curiosity about integrals, series, and identities into well-structured mini projects.

Project pattern 1

Pattern hunting with tables

Many projects start with a simple question: “What happens if I vary this index?” Use tables to generate data and search for patterns before touching proofs.

(* Example: integrals of x^n on [0,1] *)

Clear[I];
I[n_] := Integrate[x^n, {x, 0, 1}]

(* Generate a table *)
Table[{n, I[n]}, {n, 0, 8}]
            

In your notebook, add a subheading “Conjecture” and describe the pattern you see in words and symbols.

Extend this pattern

• Replace x^n with x^n Log[x] and repeat the table.
• Try different intervals, such as {x, 0, 2}.
• Record which choices lead to simple formulas and which do not.

🧠Level: 300–400 🕒Half-day project 📊Good first mini project

Project pattern 2

Designing an integral family notebook

The ODE–Integration Bee and many advanced exercises involve families of integrals depending on parameters. Your notebook should tell the story of such a family clearly.

(* Template: integral family notebook *)

(* Title: A family of Gaussian-type integrals *)

(* 1. Define the integrand with parameters a and b. *)
Clear[f];
f[a_, b_, x_] := Exp[-a x^2] Cos[b x];

(* 2. Define the integral as a function of a and b. *)
Clear[I];
I[a_, b_] := Integrate[f[a, b, x], {x, 0, Infinity}]

(* 3. Compute special cases. *)
Table[{a, b, I[a, b]}, {a, 1, 3}, {b, 0, 2}]
            

Separate your notebook into clearly-labelled sections:

• “Definition of the family”.
• “Sample values and patterns”.
• “Conjectures and open questions”.

Whenever Mathematica returns an expression with special functions, add a short comment: “How does this relate to what I know from analysis or special functions?”

📐Connects to analysis & transforms 🐝Supports Integration Bee solutions

Project pattern 3

Identity notebooks: guess, test, refine

An identity notebook is a controlled environment where you test formulas before you attempt a proof. The process is: guess → numeric test → simplify → refine.

(* Example: trigonometric identity *)

lhs[x_] := Sin[x]^2 + Cos[x]^2;
rhs[x_] := 1;

(* 1. Simplify the difference symbolically. *)
Simplify[lhs[x] - rhs[x]]

(* 2. Numeric check at sample points. *)
Table[{t, N[lhs[t]], N[rhs[t]]}, {t, 0, 2 Pi, Pi/6}]
            

Replace this with identities from your analysis or algebra courses: product-to-sum, partial fraction decompositions, or special cases of more advanced transforms.

Structure of an identity notebook

• One section per identity or family of identities.
• A clear statement in words and symbols at the top.
• A symbolic block (Simplify or FullSimplify).
• A numeric testing block with a table of values.

Project pattern 4

ODEs, recurrences, and visual comparison

Mathematica can solve simple ODEs and recurrences symbolically and numerically. This is useful for comparing exact and approximate solutions.

(* ODE: y' = y, y(0) = 1 *)

sol = DSolve[{y'[x] == y[x], y[0] == 1}, y, x];
ySol[x_] = y[x] /. First[sol];

(* Numeric solution on [0, 4] *)
numSol = NDSolve[{y'[x] == y[x], y[0] == 1}, y, {x, 0, 4}];

(* Plot both for comparison *)
Plot[
  {ySol[x], y[x] /. First[numSol]},
  {x, 0, 4},
  PlotLegends -> {"Exact", "Numeric"}
]
            

Now replace this with ODEs or recurrences from your course notes and see how symbolic and numeric solutions behave on the same graph.

📈For ODE and numerical analysis 🕒Weekend exploration

How to turn this into a serious project

A good Mathematica project does not have to be huge. It has to be well organised and honest about what you know and what you are still guessing.

Notebook skeleton

(* Project title: ___________________________________ *)
(* Name: ______________   Course / context: _________ *)
(* Date started: _______  Date last updated: ________ *)

(* 1. Introduction: 3–6 sentences about the question. *)

(* 2. Data section: tables, plots, and numerical experiments. *)

(* 3. Conjectures: clean statements of what seems to be true. *)

(* 4. Links: which course concepts or problems does this touch? *)

(* 5. Open questions: what you still do not understand. *)

Project checklist

The notebook has a clear title and introduction at the top.
Every significant block of code has a comment explaining its purpose.
Numerical checks are used to support (not replace) reasoning.
Plots are labelled well enough that another student can read them.
Conjectures are separated from known results.

Connecting to LaTeX & presentations

When a project stabilises, you may want to write it up in LaTeX or present it at a NAMSSN event. The Mathematica Lab and LaTeX Lab are meant to work together:

Exporting expressions

Use TeXForm[expr] in Mathematica to obtain a LaTeX representation of an expression. Paste that into the LaTeX Lab editor to see how it looks in a proper document.

Slides & posters

For mini-seminars or Bee-related events, copy only the most informative plots and tables, and summarise your discoveries in 3–5 key points on slides or posters.

Over time, NAMSSN UI can collect the strongest student notebooks and use them as reference projects for future generations — a living archive of “Mathematica for serious undergraduates”.