# Math 21a: section knill Fall 2022

## 21a Fall 2022

# Open AI I, See part II and See part III

In 2004, a few students have experimented with me building chat bots who can understand mathematics in in 2004. It was hard already just to feed in all the math information. Of course this is better in a large scale effort like Open AI. Things start to be impressive. I just tried out Open AI. Here are some questions and answers from the bot:------------------------------------------------------------------- Why is multivariable calculus important? Multivariable calculus is important because it allows us to analyze and solve problems involving functions of multiple variables. It is used in many areas of mathematics, science, and engineering, including physics, economics, and computer science. It is also used to solve problems in areas such as optimization, probability, and statistics. Multivariable calculus is essential for understanding the behavior of complex systems and for making predictions about the future. ------------------------------------------------------------------- Find the cross product of (3,4,5) and (3,4,9) (-27, 18, -9) ------------------------------------------------------------------- Find the maximum of f(x,y) = x^2 y + y x^2 The maximum of f(x,y) is unbounded. ------------------------------------------------------------------- Find the maximum of f(x,y) = x^2 -y^2 under the constraint x^2+y^2=1 The maximum of f(x,y) = x^2 -y^2 under the constraint x^2+y^2=1 is 1. This can be seen by setting x=1 and y=0, which satisfies the constraint and gives a maximum value of f(x,y) = 1. ------------------------------------------------------------------- Find the three dimensional cube of maximal volume with fixed surface area 1 The three dimensional cube of maximal volume with fixed surface area 1 is a cube with side length 0.5. This is wrong. It holds for 2 dimensional cube only. ------------------------------------------------------------------- What does Stokes theorem tell? Stokes' theorem is a statement about the integration of differential forms on manifolds, which can be informally thought of as a higher-dimensional generalization of the fundamental theorem of calculus. It states that the integral of a differential form over the boundary of some orientable manifold is equal to the integral of its exterior derivative over the whole manifold. ------------------------------------------------------------------- Tell me a joke about Stokes theorem Q: What did Stokes say when he proved his theorem? A: "I've got a theorem!" ------------------------------------------------------------------- Who found Green's theorem Green's theorem was discovered by George Green in 1828. ------------------------------------------------------------------- In which direction does the gradient point? The gradient points in the direction of the steepest increase in the function. ------------------------------------------------------------------- Tell me a joke about the gradient Q: What did the gradient say when it was feeling down? A: I'm just a little blue. ------------------------------------------------------------------- Tell me a joke about multivariable calculus Q: What did the calculus student say when they got stuck on a multivariable problem? A: I'm in way over my head! ------------------------------------------------------------------- Why is the gradient perpendicular to the level surface? The gradient of a function is a vector that points in the direction of the greatest rate of increase of the function. The level surface of a function is the set of points where the function has the same value. Since the gradient points in the direction of the greatest rate of increase, it is perpendicular to the level surface, since the level surface has no rate of increase. ------------------------------------------------------------------ Write me a Mathematica program listing the first 1000 primes isPrime[n_]:= Module[{i}, For[i = 2, i <= Sqrt[n], i++,If[Mod[n,i]== 0,Return[False]]];True] s={}; n=2; While[Length[s]<1000,If[isPrime[n],s=Append[s,n]];n++]; s This can be done faster like Map[Prime,Range[1000]] ------------------------------------------------------------------