Recent content by mnb96

I see...so basically I have to interpret the text according to the following analogies: prolongation (or jet) of f at x ↔ tangent vector of f at x jet space of f at x ↔ tangent space of f at x jet bundle of f ↔ tangent bundle of f This would explain the reason for the...

Thanks jambaugh for your help, and especially for pointing me to Olver's work. I checked Olver's book "Classical invariant theory", and I found there the same confusing "double definition" that seems to propagate in other texts as well. See excerpt below: In the first highlighted sentence he...

Hello, I am reading some material related to jet spaces, which at first glance seem to be a generalization of the concept of tangent space. I am confused about what is the correct definition of a jet space. In particular, given a map ##f: X \rightarrow Y## between two manifolds, what is the...

Hi, I am studying some material related to Grassmannians and in particular how to represent k-subspaces of ℝn as "points" in another space. I think understood the general idea behind the Plücker embedding, however, I recently came across another type of embedding (the "Projection embedding")...

Hi fresh_42, you gave a very interesting counterexample of my statement that is actually too inspiring to close the discussion here :) In fact, let's define the "product of two subspaces" as ##UV=\left \{uv\;|\; u\in U, \, v\in V \right \}##, and notice that in your construction ##H^2=0##. In...

Hi, consider a (finite dimensional) vector space ##V=U\oplus W##, where the subspaces ##U## and ##V## are not necessarily orthogonal, equipped with a bilinear product ##*:V\times V \rightarrow V##. The subspace ##U## is closed under multiplication ##*##, thus ##U## is a subalgebra of ##V##...

Thanks. We are slowly getting to the point of my question: Is there a way to quantify the amount of change in capacitance as a function of the position of the piece of glass? What I want to obtain is basically a scalar field that would represent the "sensitivity" of the original capacitor to...

Ok. Any hint about my original question about the change in capacitance being dependent on the position of the piece of glass?

Hi NFuller, thanks for your reply. I have the feeling that you answered a different question than the one I asked. You basically explained why capacitance in a "static" configuration does not in general depend on E. The scenario I was describing in my OP was the following: I was considering...

Hello, Let's consider a capacitor simply made of two conductors with arbitrary shape in the vacuum (http://www.kshitij-iitjee.com/Study/Physics/Part4/Chapter26/3.jpg). Now, if I place a small piece of dielectric material (for example a tiny sphere of glass) between the two conductors, the...

Very interesting and satisfactory answer! I didn't know about the relationship between compact support and analyticity of the FT. I am still wondering two things: 1) Is there an actual difference in this case between "countably many isolated zeroes" and "zero in a set of Lebesgue measure...

Hi stevendaryl, yes, I think the fact that the FT of a sinc function is rectangle function (and vice-versa) is a well-known result. Despite that, it is a useful remark. In fact, the rectangle function is one example of function whose FT has zeros on a set of Lebesgue measure zero. Now, one...

Hello, after working on this problem, I would have an additional question related to it. Let's consider a similar scenario where we have four conductors such that V1=V2=1 and V3=0 (as in the original post), but now the 4th conductors is "floating" instead of being grounded (i.e. it is...

Hello. Thanks for the reply. This is not a homework problem. I don't see unfortunately any clear connection between Plancherel theorem and the zeros of a Fourier transform. Maybe someone else could point out other possible directions to approach the problem? It would be even ok to restrict...

Hello, for a function f∈L2(ℝ), are there known necessary and sufficient conditions for its Fourier transform to be zero only on a set of Lebesgue measure zero?