3
votes
Accepted
How to prove that $\vec{\nabla}\times\vec{\nabla}\times(e^{i\vec{k}\cdot\vec{x}}\vec{\omega})=k^2(e^{i\vec{k}\cdot\vec{x}}\vec{\omega})$
Let $\omega = \langle \omega_1,\omega_2,\omega_3\rangle$.
Using the first identity you have listed
$$\begin{align}\nabla \times \nabla \times (e^{ikx}\omega) = \nabla(\nabla \cdot (e^{ikx}\omega)) - \...
- 2,046
3
votes
Accepted
Can an exact vector field have loops as solutions?
First of all notice that if $\mathbf{v}$ has a zero, say in $\mathbf{x}_0$, then a solution trivially exists by choosing $f(t)\equiv\mathbf{x}_0$.
Your observation is correct. Indeed, we find for any ...
- 1,342
2
votes
Accepted
Existence of $\varphi$-related smooth vector field.
Here's a counterexample, using the idea you described in your post.
Take $N = (0,\infty) $ and $M = \mathbb R$, and take $\varphi: N \to M$ to be the natural embedding $t \mapsto t$. Take $X$ to be ...
- 31.8k
2
votes
Accepted
Define vector space $V=\{ (a,0,0,b):a,b\in\mathbb{R} \} \subseteq \mathbb{R^4}$. Is $W=\{(a,b,c,d)\in\mathbb{R^4}:ab=c\}$a subspace of $\mathbb{R^4}$?
You are correct. The set $W$ is not a subspace of $\mathbb{R}^4$.
Note that $(1,2,2,d)\in W$, because $1\cdot2=2$ and $(1,1,1,d)\in W$, because $1\cdot1=1$. However, $(1,2,2,d)+(1,1,1,d)=(2,3,3,2d)$ ...
- 1,005
2
votes
Accepted
Prove the existence of a smooth global frames on $\mathbb{S}^7$.
Let $J_1,\ldots,J_7$ be the right multiplication by $(i,0), (j,0), (k,0), (0,1), (0,i), (0,j)$ and $(0,k)$ respectively.
Let $X=(p,q) \in \Bbb O$.
Then
$$
J_1X
= (p,q)\cdot(i,0)
= (ip, -qi),
$$
and ...
- 18k
2
votes
How to show there exists a smooth local frame $(\tilde{X_1},\cdots,\tilde{X_n})$ on some neighborhood of $A$ such that $\tilde{X_i}|_A=X_i$?
From J. M. Lee's book "Introduction to smooth manifolds" we have the next proposition:
10.12 Let $\pi:E\to M$ be a (smooth) vector bundle. Let $C$ be a closed subset of $M$ and $s:C\to E$ a (...
- 361
2
votes
Accepted
Behaviour of solution of differential equation at infinity
It does go to $+\infty$, but as $x$ approaches a finite value, namely $\ln(2)$.
- 441k
2
votes
Accepted
Prove that $\gamma$ is constant iff $\gamma′(t) = 0$ for all $t \in J$ if there is always an interval containing $t$ in which $ \gamma$ is const
Your argument for $\implies$ looks fine.
For $\impliedby$, I agree that we should use part (a), and I also like the fact that you're thinking about how $\gamma$ looks in a chart. Here's a way to ...
- 31.8k
1
vote
Accepted
Divergence of two orthogonal vector fields
We could have $b_1=0$, and then $b_2$ can be whatever it wants.
For a less trivial example, you could pick $b_1$ to be a vector field that "rotates" around the origin of the Euclidean plane (...
- 197k
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