前段时间2024年阿里巴巴全球数学竞赛初赛的成绩出了, 意料之内地没有进决赛, 还是有一些失望叭(；′⌒`) 但终归还是自身实力不济, 只能来年有空再战叭~ 最近在学习矢量分析, 查阅了各种资料后, 总算掌握了爱因斯坦求和约定与Levi-Civita符号这两个强有力的矢量分析工具, 特此记录.
参考材料
1. [工數筆記] 愛因斯坦求和約定（Einstein summation convention）
2. Gradient of a dot product
3. 预备知识—矢量场论复习
4. how-to-prove-that-nabla-a-cdot-b-a-cdot-nabla-bb-cdot-nabla-aa-times
5. 矢量分析简介
1. 基本定义
$$\delta_{ij} = \left\{\begin{matrix}
1, & i = j, \\
0, & i \ne j,
\end{matrix}\right. \\ \epsilon_{ijk} = \left\{\begin{matrix}
1, & \text{ if } (i, j, k) \in \{ (1, 2, 3), (3, 1, 2), (2, 3, 1), \} \\
-1, & \text{ if } (i, j, k) \in \{ (3, 2, 1), (1, 3, 2), (2, 1, 3), \} \\
0, & \text{ if } i = j\ or\ j = k\ or\ j = i.
\end{matrix}\right.$$
2. 常见表达式及其对应的爱因斯坦求和约定与Levi-Civita符号
$$\begin{matrix}
\nabla \textbf{v} & | & \frac{\partial}{\partial x_i} \textbf{e}_i v_i \textbf{e}_i = \frac{\partial}{\partial x_i} v_i \\
\nabla \cdot \textbf{v} & | & \frac{\partial}{\partial x_j} \textbf{e}_j v_i \textbf{e}_i \\
\nabla \times \textbf{v} & | & \epsilon_{ijk} \frac{\partial}{\partial x_i} \textbf{e}_i \textbf{e}_j v_k \textbf{e}_k
\end{matrix}$$
3. 相关例题
例3.1 $$\nabla(\textbf{u}\cdot \textbf{v}) = (\nabla\textbf{v})\cdot\textbf{u}+(\nabla\textbf{u})\cdot\textbf{v}.$$证: 因为$\nabla \textbf{u} = (\frac{\partial u_j}{x_i})$, $\nabla \textbf{v} = (\frac{\partial v_j}{x_i})$, 故
$$\begin{equation}
\begin{aligned}
&(\nabla\textbf{v})\cdot\textbf{u}+(\nabla\textbf{u})\cdot\textbf{v} \\ =& ((\frac{\partial v_j}{x_i} u_j + \frac{\partial u_j}{x_i} v_j) e_i) \\ =& (\frac{\partial (u_j v_j)}{x_i} e_i) \\ =& \nabla(\textbf{u} \cdot \textbf{v}).
\end{aligned}
\end{equation}$$
例3.2 $$(\nabla \times \textbf{v})\times\textbf{a}=[\textbf{v}\nabla-\nabla\textbf{v}]\cdot\textbf{a}.$$证: $$\begin{equation}
\begin{aligned}
&(\nabla \times \textbf{v})\times\textbf{a} \\ =& \epsilon_{ijk} (\nabla \times \textbf{v})_i \textbf{e}_j a_k \\ =& \epsilon_{ijk} \epsilon_{ipq} (\frac{\partial}{\partial x_q} v_p) \textbf{e}_j a_k \\ =& (\delta_{jp} \delta_{kq} – \delta_{jq} \delta_{kp}) (\frac{\partial}{\partial x_q} v_p) \textbf{e}_j a_k \\ =& (\frac{\partial}{\partial x_k} v_j) \textbf{e}_j a_k – (\frac{\partial}{\partial x_j} v_k) \textbf{e}_j a_k \\ =& a_k (\frac{\partial}{\partial x_k} v_j e_j) – (\textbf{e}_j \frac{\partial}{\partial x_j} v_k) a_k \\ =& a_k \textbf{e}_k (\textbf{e}_k \frac{\partial}{\partial x_k} v_j e_j) – (\textbf{e}_j \frac{\partial}{\partial x_j} v_k \textbf{e}_k) a_k \textbf{e}_k \\ =& (a_k \textbf{e}_k \cdot \textbf{e}_k \frac{\partial}{\partial x_k}) v_j e_j – (\nabla \textbf{v}) \cdot \textbf{a} \\ =& (\textbf{a} \cdot \nabla) \textbf{v} – (\nabla \textbf{v}) \cdot \textbf{a}.
\end{aligned}
\end{equation}$$
例3.3 $$\nabla(\textbf{u}\cdot\textbf{v})=\textbf{u}\times(\nabla\times\textbf{v})+\textbf{v}\times(\nabla\times\textbf{u})+\textbf{u}\cdot(\nabla\textbf{v})+\textbf{v}\cdot(\nabla\textbf{u}).$$证: $$\begin{equation}
\begin{aligned}
Rhs =& \textbf{u}\times(\nabla\times\textbf{v})+\textbf{v}\times(\nabla\times\textbf{u})+\textbf{u}\cdot(\nabla\textbf{v})+\textbf{v}\cdot(\nabla\textbf{u}) \\ =& (\nabla \textbf{v}) \cdot \textbf{u} – (\textbf{u} \cdot \nabla) \textbf{v} + (\nabla \textbf{u}) \cdot \textbf{v} – (\textbf{v} \cdot \nabla) \textbf{u} + \textbf{u} \cdot (\nabla \textbf{v}) + \textbf{v} \cdot (\nabla \textbf{u}) \\ =& (\nabla \textbf{v}) \cdot \textbf{u} + (\nabla \textbf{u}) \cdot \textbf{v} \\ =& \nabla(\textbf{u}\cdot\textbf{v}).
\end{aligned}
\end{equation}$$
例3.4 $$\nabla\times(\textbf{u}\times\textbf{v}) = \textbf{v}\cdot(\nabla\textbf{u})-\textbf{v}(\nabla\cdot\textbf{u})+\textbf{u}(\nabla\cdot\textbf{v})-\textbf{u}\cdot(\nabla\textbf{v}).$$证: $$\begin{equation}
\begin{aligned}
Lhs =& \nabla\times(\textbf{u}\times\textbf{v}) \\ =& \epsilon_{ijk} \frac{\partial}{\partial x_i} \textbf{e}_j (\textbf{u} \times \textbf{v})_k \\ =& \epsilon_{ijk} \frac{\partial}{\partial x_i} \textbf{e}_j \epsilon_{kpq} (u_p v_q) \\ =& (\delta_{ip} \delta_{jq} – \delta_{iq} \delta_{jp}) \frac{\partial}{\partial x_i} \textbf{e}_j (u_p v_q) \\ =& \frac{\partial}{\partial x_i} \textbf{e}_j (u_i v_j) – \frac{\partial}{\partial x_i} \textbf{e}_j (u_j v_i) \\ =& v_j \textbf{e}_j \frac{\partial}{\partial x_i} u_i + u_i (\frac{\partial}{\partial x_i} v_j \textbf{e}_j) – v_i (\frac{\partial}{\partial x_i} u_j \textbf{e}_j) – u_j \textbf{e}_j \frac{\partial}{\partial x_i} v_i \\ =& \textbf{v} \cdot (\nabla \textbf{u}) + \textbf{u} (\nabla \cdot \textbf{v}) – \textbf{v} (\nabla \cdot \textbf{u}) – \textbf{u} \cdot (\nabla \textbf{v}).
\end{aligned}
\end{equation}$$