Today I want to talk about hypersurfaces.
The hypersurfaces is a mathematical concept widely used in physics with a very bombastic name, mainly because of the hyper prefix denotes something that is very large, and therefore very complicated.
At the end, like every mathematical concept, is not as fierce as they paint and following definitions hair stand on end or that I would play pasaros hours giving “reverse” between palaver to understand them, understand (like everything in mathematics) that is a chorrada authentic.
If the definition of hypersurface reference to topological spaces homeomorphic (as said, a word) is done and that means simply is continuous (which has no points that pyran to infinity so for the good) and, above all, Conversely, it is also continuous, so we understand a hypersurface is but a surface, a plane, or space (I think that would hyperspace and not go faster than light) represented or painted or He wants to know in a simple way.
That is to say without going into too much mathematics to not lose you, it is to take something, a very complicated area and represent more easily, such as taking a torus (a donut, go) and represent it in a two-dimensional plane, one surface.
For example, the other day when I commented Minkowski space, that of the cones of the future and the past as the speed of light, you indicate that the origin was in a plane. Well, if not a plane, it is a 2-dimensional hypersurface representing whether a 3-dimensional surface. That is, all standard in 3-dimensional space is represented there in 2 with a new axis is time. This helps us understand the light cone, as is formed and above all it means.
The hypersurfaces are widely used in physics, differential geometry. It is therefore essential for a physical not only have the “cursed analysis” in the head and if the “simple algebra”.
And you say, why ?. As for something a theoretical physicist will have in the mouth all day tensioners.
Now is the moment beatings, so you can jump because with what I have told you all this time is that you know what a tensor.
Differential geometry is necessary to Riemannian geometry (he never if I write well and so I will) because its non-Euclidean way better define the possible geometry of space time and help, using tensioners, to maintain and calculate properties the objects simply. As we know that space is not flat and the actual geometry of the universe is unknown, through geometry “Riemanna” in local areas where the curvature is practically zero helps us to simplify algebraic calculations tensioners that apply through a very simple rules. In addition, this geometry is so simple that step into space Euclidio (the life) is almost trivial can thus, the data, simply match them with reality. We’re going to move from Rienmann to lifelong is easy.
Therefore, differential geometry and Riemann geometry are basic to relativity and the equations of string, where, through hypersurfaces (which are mixed together) can come to calculate the equations of the string (and their interactions).
And, as a final colophon. If you want to study physics, no despise on algebra, something that teachers often give these subjects do you learn either through ignorance or because they can not give importance to these matters as physicist future.