Monday, October 11, 2010

"well, i've been afraid of changes .."

an excerpt from 19 sept 2010:

awesome: the new edition of the stεwart calcμlus textbook does multιvariate limits with δ's and ε's!

argh, never mind. this is an acquired taste.

my students don't understand the significance of ε and δ at all: i might as well have writ arcane sigils on the chalkboard, as if committing some black magic.

come to think of it, it took my analysis students quite a while to get the hang of limits, without the limit notation.
i even tried to draw a diagram, labeling the difference in height as ε and the difference in "horizontal distance" as δ.

i fathom that the students must have been wondering why i didn't write Δx's, Δy's, and Δz's.
[sighs]

next time, i'll try and formulate these limit questions (at least, when the limit exists) as some kind of approximation problem:
example: consider the function
$$f(x,y) = \left\{\begin{array}{rl} \frac{xy^3}{x^2+y^4}, & (x,y) \neq (0,0), \\ 0, &(x,y)= (0,0). \end{array}\right.$$ if $f$ is continuous at $(0,0)$, then how close must $(x,y)$ be to $(0,0)$ so that $f(x,y)$ is within a distance of $0.001$ from $f(0,0)=0$?
as you can see, it's an ε-δ computation with a fixed ε=0.001. (from experience, students favor explicit numbers over abstractions, any day.)

what does it matter, anyway? for engineers that have no need for mathematιcal anaΙysis in their careers, this could be the form of the problem that is the most useful to them.