My main concern is how to display complex relations.
For example, to display a real relation, we simply display a graph of all the valid points in an \( \mathbb{R}\times\mathbb{R} \) space. This allows the graph to be neither one-to-one nor onto. The inverse relation is displayed simply by reflecting the graph around the line x=y, so to speak.
But to display a complex relation, we need a graph in a \( \mathbb{C}\times\mathbb{C} \) space, which is essentially a 4-D space. We could use a 3-D graph to display one of the complex variables against the real part, or the imaginary part, or the modulus, or the argument, of the other complex variable. Indeed, I've seens such 3-D graphs at wikipedia and other sites.
And eventually I plan to do such. For now, I'm trying to stick to 2-D graphs, and this leaves me drawing contour lines (real and imaginary). It works, but it takes some mental acrobatics to "read" the graph, and it's hard to pick out such elementary features as fixed points. It'd be nice to be able to develop a "sixth sense" for the 4-D structure of a C x C relation.
As for more basic concepts, I've studied a bit of tensor calculus over the years, so I'm familiar with Riemann surfaces and change of coordinates and the such. (I've studied special and general relativity pretty extensively over the past decade or two.) Once I came to grips with the structure of the slog, it didn't bother me as much that it has such oddities as singularities that are present in some branches and not present in other branches. Indeed, it reminds me of studying gravitational lensing, and other oddities of relativity theory. We're not in a Lorentzian space, but a complex space is weird in its own ways. For example, complex transformations preserve angles, at least at infinitesimal scales.
But topology is a tougher subject. I never appreciated that until this last year, when I tried to teach myself some basic topology. To be honest, differential equations and tensor calculus seem like a cakewalk compared to trying to understand the first chapter of my topology book!
For example, to display a real relation, we simply display a graph of all the valid points in an \( \mathbb{R}\times\mathbb{R} \) space. This allows the graph to be neither one-to-one nor onto. The inverse relation is displayed simply by reflecting the graph around the line x=y, so to speak.
But to display a complex relation, we need a graph in a \( \mathbb{C}\times\mathbb{C} \) space, which is essentially a 4-D space. We could use a 3-D graph to display one of the complex variables against the real part, or the imaginary part, or the modulus, or the argument, of the other complex variable. Indeed, I've seens such 3-D graphs at wikipedia and other sites.
And eventually I plan to do such. For now, I'm trying to stick to 2-D graphs, and this leaves me drawing contour lines (real and imaginary). It works, but it takes some mental acrobatics to "read" the graph, and it's hard to pick out such elementary features as fixed points. It'd be nice to be able to develop a "sixth sense" for the 4-D structure of a C x C relation.
As for more basic concepts, I've studied a bit of tensor calculus over the years, so I'm familiar with Riemann surfaces and change of coordinates and the such. (I've studied special and general relativity pretty extensively over the past decade or two.) Once I came to grips with the structure of the slog, it didn't bother me as much that it has such oddities as singularities that are present in some branches and not present in other branches. Indeed, it reminds me of studying gravitational lensing, and other oddities of relativity theory. We're not in a Lorentzian space, but a complex space is weird in its own ways. For example, complex transformations preserve angles, at least at infinitesimal scales.
But topology is a tougher subject. I never appreciated that until this last year, when I tried to teach myself some basic topology. To be honest, differential equations and tensor calculus seem like a cakewalk compared to trying to understand the first chapter of my topology book!
~ Jay Daniel Fox
