线性代数网课代修|ENGO361代写|ENGO361英文辅导|ENGO361

如果你也在线性代数linearalgebra这个学科遇到相关的难题,请随时添加vx号联系我们的代写客服。我们会为你提供专业的服务。 linearalgebra™长期致力于留学生网课服务,涵盖各个网络学科课程:金融学Finance,经济学Economics,数学Mathematics,会计Accounting,文学Literature,艺术Arts等等。除了网课全程托管外,linearalgebra™也可接受单独网课任务。无论遇到了什么网课困难,都能帮你完美解决! The proof is concluded if we prove that for any \(x\in M\) and any \(0\neq V\in T_{x}M\) there exists a harmonic function \(H\) such that \(\nabla H(x)=V\). We prove this fact in Proposition A.5 here below. **Proposition**.: _For any \(x\in M\), \(0\neq V\in T_{x}M\) there exists a harmonic function \(H\) in \(M\) with \(\nabla H(x)=V\)._One can check that the proposition is true when \(M\) is a simply connected surface. In fact, \(M\) can be conformally mapped to \(B(0,1)\subset\mathbb{R}^{2}\). In particular, for any fixed \(x\in M\), there is always a unique (up to rotations) \(\phi:M\to B(0,1)\) conformal diffeomorphism such that \(\phi(x)=0\). Then, taking \(H_{i}=x_{i}\circ\phi\) (\(x_{i}\) coordinate functions in \(\mathbb{R}^{2}\)), we have that \(\Delta H_{i}=0\) and \(\{\nabla H_{i}(x)\}_{i=1}^{2}\) span \(T_{x}M\). In general, the proposition is true for any domain of \(\mathbb{R}^{n}\) by just taking the coordinate functions (in a coordinate system where \(x\) is the origin). Also, it is immediately true for any surface of genus \(g\) and \(k\) boundary components by the Uniformization theorem. For a higher dimensional Riemannian manifold we need a slightly more involved proof.Proof of Proposition a.5.: It is a standard fact to prove that for any \(x\in M\) and any \(0\neq V\in T_{x}M\), there exists a neighborhood of \(x\) (which can be taken open, smooth) and a smooth \(\varphi:U\to\mathbb{R}\), continuous on \(\overline{U}\), such that \(\Delta\varphi=0\) in \(U\). For example, just take (a linear combination of) harmonic coordinates in a neighborhood of \(x\).Let now \(U\subset M\) be a smooth open subset of \(M\). For any \(m\geq 2\) let\[X=\{v\in H^{m}(U):\Delta v=0\}\]and\[Y=\{u\in H^{m}(M):\Delta u=0\}.\]We prove that, given \(v\in X\) and \(\epsilon>0\) there exists \(u\in Y\) such that\[\|v-u_{|_{U}}\|_{H^{m}(U)}

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