4
votes
Accepted
Stokes theorem 2 sides not matching with Magnetic waves
For those not wanting to translate in and out of cylindrical coordinates, $\vec H$ corresponds to the $1$-form $\omega = \frac12\cos(\phi/2)dr - \sin(\phi/2)(r\,d\phi)$, and $d\omega = -\frac34\sin(\...
- 111k
3
votes
Accepted
Is a normal vector field surjective?
Here's an approach you should think through: Choose $w\in S^{n-1}$ and consider the function $f\colon \Omega\to \Bbb R$ defined by $f(x)=x\cdot w$. By compactness of $M=\partial\Omega$, the function ...
- 111k
2
votes
Christoffel Symbol Regularity
In general I’d only expect this locally. To get the full statement, you’d need bounds on the inverse metric, which roughly translate to ‘lower bounds’ on the metric. Then, you’d need some result about ...
- 49.7k
2
votes
Accepted
Energy minimization doesn't seem to yield a geodesic
By definition, geodesic can be defined through minimization of the following functional,
$$L=\int_{0}^{1}dt\,\sqrt{g_{ij}(t)\dot{\xi}^i(t)\dot{\xi}^j(t)}$$
and the resulting equation for geodesic is
$$...
- 486
2
votes
Accepted
Computing $\left(\frac{\partial x^{i}}{\partial y^{j}}\right)_{p}$ for $\mathbb{R}$ valued component functions.
There is no need for the chain rule. As you showed$$ x \circ y^{-1} : \mathbb R^2 \to \mathbb R^2 : (u,v) \mapsto(u,v-u^3).$$
Therefore define a function $$ g : \mathbb R^2 \to \mathbb R : (u,v) \...
- 3,022
1
vote
Accepted
Computation of Riemannian Hessian of the Stiefel manifold
$\newcommand{\rgrad}{\mathrm{rgrad}}$
$\newcommand{\grad}{\mathrm{grad}}$
$\newcommand{\hess}{\mathrm{hess}}$
$\newcommand{\rD}{\mathrm{D}}$
$\newcommand{\rP}{\mathrm{P}}$
Try this :-).
https://link....
1
vote
Accepted
What is the sectional curvature of this explicitely given n-dimensional submanifold of R^(n+1)?
$\newcommand{\Two}{\mathsf{I}\mathsf{I}}$
$\newcommand{\rD}{\mathrm{D}}$
You can use the second fundamental form and the Gauss-Codazzi equation. For embedded manifolds in a vector space, the Gauss-...
1
vote
Trying to prove that for a curve $C$, that is parameterized by $\gamma (s)$, is spherical, i.e. every point of $C$ lies on a sphere.
Hint: The condition $\gamma$ is spanned by $n$ and $b$ means that
$$t(s)\cdot\gamma(s) = 0$$
for all $s$.
- 32.4k
1
vote
Accepted
Question about the definition of a vector field on a manifold
Syntactically, if $X:M\to TM$ is a vector field and $p\in M$, then $X(p)=(p,v)$ for some $v\in T_p M$ because $TM$ is a disjoint union of all $T_p M$'s, i.e. formally an object like $\bigcup_p \{p\}\...
- 824
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