Jaivir Baweja
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 Feb10 comment Are these vectors in the span of $\mathbb R^3$? Solving the system to be linearly independent (definition of linear subspace), we get the basis vectors \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}, \begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix}, and \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}, which form the canonical basis (i.e. coordinates are the same) for $\mathbb{R}^3$. Jan30 comment On a theorem on Lie derivatives I follow the first part of your question, but I think that since we have that $i_X: \Omega^{n} \to \Omega^{n-1}$ is definition of interior product, the result follows since $i_X df= f$ and since all zero-forms are trivially zero (need it for the Lie algebra) LHS vanishes. Jan27 revised Minimal polynomial over the field $\Bbb Q$ Edited in $\LaTeX$. Jan27 suggested approved edit on Minimal polynomial over the field $\Bbb Q$ Jan25 awarded Commentator Jan25 comment How can I prove that “If $M$ is contractible differentiable manifold, then $M$ is orientable?” @user- Please see FAQ page on how to ask a good question. Jan25 comment How can I prove that “If $M$ is contractible differentiable manifold, then $M$ is orientable?” By the Poincare lemma, all closed forms are exact. Then by the natural projection $\pi: TM \to M$.., compute the homology groups and show isomorphic to $\mathbb{Z}. Jan25 comment Lim Sup/Inf for real valued functions The supremum is the least upper bound on$f(x)$. Therefore by definition of limit we have that it is least upper bound on a point. I think that it only exists if$\textup{lim} \ \ textup{sup} f(x)= \textup{lim} \ \textup{sup} f(c)$. Jan21 revised Prove using the definition of Lebesgue outer measure Formatting with$\LaTeX$Jan21 suggested approved edit on Prove using the definition of Lebesgue outer measure Jan21 awarded Autobiographer Jan21 revised Sketching complex numbers in coordinate system Improved$\LaTeX$formatting so equations easier to understand Jan21 suggested approved edit on Sketching complex numbers in coordinate system Jan21 accepted Prerequisites for understanding the Hodge conjecture Jan21 comment Curvature of Hyperbolic Space Given the formula for the sectional curvature, we can make the tangent vectors orthonormal by taking$u= \frac{x^i}{4\delta_{ij}- 8x^i 8x^j}, v= \frac{x^j}{4\delta_{ij}- 8x^i 8x^j}$. Now we can simply use the standard Riemann curvature$K(u,v)= \left \langle R(Du,Dv)Dv,Du \right \rangle$, since the tangent vectors are along the tangent space. This definition still involves the Riemann curvature tensor as required in your question. Jan21 revised Specific help in showing that Poisson Bracket is part of this Lie Algebra Changed the {f,g} to$\LaTeX$Jan21 suggested approved edit on Specific help in showing that Poisson Bracket is part of this Lie Algebra Jan21 comment Multilinear Functions In a multilinear function, each of the seperate variables are linear. For example$f(cx, by, aw)= cf(x)+ bf(y)+af(w)$, where the property that$f(a+b)=f(a)+f(b)$of a linear map was used along with the other property. Jan21 comment How should I prove a set is convex? Since$S$is contained in a real vector space, we have to show that for all$t \in [0,1]$,$(1-t)x+ty=1$. The condition$x^Tx=1$implies that matrix is unitary, and so letting$x_1=x$and$x_2=y$in the equation above, you can check that it is convex. Jan21 answered Given an algebraic curve$F(x,y)=0$, why do the partial derivatives of$F(x,y)\$ being zero at a point imply the plane curve has a singularity?