Comment by Jackson on Prove that the sum of all simple roots is a root
Thanks for the response. I'm still confused by the strict inequality: why do we get that $(\alpha_j,\alpha_l) < 0$ by construction? I'm very new to root systems so I am probably missing something...
View ArticleComment by Jackson on Vector fields as rank 1 contravariant tensor fields
I understand how the field acts it just seems strange to me that a vector field can be defined as $X : C^{\infty}(M) \rightarrow C^{\infty}(M)$ and $X : \Omega^1(M) \rightarrow C^{\infty}(M)$. Is that...
View ArticleComment by Jackson on Integral of Laguerre Polynomial
@PlokavianNerveGas you guessed it. I'm trying to prove an identity involving operators acting on the ground state of the hydrogen atom.
View ArticleComment by Jackson on What is the probability of two out of three events...
why is the first term not $P(ABC^c)$?
View ArticleAnswer by Jackson for Understanding a proof by induction
Suppose that the assertion holds for all integers less than or equal to $n$. We want to show that it holds for $n+1$. By assumption, we have$$1+3+\cdots+(2n-1) = n^2$$Adding $(2n+1)$ to both sides, we...
View ArticleProve that the sum of all simple roots is a root
Let $\Delta$ be an indecomposable root system in a real inner product space $E$, and suppose that $\Phi$ is a simple system of roots in $\Delta$, with respect to an ordering of $E$. If $\Phi =...
View Articlearea of arbitrary surface element
I am a physics student with a minimal background in differential geometry and I am trying to determine an area element on an arbitrary surface. Suppose we have a surface parameterized by a function...
View ArticleProof that the Lorentz Group SO(3,1) is a manifold
I am trying to prove that the Lorentz group $SO(3,1)$ is a Lie group. To prove that it is a manifold, I was thinking of proving that it is a closed subgroup of $GL(4,\mathbb{R})$. Firstly, I have not...
View ArticleClik here to view.
Centroid of wedge
I am going crazy trying to figure out what I am doing wrong on this basic problem. I need to find the $y$ coordinate of the center of mass of a pan of water that is sloshing back and forth. Let the...
View ArticleChange of variables in partial derivative
I am stuck on a simple exercise in quantum mechanics because I can't figure out how to modify a partial derivative under a change in variables. If I have a Hamiltonian in two variables $x_1$ and $x_2$,...
View ArticleSuppose that $T$ is injective, how to prove that $T^* T$ is injective?
Assume $(V,\langle \ , \ \rangle_V)$ and $(W,\langle \ , \ \rangle_W)$ are finite dimensional inner product spaces and $T : V \rightarrow W$ is an injective linear transformation. Prove that $T^*T : V...
View ArticleUniqueness of identity matrix for one matrix
I know that there is only one matrix $I$ such that for all matrices $M$, $M = I M = M I$. But in general, suppose I have a particular matrix $M$, and the matrix equation$$M = M T$$Can I conclude from...
View ArticleLinear approximation of quotient
I am confused as to how to proceed with the following linear approximation:$$\frac{(2.01)^2}{\sqrt{.95}}$$I know that we need to define a function such that $f(x) = \frac{(2.01)^2}{\sqrt{.95}}$ and...
View ArticleAnswer by Jackson for Difference between infinitesimal parameters of Lie...
I think that I answered my own question, with the help of the commenters. My confusion stemmed from the fact that I didn't see why $\epsilon^{\mu \nu}$ had to be antisymmetric, and mistook the...
View ArticleDifference between infinitesimal parameters of Lie algebra and group...
I am getting myself confused regarding the differences between the infinitesimal generators of Lie group and the elements of the Lie algebra, likely due to the fact that I am studying from a physics...
View Article$\text{SL}(2, \mathbb{R})$ mapping on the boundary and interior of the disk
I am trying to show that $\operatorname{SL}(2, \mathbb{R})$ can be used to map an arbitrary point in the interior of the disk and an arbitrary point on its boundary to any pair of points, one in the...
View ArticleComment by Jackson on Can vector fields be thought of as rank-1 contravariant...
I understand how the field acts it just seems strange to me that a vector field can be defined as $X : C^{\infty}(M) \rightarrow C^{\infty}(M)$ and $X : \Omega^1(M) \rightarrow C^{\infty}(M)$. Is that...
View Article
