Part IV (14) Advanced Calculus This topic is also part of calculus but concentrates on surfaces, volumes, and maybe the math of flows of heat, water, etc. Co

Part IV (14) Advanced Calculus This topic is also part of calculus but concentrates on surfaces, volumes, and maybe the math of flows of heat, water, etc. Commonly this part of calculus makes heavy use of vectors and matrices. Also covered can be Fourier series and integrals. A pure tone in music, say, from an organ, has a fundamental frequency, say, 440 cycles per second and also overtones at natural number multiples of that fundamental frequency. Each of the overtone frequenciess works like an axis of orthogonal coordinates and is a terrific way to analyze or synthesize such a tone. The Fourier transform is closely related and has an orthogonal coordinate axis for each real number. Engineering is awash in linear systems that do not change their properties over time -- time invariant linear systems. Such systems merely adjust some Fourier coefficients. This is a powerful result. (15) Measure Theory This topic develops integral calculus with a lot more generality. (16) Probability Theory In probability we imagine that we do some experimental trials and observe the results. We have a sample space, our set of trials. We regard the set of all trials like a region with area, and we say that the area is 1. Maybe we flip a coin and get Heads. The set of all trials that yield Heads is one event Heads. The event Heads has area, probability P(Heads) With a fair coin we have P(Heads) = 1/2 A random variable X is some outcome of a trial. Usually X is a number. In this case we can write P(X >= 2) for the probability that random variable X is >= 2. Events A and B are independent provided P(A and B) = P(A)P(B) In probability we can have algebraic expressions with random variables, have a distance between random variables, have a sequence of random variables converge to a random variable, etc. Given a real valued random variable X, for real number x we can let F_X(x) = P(X <= x) Then F_X is the cumulative distribution of X, and f_X(x) = d/dx F_X(x) is the probability density of X. One of the major results is the central limit theorem that shows a distribution converging to the Gaussian bell curve. Two more results are the weak and strong laws of large numbers that show convergence to average values (means, expectations). One more result, amazing, is the martingale convergence theorem. The Radon-Nikodym theorem of measure theory provides an amazing general theory of conditional expectation which is one way to look at the value of information. In statistics we are given the values of some random variables and try to infer the values of something related. The more serious work in probability and statistics is done on the foundation of measure theory. (17) Optimization We have a furniture factory and a new shipment of wood. We have some freedom in what we do with this wood. An optimization problem is how to use the freedom to make the greatest revenue from the wood. More generally we have some work to do with some freedom in how we do the work, and an optimization problem is how best to exploit the freedom to have the best outcome, profit, for the work. If the profit is linear and if the work and freedom are described by linear equations, then we have the topic of linear programming. Linear programming has played a role in the Nobel prizes in economics. The main means of solving linear programming problems is the simplex algorithm which is a not very large modification of Gauss elimination. In practice the simplex algorithm is shockingly fast; in theory its worst case performance is exponential. In the furniture making, we can't sell a fraction, say, 1/3rd, of a dining room chair. So, we want our solution to consist of integers. The simplex algorithm does not promise us integers. So we have a problem in integer linear programming. In practice from the simplex algorithm we may get to make 601 + 1/3rd dining room chairs and round that to 601 or just 600. No biggie. If we insist on integers, then, surprisingly, even shockingly, we have one of the early examples of a problem in NP complete. In practice there are still ways to proceed; often we can get the optimal integer solutions we want easily; sometimes such solutions are challenging; sometimes we have to settle for approximations; often the approximations can be quite good, like forgetting about 1/3rd of a chair; in theory the ways have worst case performance that is exponential and nearly useless. One integer linear programming problem I attacked has 600,000 variables and 40,000 constraints. I wrote some software that in 600 seconds on a slow computer found an integer solution within 0.025% of optimality. The solution made heavy use of the simplex algorithm, non-linear duality theory, and something called Lagrangian relaxation. In addition to optimization, the simplex algorithm and related math make nice contributions to geometry, e.g., prove theorems of the alternative used in proving the famous Kuhn-Tucker conditions necessary for optimality. Since it is possible that the work we are trying to optimize is complicated, its mathematical description may also be complicated and the optimization effort, difficult. Optimization problems are common in animal feeding, cracking crude oil, transportation problems, and much more. Game theory can be regarded as a special case. One proof of the famous Nash result (featured in the movie about Nash, A Beautiful Mind) is via linear programming. Some problems are to find curves, say, how to climb, cruise, and descend an airplane to minimize fuel burned or the best curve for a missile to attack an enemy fighter jet. Newton found the brachistochrone curve solution to the problem of the frictionless curve that would let a bead slide down in minimum time. Relevant fields are calculus of variations and deterministic optimal control. One famous result is the Pontryagin maximum principle. Some problems have uncertainty; so the optimization may be to find the solution with the best expected outcome. For such problems there is the field of stochastic optimal control.