Harvard Multivariable Calculus

Welcome to Math S 21a, Summer 2020. The course website is here: http://people.math.harvard.edu/~knill/teaching/summer2020/index.html … Video parts are here: https://www.youtube.com/playlist?list=PLlBvCN3MG_bamPmQyD_P8naVsPp__j7cM … Tomorrow, is the first class. I'm excited. You will also see Anna, Elliot, Travis, Michael and Jesus who teach with me this summer.pic.twitter.com/Vg7CJkJpjb

Welcome to Harvard Mulltivariable 2019! The website is http://sites.fas.harvard.edu/~math21a/  On Wednesday, September 4th, in Science Center B, there is the intro meeting for the course. Bright and early at 8:15 AM.pic.twitter.com/0M3VRwATSG

Multivariable Summer School is going to an end. Have a great rest of the summer and keep in touch! The following picture was taken before the final exam. Last push!pic.twitter.com/lxXn7pNgSb

Welcome to the Harvard Summer School Math S 21a course! The course website is here: http://www.math.harvard.edu/~knill/teaching/summer2019/index.html … First class is on Tuesday morning bright and early at 8:30 AM in Hall E. See you then in the Red Room. (RedRum, RedRum ... )pic.twitter.com/8lUGV82H6X

Clairaut counter example animation: http://www.math.harvard.edu/~knill/teaching/math22a2018/exhibits/clairaut/index.html … The functions which are given in polar coordinates r² sin(2t) or r² sin(4t) appear to be similar, but while for the first, the Clairaut theorem holds, the second one is the standard counter example.pic.twitter.com/w3N3djSY5T

This semester, I teach Math 22a, which is new in the calculus mix this semester. The website is :http://www.math.harvard.edu/~knill/teaching/math22a2018/index.html … 22a should have the same workload than 21a. What is different?: A) Early links between linear algebra and calculus B) more mathematical C) introduces proofs.pic.twitter.com/fXPQOzmyB7

In the final exam: http://www.math.harvard.edu/~knill/teaching/summer2018/final/final.pdf … (PDF) we had a Lagrange problem with some l⁴ curve. It is an exciting problem to understand such convex billiards. One currently does not know how to prove that the Kolmogorov-Sinai entropy is positive. http://www.math.harvard.edu/~knill/teaching/summer2018/exhibits/lp …pic.twitter.com/2OpVMSu2Eg

Some of the Mathematica projects of the Math S 21a Harvard Summer school class. As usual in the summer (covering in 6 weeks all multivariable), there was little time for this project. But the class did nevertheless did produce some nice things: http://www.math.harvard.edu/~knill/teaching/summer2018/exhibits/exhibits … Enjoy.pic.twitter.com/TgMTOHtNiQ

Welcome to Harvard Math S 21a! The course site is on: http://www.math.harvard.edu/~knill/teaching/summer2018 … On the handouts page, you find the handouts from the first week: like a summary of all lectures: http://www.math.harvard.edu/~knill/teaching/summer2018/handouts/00-introduction.pdf … or some relation between discrete and continuum calculus http://www.math.harvard.edu/~knill/teaching/summer2018/handouts/00-quantum.pdf …pic.twitter.com/GUc0X1mqFy

In the Archimedes Revenge problem (see problem 13 in http://www.math.harvard.edu/~knill/teaching/summer2017/final/final.pdf …), a gift card was promised for students who can solve it. There has been one student of that class Runze Li, who solved it correctly until December 31th 2017: http://www.math.harvard.edu/~knill/teaching/summer2017/exhibits/revenge/index.html … Congratulations!pic.twitter.com/DBsfGkH0t5

A selection of Mathematica projects done this semester: http://www.math.harvard.edu/archive/21a_fall_17/exhibits/projects/index.html … great job, students of Math 21a, Fall 2017 ! More exhibits: http://www.math.harvard.edu/archive/21a_fall_17/exhibits.html …pic.twitter.com/Vp9yh9EkpI

The sculpture has appeared in our final exam. It is a death star, neutralized by a cone! The artist has called it life force II. May the force be with us! http://sites.fas.harvard.edu/~math21a/exhibits/lifeforce/index.html …pic.twitter.com/uE8cbRSTP4

Some class submissions of the optional tangent art problem in HW 14. Only a small sample http://sites.fas.harvard.edu/~math21a/exhibits/tangent/index.html …pic.twitter.com/RasPGsapQ6

Given two lines r(t) = P + t v and s(t) = Q + t w. Is the midpoint (P+Q)/2 equidistant from the lines?(TF20 problem) http://sites.fas.harvard.edu/~math21a/exhibits/tf …pic.twitter.com/g6lrMq9gjr

A large resolution rendering from an illustration used for the first midterm. It was rendered in Mathematica using a NASA model of Cassini.pic.twitter.com/SVmp6qfntK

Become a good friend of the conic cat featuring a parabola, a circle (a special ellipse), a hyperbola and a singular conic, the whiskers !pic.twitter.com/OeHbOjw301

Check out the exhibit page http://www.math.harvard.edu/~knill/teaching/summer2017/exhibits.html … of this summer. This is the end of the course.pic.twitter.com/9DUWBkYtE0

Archimedes Revenge: verify that the volume of the solid x^2+y^2-z^2<1, y^2+z^2-x^2<1, z^2+x^2-y^2<1 is log(256). See http://goo.gl/r48Xxz pic.twitter.com/gO6o5Pr058