i'm back

hi. it's been a while, but i thought it was time to start posting again. i'm reading Steven Pinker's new book The Stuff of Thought, and he mentions the classic short film by Charles and Ray Eames, Powers of Ten. here it is for your viewing pleasure

class summaries, June 4 and 5

welcome to number theory! so far we've covered the integers, induction, fibonacci numbers, the division algorithm, and representations of integers in various bases.

hiatus

no posts for a while. i'm enjoying May, catching up on some yard work, and trying to finish off that paper on multi-dimensional persistence. summer school starts June 5; see you then.

requiem, part 2

as a proud alumnus of Virginia Tech, i've spent the last two days in a state of utter dismay. my heart hurts. i'm horrified beyond belief. please say a little prayer for the victims and their families.

requiem

R.I.P. K.V.

so it goes.

class summaries: April 4

calc III: vector-valued functions and space curves.

topology: cellular homology.

class summaries: April 2

calc III: quadric surfaces, cylindrical and spherical coordinates.

topology: adjunction spaces and cell complexes.

class summaries: March 30

calc III: lines and planes.

topology: homotopy invariance of homology.

class summaries: March 28

calc III: cross products. long live torque!

topology: the homology of a point, the 0-th homology group. algebra rules!

class summaries: March 26

calc III: vectors and dot products.

topology: introduced singular homology.

class summaries: March 23

calc III: 3-d coordinate systems; handed back tests.

topology: finished the proof of the Poincare-Hopf Theorem. next up: singular homology!

class summaries: March 21

calc III: test.

topology: more of the proof of the Poincare-Hopf theorem. will finish this next time.

class summaries: March 19

well, spring break is over and it's back to the grind.

Calc III: review for test #2. remember, it's on Wednesday, March 21.

Topology: trying to prove the Poincare-Hopf Theorem. lots of technicalities. nondegenerate zeros are better, so we'll try to reduce to that.

class summaries: March 7

calculus III: areas and lengths of polar curves. the cardioid has length 8(!).

topology: the index of a vector field on a manifold.

class summaries: March 5

calculus III: polar graphs, etc. almost done with chapter 11!

topology: tangent vector fields on manifolds; there's a nowhere-vanishing vector field on the n-sphere if and only if n is odd.

Colbert on topology

Let's start a petition for Colbert to win the Fields Medal at the 2010 ICM!

Click here.

class summaries: March 2

calculus III: introduced polar coordinates. graphed some stuff.

topology: proved that the Brouwer degree is a smooth homotopy invariant.

class summaries: February 28

calculus III: lengths of parametrized curves; surface area of solids of revolution

topology: more about oriented manifolds; Brouwer degree

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!

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.

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.

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.

new blog to visit

visit Crooked Letter Blog now! (or else!)

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)= P_n(x) + R_n(x), where R_n(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.

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!

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'!