Prerequisites and Appendix #

$\Delta u = \sum_{i=1}^{n} u_{x_i x_i}$

$\nabla f(x) = \sum_{i=1}^{n} \frac{\partial f}{\partial x_i} e_i = (\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \ldots, \frac{\partial f}{\partial x_n})$

Derectional Derivative is defined as $D_v f(x) = \nabla f(x) \cdot v$ which is the derivative of $f$ at $x$ in the direction of $v$.

Note that the directional derivative maximal at the direction of the gradient, and the maximal value is $|\nabla f(x)|$. This is because $\nabla f(x) \cdot v = |\nabla f(x)| |v| \cos \theta$ where $\theta$ is the angle between $\nabla f(x)$ and $v$. The maximum value is achieved when $\theta = 0$, which means $v$ is in the same direction as $\nabla f(x)$.

Lipschitz continuity: A function $f$ is Lipschitz continuous if there exists a constant $L$ such that for all $x, y$ in the domain of $f$, we have $|f(x) - f(y)| \leq L |x - y|$.

Uniform continuity: A function $f$ is uniformly continuous if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x, y$ in the domain of $f$, if $|x - y| < \delta$, then $|f(x) - f(y)| < \epsilon$.

Divergence Theorem: Let $V$ be a vector field defined on a region $D$ in $\mathbb{R}^n$ with a piecewise smooth boundary $\partial D$. Then the divergence theorem states that the flux of $V$ across the boundary $\partial D$ is equal to the integral of the divergence of $V$ over the region $D$. Mathematically, it can be expressed as: $$\int_{\partial D} V \cdot n , dS = \int_{D} \nabla \cdot V , dV$$ where $n$ is the outward unit normal vector to the boundary $\partial D$, and $dS$ and $dV$ are the surface area element and volume element, respectively.

Harmonic functions: A function $u$ is harmonic in a domain $D$ if it satisfies Laplace’s equation $\Delta u = 0$ in $D$ and the derivatives are defined.

Uniformly bounded: A function $f$ is uniformly bounded if there exists a constant $M$ such that for all $x$ in the domain of $f$, we have $|f(x)| \leq M$.

Partial Chain Rule: $\frac{\partial f}{\partial x_i} = \sum_{j=1}^{m} \frac{\partial f}{\partial y_j} \frac{\partial y_j}{\partial x_i}$ where $y_j$ are functions of $x_i$.

Hyperplane: A hyperplane in $\mathbb{R}^n$ is a subspace of dimension $n-1$. It can be defined as the set of points $x$ in $\mathbb{R}^n$ that satisfy a linear equation of the form $a_1 x_1 + a_2 x_2 + \ldots + a_n x_n = b$, where $a_i$ are constants and not all zero, and $b$ is a constant.

Curl: The curl of a vector field $F = (P, Q, R)$ in three-dimensional space is defined as: $$\nabla \times F = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)$$ The curl measures the rotation of the vector field at a point. If the curl is zero, the vector field is said to be irrotational. In the form of a determinant, the curl can be expressed as: $$\nabla \times F = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix}$$