# Tagged Questions

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### What does it mean by saying that $u^n, J^n$ “$C^{\infty}$ converges” to u, J?

The question arose while reading the big book of McDuff & Salamon. Here $\Sigma$ is Riemann surface and M is compact symplectic manifold. Let $u^n(n\in \mathbb N), u : \Sigma \rightarrow M$ be ...
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### How do I convert OS coordinates (X and Y) to longitude and latitude coordinates?

How do I convert OS coordinates (X and Y) - Eastings and Northings to longitude and latitude coordinates? For example X and Y below X (Eastings): 347904 Y (Northings): 287484
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### How do I figure out the grid references necessary to create a circular 1 mile radius around one location?

How do I figure out the grid references necessary to create a circular 1 mile radius around one location? For instance: Grid Reference: SO 47904 87484 X (Eastings): 347904 Y (Northings): 287484 ...
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### How do I figure out the longitude and latitude coordinates necessary to create a circular 1 mile radius around one location?

How do I figure out the longitude and latitude coordinates necessary to create a circular 1 mile radius around one location? I don't mind how many coordinates that takes. For instance: Latitude = ...
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### Question about statement of Rank Theorem in Rudin

Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ mapping of an open set $E\subset \mathbb{R}^n$ into $\mathbb{R}^m$, and $F'(x)$ has rank $r$ for every $x\in E$. ...
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### How to choose $f\in C^{2}(\mathbb{R})$ with compact support and takes value 1 on connected compact set?

Let $0< \delta < \pi$. My questions: (1) How to construct(choose/method) $f\in C^{2}(\mathbb R)$(= First two derivatives ($f' \ \text{and} \ f''$) of $f$ on $\mathbb R$ exists and are ...
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### Are diffeomorphic sets smoothly deformable into each other?

Given connected, bounded and open sets $U, V\subset \mathbb{R}^n$ and an orientation preserving diffeomorphism $F:U\to V$, is there always an isotopy $H:[0,1]\times \mathbb{R}^n\to\mathbb{R}^n$, s.t. ...
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### Finite length of a spiral problem [duplicate]

consider a "spiral" $\alpha(t)=r(t)\left(\cos(t),\sin(t)\right)$, where $r$ is $\mathcal{C}^1$ and $0\le r(t) \le 1$ for all $0 \le t$ Show that if $\alpha$ has finite length on $[0,\infty)$ and ...
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### Change of Variables from Sphere to Plane

Say we are in the space $\mathbb{R}^{n}$. Consider the $n-1$ dimensional surface measure $dS$ on the boundary of the upper half sphere. We can define the coordinates on this sphere by ...
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### Orientability of surfaces in $\mathbb{R}^3$

I have a question: I'm currently reading a few things about orientability and understood this concept as the answer to the question: Given a surface and a unit normal vector field on it: Is there a ...
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### The unsolved extension problem on manifolds.

I have been struggeling for quite a while with this problem: Let $M \subset \mathbb{R}^n$ be a compact $C^k-$ submanifold and $\phi_i: B_i(0) \rightarrow M$ be the associated set of charts ...
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### How do 1-d compact submanifolds look like?

I was wondering whether it is true that every 1-d compact submanifold $\subset \mathbb{R}^n$ that is $C^1$ is a closed curve that is also $C^1$, cause I cannot think of more examples. Therefore, I ...
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### Partition of unity. Does this one exist?

Let $X:=\mathbb{R^n}$ be given and $M \subset X$ be a compact set in it. Then my question is: Are there $\alpha_i \in C^{\infty}(X,\mathbb{R})$ such that $supp(\alpha_i) \subset N$, where $N$ is an ...
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### partial derivatives continuous $\implies$ differentiability in Euclidean space

I am given this theorem: If $f \in C^1(A,\mathbb R^m)$, i.e. every partial derivative of $f$ is continuous on $A$, and $A$ is open in $\mathbb R^n$, then $f$ is differentiable on $A$. Is the ...
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### A little confusion about compactness and connectedness

This question may be a bit simple or even naive for some people but it indeed confuses me for a long time. Thank you all if you provide any explanation. I know concepts: compactness means any open ...
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### Parametrization of level sets of a smooth function

Let $H:\mathbb{R}^2\rightarrow\mathbb{R}$ be given by $H(q,p)=p^2/2+3q^2/2$ (single-well potential). This function has a critical point at $(0,0)$. Define $T:\mathbb{R}^+\rightarrow \mathbb{R}$ by, ...
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### Sobolev maps between manifolds.

Let $M, N$ be smooth compact Riemannian manifolds. I have a reference that defines the $k$th Sobolev space of maps from $M$ to $N$, denoted $H^k(M, N)$, by saying that one only needs to check that ...
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### Derivative with respect to the Frenet frame

Given a regular smooth curve $\alpha: I \to \mathbb{R}^3$, we have that the Frenet Frame $\vec{t},\vec{n},\vec{b}$ is an orthonormal basis of $\mathbb{R}^3$ at each point $s \in I$, and hence we can ...
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### Analysis on Manifolds, Lipschitz Vector fields

Let (M,g) be a smooth compact Riemannian manifold, $\phi:M\rightarrow M$ and $L^n_x=D\phi^n_x:T_xM \rightarrow T_{x^n}M$. Let $(L^n_x)^*$ be its conjugate with respect to the inner product induced by ...
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### Expressing $\frac{d}{dt}\left(\int_{D(t)}u(x,t)dx\right)-\int_{D(t)}u_t(x,t)dx$ as a surface integral?

the following question was the last problem on the Fall 2010 qualifying exam at UCLA. Define $D(t)=\{x^2+y^2\leq r^2(t)\}\subseteq\mathbb{R}^2$ where $r(t)\colon\mathbb{R}\to\mathbb{R}$ is ...
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### Definition of weak solutions from geometrical point of view

Why are weak solutions defined like: A function $u \in H^1(\Omega)$ is a weak solution of $$Lu:=\operatorname{div}(A\nabla u)+b\cdot\nabla u+cu=f+\operatorname{div}F, \;\text{in } \Omega$$ if ...