# Tagged Questions

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### Hadamard's Lemma in multidimensional real analysis

This is Hadamard's Lemma: Let $U \subset \Bbb R^n$ be an open set, let $a \in U$ and $f: U \to \Bbb R^p$. Then the following assertions are equivalent. The mapping $f$ is differentiable at $a$. ...
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I found a textbook that says that $r \in [0,\infty), \theta \in [0,\pi], \phi \in [-\pi,\pi]$ are the spherical coordinates. Then he goes on with: This corresponds to a unitary map $L^2(\mathbb{R}^3) ... 0answers 33 views ### Question on Inductive Proof of Implicit Function Theorem I am struggling with an inductive proof of the implicit function theorem, concretely with the final part of construction of a function, up to this final point everything is perfectly clear to me. ... 0answers 44 views ###$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $In a metric space$(M,d)$the triangle inequality$d (x, z) \le d(x, y) + d (y, z)$gives us's the inequalitie $$\quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}}$$ ... 2answers 84 views ### Why are open sets used in definitions in differential geometry? I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ... 0answers 26 views ### Proving that a function is a$C^\infty$submanifold in$\Bbb{R}^2$of dimension 1 We need to prove that for all$c\in\Bbb{R}$the set$\{x\in\Bbb{R}\,\colon\, g(x)=c\, \}$is a$C^\infty$submanifold ($g\,\colon\,\Bbb{R}^2\rightarrow \Bbb{R};(x_1,x_2)\mapsto x_1^3-x_2^3$) in ... 3answers 62 views ### A diffeomorphism with negative Jacobian swaps the orientation? Let C be a simple close oriented curve$C^1$in$\mathbb{R}^2$and let$g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$a diffeomorphism such that$\forall (x, y) \in C$it holds that the determinant of the ... 0answers 27 views ### arbitrary reparametrization Let$\alpha: (a,b)\rightarrow \mathbb{R}^n$of class$C^{\infty}$with$\Vert\alpha^{\prime}\Vert>0 $then if$\{ k,m,n\} \subset \mathbb{R}_+$there repametrizacion$\beta: (m,n)\rightarrow ...
The eigenfunctions of the Laplace-Beltrami operator of the flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and their multiplicity are well-known. What happens if we change the sides of the torus ...