Monday, February 26, 2007

class summaries: February 26

Calculus III: slopes, second derivatives, and areas under parametrized curves. the area under one arch of a cycloid is three times the area of the rolling circle!

topology: introduced oriented manifolds: sphere good, Mobius band bad!

Friday, February 23, 2007

Class summaries: Feb. 23

Calculus III: finally back to calculus as you know it. introduced parametrized curves--section 11.1 in the text.

topology: finished proof of homogeneity lemma and proved that the mod 2 degree of a function is a homotopy invariant.

Wednesday, February 21, 2007

class summaries: February 21

Calculus III: visited the computer lab in A-14 to discuss how to use Mathematica. the first project is due next Friday, March 2. project 2 will be due on March 30. the notebooks may be downloaded here. quiz on Taylor series (sections 12.10 and 12.12) on Friday.


Topology: smooth homotopy and homogeneity.

Monday, February 19, 2007

Class summaries: February 19

Calculus III: Finished up Taylor series. Talked about exponentiation of complex numbers. From now on, this course will look more like the calculus you're used to.


Topology: Finished the proof of Sard's theorem and began talking about smooth homotopy of functions.

Saturday, February 17, 2007

Wednesday, February 14, 2007

Class summaries: February 14

Calculus III: We talked about the Taylor polynomials associated to a function
f(x) and discussed Taylor's formula: f(x)= Pn(x) + Rn(x), where Rn(x) is an explicitly defined error term. This is amazing since it allows us to use a polynomial (these don't suck, remember) to approximate any function we like, and we can even put a bound on the error associated with doing this. Nice!


Topology: got two-thirds of the way through the proof of Sard's theorem: the set of critical values of a smooth function has measure zero.

Monday, February 12, 2007

Class summaries: Feb. 12

Calculus III: we explore the well-known fact that most functions suck. to deal with this, we start with linear and quadratic approximations to the function in question, and then move on to Taylor series. the possibilities are endless.


Topology: classification of smooth 1-manifolds. everything's a circle or an interval!

Sunday, February 11, 2007

the revolution will not be televised

welcome to "Ask Dr. K", a place to talk about math and anything else that might come up.

for my students:

watch here for class information, daily summaries, etc. you may post questions and comments here and i'll try to answer them.

let's get rockin'!