Of course we're talking about XY not being a vector field on our manifold M, and [X,Y] being a vector field on our manifold M
The Question:
Let A,B be n*n matrices, and let X(x) = Ax and Y(x) = Bx be vector fields of ℝ^n . Calculate [X,Y].
I've just been introduced to the concept of the Lie bracket of two vector fields in differential geometry. However, I have no intuition for what the bracket of the above two vector fields should look like in terms of A and B. From quantum mechanics, my intuition wants to say that the answer is AB - BA (probably just because the notation is the same as that of the commutator for quantum operators), and I've tried to prove that by inserting A and B into the formula given on Wikipedia (under "in coordinates): https://en.wikipedia.org/wiki/Lie_bracket_of_vector_fields
However, when I try this, my work dissolves into a mess of i's and j's that I can't build back up into multiplication of the two matrices. I've tried to compute the bracket using other expressions that involve it (e.g. the one under "Vector fields as derivations" on the Wikpedia page), but this one also includes an arbitrary scalar function that I'm not sure how to get rid of.
Is there any "standard" way to compute the bracket that I don't know of? I would appreciate any help. Thanks.
I've taken a course in differential geometry (mostly curves and surfaces in 3D Euclidean space though, not general Riemannian geometry) but upon further independent reading I've realized that I lack the level of understanding I need when it comes to the different kinds/notions of derivations on manifolds (I know the words, but feel as if I don't really get it).
If someone is willing to give an explanation (I'd welcome both general overviews, and deeper more far-reaching things) I'd be very grateful.
(Towards the future (PhD) I'm aiming towards geometric analysis, algebraic geometry, those kinds of things; if that helps with deciding explanatory angles)
thee title is all of it
Title
genuinely stuck. i know that the line integral of the vector field F along C is going to be 0, which implies that the field dotted with the tangent vector of the path at a given point will be 0 and thus the field is normal to the contour at that point. however, i have no idea how to show that this is true for at least two points
I am trying to figure out the best order of operations when it comes to compositing my vector blur passes. I generally consider doing motion blur after I comp my depth pass, but that was before I started using vector blurs. I'm concerned some of the objects which are clearly defined in the pass will not work right if they come after the DOF in the node graph. But the same could also be said the other way around. Vector blurs are new to me, so I'd appreciate some guidance on what order you would comp these in. Thanks!
Sorry if this is the wrong sub to post this and do guide me to the more appropriate sub if any.
I always believed the typical way of writing divergence and curl of a vector field X (div = ∇. X and curl = ∇ x X) was a matter of notation, till I was introduced to the idea that functions satisfy the required properties of a vector space. Since then I always thought ∇, represented the unit vector (∂/∂x, ∂/∂y, ∂/∂z), and that dot and cross product with vectors of a vector field nicely yields these properties.
But how exactly am I operating between them? Do they not clearly belong to different vector spaces? How can I take the dot/cross product of 2 vectors belonging to different vector spaces?
PS: I understand it might just be a matter of notation, but would appreciate it if any consistent perspective could be given on this in general!
Is there any theory behind this or is it just something observed experimentally?
I'm in mathematical ecology and have to solve some logistic growth equations (like verhulst-pearl), trying to figure out how to graph vector fields in r but not sure how to vectorize the equation to use the pracma package (unless there is another package that would better plot the vector field)
Many thanks!
My first generative piece! Check out the site. Refresh the page to generate a new piece.
https://preview.redd.it/tgfug9l0dib81.png?width=1844&format=png&auto=webp&s=4003a24ffb3c1c1f6d4f66b8126f25c9ac897b38
Sorry if this is the wrong sub to post this and do guide me to the more appropriate sub if any.
I always believed the typical way of writing divergence and curl of a vector field X (div = ∇. X and curl = ∇ x X) was a matter of notation, till I was introduced to the idea that functions satisfy the required properties of a vector space. Since then I always thought ∇, represented the unit vector (∂/∂x, ∂/∂y, ∂/∂z), and that dot and cross product with vectors of a vector field nicely yields these properties.
But how exactly am I operating between them? Do they not clearly belong to different vector spaces? How can I take the dot/cross product of 2 vectors belonging to different vector spaces?
PS: I understand it might just be a matter of notation, but would appreciate if any consistent perspective could be given on this in general!
