Harvard Multivariable Calculus @math21a
Harvard Multivariable Calculus Tweets. Also for the Summer course. And Math 22 math.harvard.edu/~knill Harvard University Cambridge Joined August 2009-
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Welcome to Math S 21a of Summer 2021. A welcome video: youtube.com/watch?v=iTr8xP… The website is here: people.math.harvard.edu/~knill/teachin…
Welcome to Math S 21a, Summer 2020. The course website is here: people.math.harvard.edu/~knill/teachin… Video parts are here: youtube.com/playlist?list=… Tomorrow, is the first class. I'm excited. You will also see Anna, Elliot, Travis, Michael and Jesus who teach with me this summer.
Welcome to Harvard Mulltivariable 2019! The website is 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.
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!
Welcome to the Harvard Summer School Math S 21a course! The course website is here: math.harvard.edu/~knill/teachin… 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 ... )
Clairaut counter example animation: math.harvard.edu/~knill/teachin… The functions which are given in polar coordinates r<sup>2</sup> sin(2t) or r<sup>2</sup> sin(4t) appear to be similar, but while for the first, the Clairaut theorem holds, the second one is the standard counter example.
This semester, I teach Math 22a, which is new in the calculus mix this semester. The website is :math.harvard.edu/~knill/teachin… 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.
In the final exam: math.harvard.edu/~knill/teachin… (PDF) we had a Lagrange problem with some l<sup>4</sup> 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. math.harvard.edu/~knill/teachin…
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: math.harvard.edu/~knill/teachin… Enjoy.
Mathematica code proving the equivalence of two formulas for the curve curvature: r[t_]:={x[t],y[t],z[t]}; s[t_]:=Sqrt[r'[t].r'[t]]; T[t_]:=r'[t]/s[t]; K1[t_]:= Sqrt[T'[t].T'[t]]/s[t]; v[t_]:=Cross[r'[t],r''[t]]; K2[t_]:= Sqrt[v[t].v[t]]/s[t]^3; FullSimplify[K1[t]^2==K2[t]^2]
Welcome to Harvard Math S 21a! The course site is on: math.harvard.edu/~knill/teachin… On the handouts page, you find the handouts from the first week: like a summary of all lectures: math.harvard.edu/~knill/teachin… or some relation between discrete and continuum calculus math.harvard.edu/~knill/teachin…
In the Archimedes Revenge problem (see problem 13 in math.harvard.edu/~knill/teachin…), 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: math.harvard.edu/~knill/teachin… Congratulations!
A selection of Mathematica projects done this semester: math.harvard.edu/archive/21a_fa… great job, students of Math 21a, Fall 2017 ! More exhibits: math.harvard.edu/archive/21a_fa…
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! sites.fas.harvard.edu/~math21a/exhib…
Some class submissions of the optional tangent art problem in HW 14. Only a small sample sites.fas.harvard.edu/~math21a/exhib…
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) sites.fas.harvard.edu/~math21a/exhib…
A large resolution rendering from an illustration used for the first midterm. It was rendered in Mathematica using a NASA model of Cassini.
Become a good friend of the conic cat featuring a parabola, a circle (a special ellipse), a hyperbola and a singular conic, the whiskers !
Check out the exhibit page math.harvard.edu/~knill/teachin… of this summer. This is the end of the course.
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 goo.gl/r48Xxz
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