We know that Euclidean geometry and Newtonian Physics are special cases that only work in a flat space-time. Got to thinking about linear algebra and matrices. Is linear-algebra a special subset of some math? And does LA only work because our local space-time is flat enough to discard any discrepancies that would result from plotting a "line" or a "plane" in a curved space-time?
5
-
$\begingroup$ So you think that linear algebra has to do with physical space? It hasn't. It deals with vector spaces, linear maps, numbers and matrices. Whether it can be applied to make mathematical models of physical phenomena is something which is outside the scope of linear algebra. $\endgroup$– egregCommented Nov 15, 2013 at 16:38
-
$\begingroup$ Thought exp. If I have two lines that meet in flat space I have no guarantee that they will meet at the same point in curved space. Similarly, I think, If I have three equations of a plane, and I want to know where they meet in 3-Space I can do this easily in a flat space, but curve space either way and now the those same 2d surfaces do not necessarily intersect in the same "location". Or am I not following you? $\endgroup$– ChrisCommented Nov 15, 2013 at 17:07
-
$\begingroup$ In the real world there are no lines and planes. Have you ever seen one? $\endgroup$– egregCommented Nov 15, 2013 at 17:15
-
$\begingroup$ a line being a vector on a surface and a plane being a surface yes I have....but that could be a naïve world view. But we know that a vector on a curved surface is a curve, similarly a surface must warp in curved space.... at some level. So I guess I was wondering if there is an algebra that compensates for these warps.... A "curvilinear" algebra if you will. $\endgroup$– ChrisCommented Nov 15, 2013 at 19:08
-
$\begingroup$ You might be interested in learning about manifolds. I haven't read it but the book Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard and Hubbard could be interesting. $\endgroup$– littleOCommented Nov 19, 2013 at 11:50
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
The keyword is Linearization. Since anything in Physics is an approximation, things that are not linear, from a foundational point of view, may nevertheless be considered as if they were linear, as a first approximation. Especially in technological applications, where robustness is more important than exactness (e.g. classical electronics), this is common practice. See: Linearization (Wikipedia).
answered Nov 19, 2013 at 11:38