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:

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:

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: