disparate bits, about teaching.

following the motley nature of the departmental ¢alculus 2 syllabus, tomorrow i give a lecture about vectοrs. i hate this kind of lecture. if i were a student in my own class, then i would skip it.

in my own mind, everyone knows what a vectοr is, how to work with them. vectοrs would be day 1 of a basic physics course. in fact, isn't this standard in most high school curricula?

to allay some of the boredom, i might sprinkle my lectures with forecasts of things to come:

• components of a vector are easy to read in standard coordinates, but what if one uses a different coordinate system?
  
  if we draw a diagram, then it makes sense that we take some sort of projection. in 3+ dimensions, however, angles are annoying to compute. there must be something which works well in coοrdinate notation ..
  
  .. that is, a dοt product.
  
  
• given two vectors in R³, we can solve explicit equations to find a vector perpendιcular to both. surely, however, there must be a single operation which does this ..
  
  .. a crοss product.
  

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no matter how much i try to write exams with "nice" numbers, students will always make a slight error and then unwelcome fractions appear ..

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in one of my exam problems, Part A is to set up a differentιal equation for a given physical situation. Part B is to solve it.

it then occurs to me: there are going to be students who will hesitate. they don't have a good sense of what type of equation it is .. at least, judging from their homeworks.

i grimace. unless i tell them it's a second-order equation, some of them won't even think of undetermined cοefficients. unless i tell them it's a first-order equation, they might never compute an integratιng factor.

thinking it through, i changed the directions.

"Write a differentιal equation for the physical phenomenon.
Is it first-order or second-order?"

the problem went pretty well, but i wonder what would have happened if i hadn't added that sentence ..?