# 物理代写|电磁学代写electromagnetism代考|PHYS3040

## 物理代写|电磁学代写electromagnetism代考|Tangential Trace Revisited

Below, the tangential trace of elements of $\boldsymbol{H}($ curl, $\Omega)$ is scrutinized, and refined generalized integration by parts à la (2.27) is established, involving two vector fields of $H($ curl, $\Omega$ ). Indeed, in the case of the tangential trace, the mapping $\gamma \top$ from $\boldsymbol{H}(\mathbf{c u r l}, \Omega)$ to $\boldsymbol{H}^{-1 / 2}(\Gamma)$ is not surjective. This seems obvious, since one has $\left(\gamma_{\top} \boldsymbol{f}\right) \cdot \boldsymbol{n}=0$ in some sense, for instance, as soon as a pointwise $\gamma_{\top} \boldsymbol{f}$ exists. But there are also more profound arguments, which allow us to prove that, even when one considers only the set of vector fields on $\Gamma$ that are orthogonal to $n$, the mapping is nevertheless not surjective $[5,65,66,72]$.

In order to prove this, together with a number of useful results, let us consider, for simplicity, the case of a polyhedral domain, still called $\Omega$, with the notations of Definition 2.1.54. We follow here the path chosen by A. Buffa and the second author in $[65,66]$, where the case of a curved polyhedron is also addressed. Again for simplicity, we assume that its boundary $\Gamma$ is topologically trivial (the notion is defined in Sect. 3.2). See [64] for a topologically non-trivial boundary: in this case, decompositions of function spaces have to be modified, with the addition of a third-finite-dimensional-vector subspace. Along the way, representative proofs, establishing the continuity of the mappings, are provided. On the other hand, the results relating the surjectivity of the mappings are stated without proof. In the more general case of a domain, the reader is referred to $[68,188]$.

Looking at the integration-by-parts formula $(2.27)$, it is clear that the normal component of $g$ does not play any role in the formula. Therefore, one can concentrate on the tangential components only.

Definition 3.1.1 Let $f$ be a smooth vector function defined on $\bar{\Omega}$. Its tangential components trace $\boldsymbol{n} \times(\boldsymbol{f} \times \boldsymbol{n}){\left.\right|{\Gamma}}$ on the boundary $\Gamma$ is denoted by $\pi_{\top} f$, and $\pi_{\top}$ is called the tangential components trace mapping.

In order to define the actual range of $\pi_{\top}$, starting from $\boldsymbol{H}^1(\Omega)$, let us introduce some spaces of vector fields, defined on $\Gamma$.

Definition 3.1.2 Let $L_t^2(\Gamma)$ be the space of tangential, square integrable vector fields:
$$\boldsymbol{L}t^2(\Gamma):=\left{v \in L^2(\Gamma): v \cdot n=0\right}$$ Let $\boldsymbol{H}{-}^{1 / 2}(\Gamma)$ be the space:
$$\boldsymbol{H}{-}^{1 / 2}(\Gamma):=\boldsymbol{L}_t^2(\Gamma) \cap H{-}^{1 / 2}(\Gamma)^3 .$$
Let $\boldsymbol{H}{|}^{1 / 2}(\Gamma)$ be the space: $$\boldsymbol{H}{|}^{1 / 2}(\Gamma):=\left{v \in \boldsymbol{H}{-}^{1 / 2}(\Gamma): \boldsymbol{v}_i \cdot \boldsymbol{\tau}{i j} \stackrel{1 / 2}{=} v_j \cdot \boldsymbol{\tau}_{i j}, \forall(i, j) \in \mathcal{N}_E\right}$$

