Part II Again, for variable t, think "There is a number; call it t". That approach to the meaning of a variable works well enough for essentially all of

Part II Again, for variable t, think "There is a number; call it t". That approach to the meaning of a variable works well enough for essentially all of math. Calculus was invented mostly by I. Newton, mostly for physics, especially for explaining the motions of the planets. Since then calculus has become a pillar of civilization, especially Western Civilization. Mostly in calculus, integration is finding the area (or more carefully defining the area) under a curve, maybe a parabola. But can also have a line integral, say, the work do when carrying 100 pounds of hay to the top of the hay loft of the barn. Here we integrate from beginning to end of the work. If we let the hay fall, then neglecting friction, etc. get the work back as energy. So, have to pay attention to the direction of the integration, from the ground up to the hay loft or from the hay loft back to the ground. The two values have opposite signs. You can teach yourself calculus: Just get 1-4, at least one good calculus texts and dig in -- read the chapters, follow the material, work nearly all the more difficult exercises, and maybe check 1-4 times a year with a good mathematician. I did that for freshman college calculus (but didn't check with anyone). I never took the course and, instead, started on sophomore calculus and made As. No problem. (3) Linear Algebra. Likely the next course would be linear algebra. It would be good to do this subject 2-4 times from more elementary treatments to some of the more advanced material. Some of the more advanced material can cover linear programming optimization, classic two person game theory, the proof of the Nash result in game theory, some treatments of the fast Fourier transform in digital signal processing, group representations in the quantum mechanics of molecular spectroscopy, error correcting codes in algebraic coding theory, numerical methods, and more. The start of linear algebra is just one or several linear equations in one or several variables. E.g., for two equations in three variables, unknowns: 2x + 3y - 6z = 0 5x - 2y + 7z = 3 For positive integers (whole numbers) m and n, given m such linear equations in n variables (unknowns), the number of solutions, depending also on the numerical constants in the equations, is none, one, or infinitely many. The main way to find all the solutions is just Gauss elimination. That covers a good chunk of a first course in linear algebra. The notation 2x + 3y - 6z = 0 5x - 2y + 7z = 3 gets to be clumsy so we write instead 2 + 3 - 6 5 - 2 + 7 write that with big, square brackets, and call it a matrix with two rows and three columns. The individual numbers are components. Of course, a matrix is conceptually close to an array with two subscripts common in programming languages. Maybe we call the matrix 2 + 3 - 6 5 - 2 + 7 A. Since matrix A has 2 rows and 3 columns, we say that it is 2 x 3 ("2 by 3"). Then we write the x, y, z in a column x y z apply the big, square brackets, and call it a matrix with 3 rows and 1 column and maybe call it v. We write the right side 0 3 also with brackets, and maybe call it b. Then we define a product, a matrix product Av so that Av = b means just the same as 2x + 3y - 6z = 0 5x - 2y + 7z = 3 From then on we work with matrices and try to avoid notation like 2x + 3y - 6z = 0 5x - 2y + 7z = 3 In this way, a course in linear algebra is commonly called a course, Linear Algebra and Matrix Theory. A matrix with just one column is a column vector; with just one row, a row vector. In short, either is called a vector. Linearity: Given m x n matrices A and B, m x 1 matrices u, v, and real numbers c, d, we have that A(cu + dv) = cAu + dAv Here we regard Au as case of function A with argument u. That is, we could have said that matrix A defines function f so that f(u) = Au With cA + dA we form products cA and dA and sums cA + dA So, we have to define both of those: The definitions are close to just obvious. The equation A(cu + dv) = cAu + dAv means that matrix A acts like a linear function. Big deal! The real world is awash in linear behavior, and linearity is an enormously powerful property mathematically. The rest of linear algebra and matrix theory is nearly all about the consequences of such linearity. The fundamental theorem of algebra is that each (high school style) polynomial in variable x can be uniquely factored into a product of terms of the form (ax + b) where the a and b are possibly complex numbers. Essentially from this fact, the more advanced parts of linear algebra use the complex numbers and not just the real numbers. At the end of a second course in linear algebra, will pay attention to the essentially geometrical notions of length and angle and, in particular, to orthogonality, that is, perpendicular. Big topics are the Gram-Schmidt process where find some orthogonal vectors and the polar decomposition. The polar decomposition result says that a matrix acting on a circle will yield an ellipse, that is, for matrix A and vector u, if we let u take on the values of all the points on a circle, then the Au will generate all the values of all the points on an ellipse. The ellipse has two axes, and they are mutually perpendicular. So, if have the vectors of the two axes, then have enough to construct the whole ellipse. In linear algebra, this situation generalizes to any finite dimension and yields the singular value decomposition, principle components, factor analysis, and more. With more advanced work, this situation becomes the spectral theory of self-adjoint linear operatiors central to quantum mechanics. (4) The Four Main Parts of Math There are four main parts of math: (A) Foundations, that is, deep in the basement with set theory. (B) Algebra, as in high school, the integers, prime numbers, the fundamental theorem of arithmetic, greatest common divisor and least common multiple, the fundamental theorem of algebra, linear algebra and matrix theory, the generalizations of number systems such as groups, rings, fields (the rational, real, and complex numbers are all fields but so is the set of integers modulo a prime number), etc. Fermat's last theorem, settled by A. Wiles, is part of algebra; so are the deep, difficult questions about prime numbers, etc. There are applications in error correcting codes and cryptography. (C) Geometry, e.g., high school plane geometry, analytic geometry, and differential geometry, a deep subject important for relativity. (D) Analysis, as in calculus, differential equations, functional analysis, e.g., Banach and Hilbert spaces, partial differential equations, e.g., as in Maxwell's equations, fluid flow, and relativity, probability, statistics, stochastic processes, optimization, and more. In differential equations we are given an equation with the derivatives of some function and want to find the function. E.g., for real valued function y(t) and constants k and b, we might have y'(t) = k y(t) (b - y(t)) where y'(t) = d/dt y(t), the first derivative of function y(t). At one time, that little differential equation saved FedEx from going out of business. The solution is a lazy S curve and a first cut at viral growth. There is topology which is partly in geometry and analysis. The main idea of topology is continuity, that is, changing without sudden jumps or some cases of wildly fast oscillations. There is algebraic geometry partly in algebra and geometry. Much of number theory has deep connections with analysis. Much of linear algebra is an introduction to functional analysis in analysis. The older applied math is mostly analysis. (5) Analysis. Usually after linear algebra will study advanced calculus and analysis. Broadly there are two approaches, (A) theory and (B) applications. The theory is mostly to give fully careful proofs of the results and first generalizations of what you saw in calculus. For the theory maybe the most respected text is R. Rudin, Principles of Mathematical Analysis, Third Edition. For this, get the third edition and not either of the two earlier editions. So, will discover that the integral learned in calculus is called the Riemann integral because B. Riemann made the theorems solid. Rudin also does the easy generalization to the Riemann-Stieltjes integral.