James S. Cook
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 Jun 14 answered Need some help understanding the condition of the implicit function theorem Jun 13 reviewed Approve Differentials - Area of a circle from its diameter Jun 13 comment How do the components of a cross product transform? @Uldreth yes, but, I think the usage has more to do with how various vectors are defined and how the equations of physics change as we transform coordinates. Jun 13 comment Linear Maps and Basis of Domain I agree with Bernard, see page 158, prop. 7.2.17 of my notes supermath.info/LinearNotes2015.pdf Jun 13 comment Why do mathematicians use $\Delta$ instead of $\nabla^2$? I'm not sure mathematicians really resist notation from other disciplines as much as we tend to change our notation over time. In my experience, physics notation was also math notation... from the time in history when that physics was written... so, other disciplines sometimes have notational timecapsules. That said, notation varies. Jun 13 comment How do the components of a cross product transform? I wish the standard calculus texts used $\epsilon_{ijk}$ to define the cross-product. It's so clean. It's funny, you also have to raise and lower indices. I'm pretty sure the Riesz rep. step is Hodge duality in your notation. Jun 13 comment How do the components of a cross product transform? @Uldreth well, I guess the point is that inversions do matter to physics. So, we don't have the luxury of ignoring those left-handed isometries. Jun 12 revised How do the components of a cross product transform? edited body Jun 12 answered How do the components of a cross product transform? Jun 12 comment Lie Groups/Lie Algebra - Applications? Oh, but, more to your reference request, and a bit off topic, but you might enjoy the paper by Baez on Octonions. It perhaps will scratch your itch. I want all my students to read math.ucr.edu/home/baez/octonions/oct.pdf Jun 12 comment Lie Groups/Lie Algebra - Applications? One of my students who went on to graduate school to study analytic number theory, or, maybe algebraic number theory. Or, well, I don't know, but, Lie algebra representation is important to him. Kind of surprising to me. On the other hand, another of my students is working towards some graduate work in differential geometry. He also needs Lie theory. Essentially, differentiation puts you in contact with derivations and derivations go hand in hand with Lie algebras. Besides that, and maybe in tune with that, Lie algebras approximate Lie groups and groups, well, they be everywhere. Jun 12 comment What is a good reference that connects calculus with differential geometry? I'm actually working on differential geometry this summer. I'll post a link if you want. I suppose creationists can also do math. At least, I know a few... Jun 12 reviewed Approve Nullstellensatz for Coordinate Ring Jun 11 reviewed Approve Finding limits of a line integral with vector fields Jun 11 comment What is a good reference that connects calculus with differential geometry? you might find supermath.info/AdvancedCalculus13.pdf useful. I try to start by treating multivariate analysis, Jacobians and all that then I spend the latter half on forms and such. Tu and Lee's manifold texts are certainly helpful, perhaps Conlon, or Munkres also should be considered. Jun 11 comment Coordinate systems on manifolds Well, I think this is my last comment here since the system gets grumpy now, but, to answer your last comment, I don't try to visualize past $n=2$. Generally, what is "good" depends on some symmetry of the example, for things built from sums of squares I see sines and cosines. For differences of squares I see cosh and sinh. Using cartesian coordinates is not usually good for explicit computations, unless, it's a plane or nice graph. There are no universal rules here. It is more of an art than an algorithm. See math.stackexchange.com/a/836833/36530 Jun 10 comment Coordinate systems on manifolds Yep, curved manifolds (those with nontrivial curvature) are locally Euclidean. The term locally Euclidean is a topological condition. See en.wikipedia.org/wiki/Topological_manifold for example. Yes, there are in principal many choices of coordinates, if we work with a maximal atlas then $\infty$-many coordinates are available. Jun 10 comment How to intuitively understand prolongations It might be helpful to look at Cartan For Beginners by Ivey and Landsberg. My intuition is that the prolongation is differentiation. Since partial derivatives commute we have to account for that by using symmetrized tensors. Also, the use of a vector space and its dual is important as it gives us a coordinate-invariant presentation of a PDE. The so-called Gauss map and Tableau work together to give us an algebraification of the PDE. I'm still working on really understanding all this myself. Jun 10 comment Coordinate systems on manifolds To answer your question more abstractly, a patch is like the parametrizations we used in calculus III to set-up flux or surface integrals. A pair of parameters $(u,v)$ gives us a point $(x(u,v),y(u,v),z(u,v))$ on the surface. So, $(u,v)$ is not "on" the surface, but, they do set-up a coordinate system on the surface in a natural way. A chart takes its domain as some subset of the manifold and its image is in $\mathbb{R}^n$. When I talk about a system of coordinates on a manifold, I'm talking about a chart. On $\mathbb{R}^3$ we have three usual charts we use in calc. III. Jun 10 comment Coordinate systems on manifolds No my $\Phi$ and $\Psi$ are "patches". The inverse maps are the coordinate "charts" on the sphere. Let me pick on an easier example, for the $z=1$ plane we have patch $\Phi(u,v) = (u,v,1)$. On the other hand, $\phi^{-1}(x,y,z) = (x,y)$ for all $(x,y,z)$ on the plane. Part of the confusion stems from the fact that we use $x,y,z$ as both fixed points and coordinate functions. I can write $\phi^{-1} = (x,y)$ this indicates $\phi^{-1}(a,b,c) = (x(a,b,c),y(a,b,c)) = (a,b)$. So, $(x,y)$ is a chart on the $z=1$ plane, but, understand, this views $x,y$ as real-valued functions with domain $z=1$.