This site contains: mathematics courses and book; covers: image analysis, data analysis, and discrete modelling; provides: image analysis software. Created and run by Peter Saveliev.

From Intelligent Perception

I would like to come up with an outline of what undergraduate mathematics curriculum ought to be. I want to go wild and ignore the reality: prerequisites, "service courses", overlapping degree requirements, transfer students, high school curriculum, SATs, GREs, etc.

It's similar to baseball, football fantasy leagues. One puts the existing players at the right positions to make the best possible team. Sadly, the team will never play...

1 Discrete functions

Consider the fact that nowadays data comes in the digital format. This data is simply large tables of numbers. Just two examples:

• experimental data: thousands of experiments each producing a list of hundreds of measurements,
• digital images: 2d and 3d arrays of pixels each with a different vector value.

Sometimes there is a continuous function or process behind the numbers but often there isn't. The issues one has to deal with are the same though: max/min, increasing/decreasing behavior, rate of change, lengths, areas, volumes, etc.

What is commonly done is to go back to continuous functions via approximation, interpolation, curve fitting, etc. This approach is not the best solution as it introduces errors in your analysis (image approximate conservation of energy!). It also requires even more advanced math. Meanwhile, a typical graduate has never seen discrete functions and now at the workplace he doesn't connect what he does with Excel every day with what he learned in his calculus class.

The bottom line: the numerical/computational aspect should be built in!

How to do this? The answer is known: introduce discrete differential forms and discrete exterior calculus.

I have been trying to introduce discrete forms in a gentle manner in some of my courses:

1. Differential Geometry,
2. ODEs and PDEs,
3. Calculus 1-3 and its prerequisites.

There is a lot of literature on discretization methods in modeling and simulation but it's quite complex. Waiting for these ideas to trickle down to undergraduate courses might take 20 years.

These have always had their discrete parts integrated from the beginning:

• Probability/Statistics and
• Algebraic Topology

With the latter it still makes a lot of sense to initially separate point-set topology from algebraic topology, as in Homology of cell complexes: course.

2 Less emphasis on closed-form formulas

Once again, data comes in the digital format. Since there is no formula that produces this data, one has to deal with it as just function. So, surprisingly, taking the applied approach means more abstraction not less: more $y = f(x)$ than $y = x^2$.

To make room for this new stuff, one has to cut something out. Fortunately, one can drop a lot of

• information about specific functions (polynomials, exponential, logarithm, etc),
• trigonometry (if I see secant one more time! The point is that I don't see it, so why teach it?),
• algebraic tricks for solving equations, manipulation, and simplification: rationalization, partial fractions etc,
• integration techniques:
  • integration by substitution is important for a good reason -- it's about change of variables; but
  • figuring out which technique to use to evaluate a particular integral (trig substitution vs integration by parts vs algebra etc) is so 19th century.

3 Proofs

Teaching proofs to students who aren't math majors is a lost cause.

Proofs still make sense when they are instructive.

Students still need more math. More math not more proofs. Consider calculus. So many new concepts but most proofs are beyond reach.

Also fewer definitions. Don't introduce a new one unless you are to use it in other courses.

4 Breadth first and bottom up

Start calculus with sequences. Compared to the epsilon-delta, I find this approach much more intuitive. Another advantage is that you can deal with series later, after you've considered convergence of functions, based on sequences. Then, you can focus on the important stuff, such as Taylor and Fourier series etc.

I have never taught this way yet so it's hard to be specific yet. But I did learn calculus this way, back in Russia years ago. I even remember the professor, Vladimir Zorich. His textbook, Mathematical Analysis, is available on Amazon. It's fairly advanced and it teaches both sequences and epsilon-delta in parallel...

Introduce functions of several variables early. Do you see a lot of functions of single variable around? Waiting until calc 3 is unwise (some never get there) and unnecessary. Partial derivatives are manageable.

More bottom up. Removing structures from consideration is hard; adding is much easier. Compare how you start with point-set topology -- by removing the geometry from the Euclidean space -- and how you start with inner product spaces -- by adding this structure to vector spaces.

See also this typical Math major.

5 Discrete calculus as a subset of calculus

...on the pre-calculus level.

Quote: "Calculus Without Limits ... deliberately minimizes the use of limits, one of the major stumbling blocks initially standing in the way of calculus students." There are no limits in discrete calculus!

1. Derivatives.

Functions and forms. The velocity problem. Change and the exterior derivative. The product rule. The chain rule. Implicit differentiation. Rates of change in the natural and social sciences. Derivatives of exponential functions. Exponential growth and decay. Related rates. Maximum and minimum values. How derivatives affect the shape of a graph. Optimization. Anti-derivatives.

2. Integrals.

Areas and distances. The definite integral. The fundamental theorem of calculus. Indefinite integrals and the net change. Integration by substitution. Areas between curves. Volume. Work. Average value of a function. Applications to physics and engineering. Applications to economics and biology. Probability.

3. Parametric curves and differential equations.

Curves defined by parametric curves. Modeling with differential equations. Separable equations. Models for population growth. Linear equations.

4. Vector functions.

Three-dimensional coordinate systems. Vectors. The dot product. The cross product. Equations of lines and planes. Vector functions and space curves. Derivatives and integrals of vector functions. Curvature. Motion in space: velocity and acceleration.

5. Calculus of multiple variables.

Functions of several variables. Partial derivatives. The chain rule. Directional derivatives and the gradient. Maximum and minimum values. Double integrals. Iterated integrals. Applications of double integrals. Triple integrals.

6. Vector calculus.

Vector fields. Line integrals. The fundamental theorem of calculus for line integrals. Green's theorem. Curl and divergence. Surface integrals. Stokes' theorem. The divergence theorem.

7. What's next?

$\pi =4$ and what to do about it.