## 物理代写|电磁学代写electromagnetism代考|Scalar and Vector Potentials

We discuss two different mathematical points of view, namely the analyst’s and topologist’s, concerning the existence of potentials for curl-free fields. We then reconcile these two points of view and define a general framework.
For the analyst [124], the main issue is the regularity of the boundary. Accordingly, the analyst’s hypothesis on $\Omega$ is:
(Ana) ” $\Omega$ is an open set of $\mathbb{R}^3$ with a Lipschitz boundary”.
For the topologist $[126,127]$, the main issue is (co)homology and, of particular interest for our purpose, the existence of single-valued potentials to curl-free smooth fields. In other words, given a vector field $v$ defined on $\Omega$ such that curl $v=0$ in $\Omega$, does there exist a continuous single-valued function $p$ such that $v=\operatorname{grad} p$ ? The answer to this question can be found in (co)homology theory, which results in the topologist’s dual hypothesis:
either (Top) $){I=0} \quad$ “given any vector field $v \in C^1(\Omega)$ such that curl $v=0$ in $\Omega$, there exists $p \in C^0(\Omega)$ such that $v=\operatorname{grad} p$ on $\Omega$ “; or $\quad(\text { Top }){I>0} \quad$ “there exist I non-intersecting manifolds, $\Sigma_1, \ldots, \Sigma_I$, with boundaries $\partial \Sigma_i \subset \Gamma$ such that, if we let $\dot{\Omega}=\Omega \backslash \bigcup_{i=1}^I \Sigma_i$, given any vector field $v \in C^1(\Omega)$ such that curl $v=0$ in $\Omega$, there exists $\dot{p} \in C^0(\dot{\Omega})$ such that $v=\operatorname{grad} \dot{p}$ on $\dot{\Omega} “$.
Here, $I$ is equal to the minimal number of required cuts $\left(\Sigma_i\right)_i$. Mathematically, $I$ is equal to $\beta_1(\Omega)$, the first Betti number. Note that according to the above, $I=0$ is an admissible value, in which case the existence of continuous single-valued potentials is guaranteed on $\Omega$, whereas $I>0$ corresponds to the case when cuts must be introduced. This is the reason why we use the notations $(\text { Top }){I=0}$ and $(\text { Top }){I>0}$ to discriminate the two cases. When $I=0$, the set $\Omega$ is said to be topologically trivial.
Remark 3.2.1 Recall that, according to homotopy theory, a connected set is simply connected if every closed curve can be contracted to a point via continuous transformations. It is often assumed that each connected component of $\Omega$ must be simply connected to guarantee the existence of the continuous single-valued potential: in other words, one usually states in $(\operatorname{Top}){I=0}$ (respectively $(\text { Top }){I>0}$ ) that $\Omega$ (respectively $\dot{\Omega}$ ) is simply connected. However, this property is only a sufficient condition and, from a topologist’s point of view [126], the correct assumption is of a (co)homological nature, cf. (Top) as stated above.

## 物理代写|电磁学代写电磁代考|切向轨迹重访

3.1.1设$f$是在$\bar{\Omega}$上定义的平滑向量函数。它在边界$\Gamma$上的切向分量跟踪$\boldsymbol{n} \times(\boldsymbol{f} \times \boldsymbol{n}){\left.\right|{\Gamma}}$用$\pi_{\top} f$表示，$\pi_{\top}$称为切向分量跟踪映射。

$$\boldsymbol{L}t^2(\Gamma):=\left{v \in L^2(\Gamma): v \cdot n=0\right}$$设$\boldsymbol{H}{-}^{1 / 2}(\Gamma)$为空间:
$$\boldsymbol{H}{-}^{1 / 2}(\Gamma):=\boldsymbol{L}_t^2(\Gamma) \cap H{-}^{1 / 2}(\Gamma)^3 .$$

## 物理代写|电磁学代写电磁学代考|标量势和向量势

.

(Ana) ” $\Omega$ 是开套的吗 $\mathbb{R}^3$ 具有Lipschitz边界”。

either (Top) $){I=0} \quad$ “给定任意向量场 $v \in C^1(\Omega)$ 这样的旋度 $v=0$ 在 $\Omega$，存在 $p \in C^0(\Omega)$ 如此这般 $v=\operatorname{grad} p$ 在 $\Omega$ ;或 $\quad(\text { Top }){I>0} \quad$ “存在不相交的流形， $\Sigma_1, \ldots, \Sigma_I$，有界限 $\partial \Sigma_i \subset \Gamma$ 这样，如果我们让 $\dot{\Omega}=\Omega \backslash \bigcup_{i=1}^I \Sigma_i$，给定任意向量场 $v \in C^1(\Omega)$ 这样的旋度 $v=0$ 在 $\Omega$，存在 $\dot{p} \in C^0(\dot{\Omega})$ 如此这般 $v=\operatorname{grad} \dot{p}$ 在 $\dot{\Omega} “$.

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