This is the working draft of an article I’m writing for kuro5hin.org, and hence subject to change; it’s been way too long since a good technical article was written there, and that’s part of why I loved the site so much. I’m attempting to recruit help from the UH math club members in the form of donations of sections on the parts of mathematics whose existence I am only aware of, e.g. number theory.
Excerpt (for the listing on the k5 section page)
Mathematics is a beautiful art in all its varied forms. Unfortunately, most of us do not get to see any further into this wonderful world than calculus. One of the factors sustaining the development of mathematics is the continuous development of unexpected applications in the physical sciences. In this article, I will give an overview of some of the mathematical sciences with direct physical applications, and hopefully a sense of their beauty as well as some of their guiding principles: less is more, generalization as a means to an end.
Algebra and Analysis: the twin pillars of mathematics
Most of the mathematical sciences can be categorized as either algebraic or analytic. The algebraic sciences are those that concern themselves with the results arising from the structure of certain spaces. For instance, linear algebra is the study of results arising from the structure of vector spaces. That is, we take a collection of objects that have operations defined on them— in the case of linear algebra, addition and multiplication by scalars— and pull out all the meaning we can find, in all the generality we can manage. The beauty of algebra is that the results apply to any system with the given structure, so a lot of results translate to very different spaces. For instance, if you have any inner-product space ( a vector space with an inner-product defined on it: a generalization of a dot product, which makes it possible to define angles between the vectors, and hence geometricize the space), then you know that there is an orthonormal basis for that space. Since the space of continuous functions is a vector space, this tells us that we can find a set of continuous functions such that every other continuous function is expressible as a sum of these functions. In addition to linear
algebra, algebra gives us concepts like rings and fields, useful generalizations of number systems. The modern approach to algebra is to specify a space in terms of what axioms it must satisfy, and then develop every
result as a logical conclusion of those axioms. The trick and the art to this is anticipating or deducing what combinations of axioms are useful.
Because of its abstracted nature, algebra’s roots reach deep into nearly every advanced mathematical discipline, and it even has some direct practical applications. Especially interesting in the latter camp is the Risch algorithm, which determines whether an indefinite integral is expressible in terms of elementary functions and if so returns the primitive; it relies heavily on algebraic tools: differential fields, extensions fields, etc. Algebra is intrinsically interesting because of the beauty of its abstract deductions ( somewhat similar to how a programming language specification is beautiful because you can see the depth of thinking put into it, and the precision with which it is worded), but its proliferation of definitions and axioms can seem
sterile unless you simultaneously see applications of them, or follow the historical development of those concepts so you can see the motivations behind them. The only interesting tidbit I have left on algebra— in fact, the reason why I came to have any interest in algebra at all— is the fact that it is the backbone of every CAS (Computer Algebra System, like Mathematica, Maple, Axiom, etc.), especially in the form of the Risch algorithm and the techniques used to solve systems of equations. It is interesting to note that Axiom is constructed to mirror the structures studied in algebra; by maintaining this high-level information, it makes it possible to write algorithms (for calculating GCDS, etc.) that will work for any such space without modification.
Analytic sciences are those that concern themselves specifically with a particular type of algebraic structure: complete fields. More intuitively, analytic sciences study spaces in which sequences can be said to converge, or equivalently every set has a least upper and greatest lower bound; such spaces are also known as Archimedean fields. The most common, and most used complete fields are simply the real numbers and the complex numbers. The rationals are an example of a space which does not have the Archimedean property: if you let S = {x in Q | x^2 <= 2}, then clearly S does not have a least upper bound, because you can find a rational number arbitrarily close to sqrt{2}.
It is from the Archimedean property that we get all the goodies that make analysis such a useful, powerful, and intuitive tool: continuity, the intermediate value theorem, the fundamental theorem of calculus... all of these follow from the Archimedean property. Informally, I think of analysis as dealing with smooth spaces, and as such, I find it much more compatible with physical intuition than algebra, in general.
