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Part I I have some opinions on the question in the OP: I'm heavily self-taught from independent study in math; that study helped me with, and at times was par

Part I I have some opinions on the question in the OP: I'm heavily self-taught from independent study in math; that study helped me with, and at times was part of my good career in, applied math and computing before my Ph.D. in pure/applied math and helped a lot for my Ph.D. For a curriculum: (1) Do the standard high school math, Algebra I, Plane Geometry (based on proofs), Algebra II, Trigonometry, and Solid Geometry (based on proofs). In each of these, it is sufficient (A) to work most of the more challenging exercises and (B) try to think a little about what is going on. For (C), it would be good occasionally, say, 1-4 times a year, to chat about your progress with a good mathematician, Ph.D., likely a college prof. To find a Ph.D. who will give you an hour or so per visit, maybe do some networking. If you don't like the first choice, e.g., if they are not encouraging and helpful, then try a different Ph.D. mathematician. (2) Do Analytic Geometry and Calculus. Use a good college textbook. Since highly polished texts have been widely available for decades, there's no need to pay big bucks for new copies of the latest. So get a good used text. Better yet, get 1-4 good texts, use your favorite one as your primary source and the others for more content, alternate descriptions, evidence of what is more/less important, etc. Sets: Now in an important sense, essentially all of math is heavily about sets. For first calculus, you don't need to know much about sets; the following should be nearly sufficient: By 1900 or so, a lot of math was known, but there was a nagging, somewhat philosophical, question of what math really is. The answer of pure math was to construct a foundation in, say, the deep basement, that would answer the question but not much change the rest. The answer was to start with sets and then define everything else in terms of sets. In particular do some somewhat tricky and obscure work to define some sets that look, work, walk, talk, etc. like the numbers on the line, the real numbers. We don't think of the real numbers this way, but the effort addressed the philosophical question -- or, if I can get you to believe in sets, then I've got you for the rest of math. Much of the early work on sets was from G. Cantor. Yes, soon there were some issues from Bertrand Russell and then Kurt Gödel and, later, Paul Cohen. For a long time in math, you can regard a set as a conceptual (imaginary) collection of some kind, e.g., all the peas in a little jar in the back of the refrigerator, all the people with an iPhone, all the real numbers on the line, all the water molecules in the earth's oceans, all the parabolas, all the trials you might do in a lab experiment, and much more -- really, from the philosophical approach to what math is for all of math. Or intuitively sets are the containers for what we are thinking about. Functions: Much of math is about functions. E.g., f(x) = 2x^2 + 3x - 4 So, in pure math, the function is f. The x is the argument of the function, i.e., a variable which is essentially always in math from, "Think of something; call it x." where at least in calculus usually the thing are thinking about is a number. Or, to "solve for x", think "I suspect there is a number such that these conditions hold; I want to find if there is such a number and if so what its actual numerical value is." Now you understand what math means when it talks about variable x and function f. Then, given variable (e.g., number) x and function f, f(x) is the value of f at x. E.g., for this set stuff, a function gets defined in terms of sets, e.g., the set of all ordered pairs (x, f(x)) for x in some set called the domain of the function. In calculus, the domain is usually the set of real numbers, that is, the points on the number line (yes, that's not the tricky set theory definition of the real numbers). So, with the ordered pairs (x, f(x)) we have a very precise definition of a function in terms of sets (ordered pairs can also be defined in terms of sets). Of course, in actual work, especially in applied math, nearly no one actually thinks of a function as a set of ordered pairs. Completeness: There is an old joke, partly appropriate, that "Calculus is the elementary consequences of the completeness property of the real number system.". Intuitively completeness is, if you are converging to something by more and more accurate approximations, then there really is something there for you to converge to. This property does NOT hold for the rational numbers -- the rational numbers are all the numbers of the form (that can be written as) p/q for p and q integers (whole numbers, positive, negative, and zero) with q not zero. Why? E.g., can use the rational numbers to approximate as closely as we please the square root of 2, but the square root of 2 can't be a rational number. Why not? Because if the square root of 2 were rational, then we would also have for integers p and q, q not zero (p/q)(p/q) = 2 or (p)(p) = 2(q)(q) So the left side has some even number (possibly 0) factors of 2 while the right side has an odd number of factors of 2, and this violates the fundamental theorem of arithmetic that each integer can be factored in only one way as a product of prime numbers. Well, a big deal about the real numbers is that they are complete; i.e., intuitively, if something converges, then there is something to converge to. Completeness is a big deal because it generalizes, especially to Hilbert space that you may come to. So, in calculus often we make approximations that can become as accurate as we please, and we want the completeness property to know that what these approximations approach, limits, can exist. E.g., with the real numbers, if we make more and more accurate approximations to the square root of 2, then we know that our approximations will converge to the actual square root of 2 which DOES exist as a real number. Sets get to be quite important in math in the last years of college math and beyond. But for calculus, nearly all of that subject, especially what learn in a first calculus course, was quite solid well before G. Cantor. So, in practice, when studying first calculus, we don't see much about sets. So, for first calculus, sets remain mostly a topic in the deep basement that addresses an old philosophical issue. So, for first calculus, sets are no big issue. Analytic Geometry: Here you study the conic sections. They are a bit amazing and important in physics, mechanical engineering, and more and something definite to work with while learning calculus. So, imagine a cone, say, an ice cream cone. Take two of them that are the same and put the points together so that they are both on the same axis through the points (I'll let you get a precise definition of axis). Now imagine a sharp sword, say, as in the old John Belushi Samurai Tailor skit and use the sword to slice the cones and look at the cut edges. Right, we assume that the sword moves in a plane. Depending on where you slice, you will get (A) just a point (where the two cones touch together), (B) two lines crossed as in an X (the cross in the X is at the point where the two cones come together), (C) a circle (we are awash in circles), (D) a parabola (to a fairly good approximation, a grand slam baseball follows a parabola), (E) an ellipse (to a quite good approximation, each planet goes around the sun in an ellipse), (E) the two parts of a hyperbola (an electron shot at a negative charge will follow one of the halves of a hyperbola as it avoids the negative charge). First Calculus in a Nutshell: Then for first calculus, there are two big topics, differentiation and integration. Differentiation is finding rate of change, intuitively the slope (as in high school algebra) of a tangent line to the graph of a curve (defined in terms of a function with domain some or all of the real numbers). E.g., for time t, let d(t) be the function that gives us our distance from home at time t. Then the derivative of function d at time t is the velocity, say, function v, at time t. So, in a car, the odometer gives d and the speedometer gives v. That is, at time t we have traveled distance d(t) and are moving a velocity v(t). Here to be simple, I do not make a careful distinction between what physics calls speed versus velocity. What differentiation does, integration reverses, undoes. So, we can use integration on function v to recover function d. We have now introduced the fundamental theorem of calculus. Right, v is the function, and v(t) is its value at time t, but in first calculus it is common to drop the distinction between v and v(t) and just say "the function v(t)" and mostly avoid mentioning v by itself.

graycat 8y ago
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.

graycat 8y ago

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.