Analysis is typically divided into Real and Complex Analysis: real analysis deals exclusively with analysis on the real number line, while complex analysis deals with analysis in the complex plane. Of course, all the results of real analysis are results in complex analysis, but not vice versa. I've not studied enough of complex analysis to see why the distinction is made: why there are real analysists, and complex analysists. Maybe one reason for the distinction lies in the fact that the complex plane is a unique mathematical entity: it is the algebraic completion of the real numbers. Simply speaking, every polynomial with real (or for that matter, complex) coefficients has a root in the complex plane. In fact, this is the historical justification for the introduction of
imaginary numbers: they made possible the formal solution of polynomials. This makes the complex plane a natural and powerful domain for the study of polynomials. As another possible reason for the distinction, perhaps a stronger one, is there is a phenomenon encountered in the complex plane that has no analog on the real line: analyticity. Analyticity is a strange property that differentiable functions possess in the complex plane: while it is possible to have a differentiable real function whose derivative is not differentiable (e.g. the integral of the absolute value function), if a function is differentiable in the complex plane, automagically derivatives of all orders exist for that function. Furthermore, in some neighborhood of the function, the function is identical with its Taylor series; this is profound because it is possible to construct a real function that is differentiable of all orders at a given point, yet in no neighborhood of that point is the function equal to its Taylor series. Analyticity, also known as holomorphicity, is a powerful and useful tool both in mathematics and engineering and physics. One more strange property of holomorphic functions: they are as delicate as
soap bubbles. If you perturb the function at a single point, it loses it's analyticity; therefore an analytic function can be defined solely by it's values on a sequence with a limit point lieing in the domain of definition of
the function. Hadamard, a famous mathematician, once said "The shortest path between two truths in the real domain passes through the complex domain"; this has held true in my brief experience: for instance, instead of the cumbersome trigonometric Fourier series, most modern expositions deal almost exclusively with complex exponential Fourier series which are a lot more palatable (more on Fourier series later). Complex analysis provides a powerful toolkit for handling a whole array of physically motivated problems: consider the use of the complex exponential (of course, a holomorphic function; in fact, an entire function, one that is holomorphic on the entire complex plane, a rare beauty) in phasors and the solution to differential
equations in physics.
Of course, there are branches of mathematics that don't fall neatly into the realms of algebra or analysis: while calculus is clearly in the analysis camp, the study of differentiable manifolds (very important!, more later) demands knowledge of both, and (classical) number theory is not usually regarded as an algebraic
theory. Really, in modern math, algebra and analysis are very comfortable bedmates.
? The answer according to Tristan Needham in his book Visual Complex Analysis:
given in Cardano’s Ars Magna:![\displaystyle x = \sqrt[3]{q + \sqrt{q^2 - p^3}} + \sqrt[3]{q - \sqrt{q^2 - p^3}}](/cz/latexrender/pictures/f2549936dfc5af28bf366bca776a6b96.png "\displaystyle x = \sqrt[3]{q + \sqrt{q^2 - p^3}} + \sqrt[3]{q - \sqrt{q^2 - p^3}}")
. For the example 
, the solution is ![x = \sqrt[3]{2 + 11i} + \sqrt[3]{2 - 11i}](/cz/latexrender/pictures/7238ca5563651264d91ba57398cba415.png "x = \sqrt[3]{2 + 11i} + \sqrt[3]{2 - 11i}")
. In general, when 
is complex, that implies the equation 
has no solution, but by graphing the line and the cubic, you can see that there is a solution in this case: 
.![\sqrt[3]{2 \pm 11i}](/cz/latexrender/pictures/080b7aa2cbf02a1a2ba80377f5d4a23e.png "\sqrt[3]{2 \pm 11i}")
is; he did so by assuming ![\sqrt[3]{2 \pm 11i} = 2 \pm ni](/cz/latexrender/pictures/505173fdb10ce5a8507961532ea4fc26.png "\sqrt[3]{2 \pm 11i} = 2 \pm ni")
(Why? I don’t know; it doesn’t seem obvious— maybe he tried different numbers besides 2, and this was one that worked). From here, he assumed the standard laws of complex arithmetic, to solve the equation 
, yielding 
and a line; there is always an intersection, so there is always a real solution), complex numbers were encountered only in the solution to quadratic equations. There, they could be ignored, since the presence of a complex number indicates that the given line and the parabola 
do not intersect.