ED Formulas

Symbol Convention%

In accordance with ISO 80000-2, the following font conventions are employed:
- Scalars and components for vectors or tensors are represented by lightface italic type ( $a, ϕ, a_{i}, T_{i j}$ ).
- Vectors are represented by boldface italic type ( $a, ω$ ).
- Second-order tensors are represented by boldface sans-serif type ( $T, I$ ).
Operators & Constants: Roman (upright) type is used for fixed mathematical constants (e.g., Pi $π$ , the imaginary unit $i$ ) and differential operators (e.g., the differential $d$ in $d x$ ).
Calculus Notation: For integrals, a thin space (\,) is used to separate the integrand from the differential operator, e.g., $\int f (x) d x$ .
Minkowski Metric: The Minkowski metric tensor $η_{μ ν}$ is defined using the mostly-plus signature convention:

η_{μ ν} = diag (- 1, + 1, + 1, + 1)

Consequently, the invariant spacetime interval is given by $d s^{2} = - c^{2} d t^{2} + d x^{2} + d y^{2} + d z^{2}$ .

1 Preparatory Math.

1.1 Vector and tensor analysis (Euclidean geometry)

1.1.1 Basics

Kronecker & Levi-Civita symbols

δ_{i j} = {\hat{e}}_{i} \cdot {\hat{e}}_{j}, ϵ_{i j k} = ({\hat{e}}_{i} \times {\hat{e}}_{j}) \cdot {\hat{e}}_{k} \Leftrightarrow {\hat{e}}_{i} \times {\hat{e}}_{j} = ϵ_{i j k} {\hat{e}}_{k}

Determinant

ϵ_{l m n} det A_{3 \times 3} = ϵ_{i j k} A_{i l} A_{j m} A_{k n} = ϵ_{i j k} A_{l i} A_{m j} A_{n k}

⟹ det A_{3 \times 3} = \frac{1}{6} ϵ_{i j k} ϵ_{l m n} A_{i l} A_{j m} A_{k n} or det A_{3 \times 3} = ϵ_{i j k} A_{i 1} A_{j 2} A_{k 3} = ϵ_{i j k} A_{1 i} A_{2 j} A_{3 k}

Contraction identities

ϵ_{i j k} ϵ_{i m n} = δ_{j m} δ_{k n} - δ_{j n} δ_{k m}, ϵ_{i j k} ϵ_{i j n} = 2 δ_{k n}, ϵ_{i j k} ϵ_{i j k} = 3! = 6

More generally*

$ϵ_{i j k} ϵ_{l m n} = det (\begin{matrix} δ_{i l} & δ_{i m} & δ_{i n} \\ δ_{j l} & δ_{j m} & δ_{j n} \\ δ_{k l} & δ_{k m} & δ_{k n} \end{matrix})$

High-dimensional case:

ϵ_{i_{1} i_{2} \dots i_{n}} ϵ_{j_{1} j_{2} \dots j_{n}} = det (\begin{matrix} δ_{i_{1} j_{1}} & δ_{i_{1} j_{2}} & \dots & δ_{i_{1} j_{n}} \\ δ_{i_{2} j_{1}} & δ_{i_{2} j_{2}} & \dots & δ_{i_{2} j_{n}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ δ_{i_{n} j_{1}} & δ_{i_{n} j_{2}} & \dots & δ_{i_{n} j_{n}} \end{matrix}) ≜ δ_{i_{1} i_{2} \dots i_{n}}^{j_{1} j_{2} \dots j_{n}} (Generalized Kronecker delta)

Axial Vector / Pseudovector

A_{i}^{'} = det (R) R_{i j} A_{i}, R \in O (3)

Symmetric-Antisymmetric decomposition

T_{i j} = T_{(i j)} + T_{[i j]} = \frac{T_{i j} + T_{j i}}{2} + \frac{T_{i j} - T_{j i}}{2}

Double Contraction (adopt to the proximity rule)

A : B ≜ A_{i j} B_{j i} = tr (A : B)

1.1.2 Cross & dot product

Scalar Triple product

a \cdot (b \times c) = b \cdot (c \times a) = c \cdot (a \times b)

Vector Triple product / BAC-CAB formula

a \times (b \times c) = b a \cdot c - c a \cdot b or (b \times c) \times a = c a \cdot b - b a \cdot c

Lagrange's identity

(a \times b) \cdot (c \times d) = (a \cdot c) (b \cdot d) - (a \cdot d) (b \cdot c)

Associative Law (the tensor remains centered in the contraction)

a \cdot T \cdot b = (a \cdot T) \cdot b = a \cdot (T \cdot b)

a \cdot T \times b = (a \cdot T) \times b = a \cdot (T \times b)

a \times T \times b = (a \times T) \times b = a \times (T \times b), T \in R^{n} \otimes R^{n} \dots \otimes R^{n}

Cross $\to$ Dot

ω \times I = I \times ω = (\begin{matrix} 0 & - ω_{3} & ω_{2} \\ ω_{3} & 0 & - ω_{1} \\ - ω_{2} & ω_{1} & 0 \end{matrix}) ≜ Ω \Leftrightarrow Ω_{i j} = ϵ_{i k j} ω_{k} ⟹ ω \times v = Ω \cdot v, \forall v \in R^{n}

1.1.3 $\nabla$

Differential operations

Coordinate Component Expansion of Differential Operators

grad Φ ≜ {\hat{e}}_{i} \partial_{i} Φ

div v ≜ ({\hat{e}}_{i} \partial_{i}) \cdot v = \partial_{i} v_{i}

rot v (or curl) ≜ ({\hat{e}}_{i} \partial_{i}) \times v = ϵ_{i j k} (\partial_{j} v_{k}) {\hat{e}}_{i} = | \begin{matrix} {\hat{e}}_{1} & {\hat{e}}_{2} & {\hat{e}}_{3} \\ \partial_{1} & \partial_{2} & \partial_{3} \\ v_{1} & v_{2} & v_{3} \end{matrix} |

\nabla^{2} Φ = \nabla \cdot \nabla Φ = \sum_{i} \partial_{i}^{2} Φ

Two critical indentities

\nabla \times \nabla φ = 0, \nabla \cdot (\nabla \times a) = 0

Leibniz Rule

\begin{aligned} \nabla (φ ψ) & = (\nabla φ) ψ + φ (\nabla ψ) \\ \nabla (a \cdot b) & = a \cdot \nabla b + b \cdot \nabla a + a \times (\nabla \times b) + b \times (\nabla \times a) \\ ⟹ a \times (\nabla \times a) = \frac{1}{2} \nabla (a \cdot a) - a \cdot \nabla a \end{aligned}

\begin{aligned} \nabla (φ a) & = (\nabla φ) a + φ (\nabla a) \\ \nabla \cdot (φ a) & = (\nabla φ) \cdot a + φ (\nabla \cdot a) \\ \nabla \times (φ a) & = (\nabla φ) \times a + φ (\nabla \times a) \end{aligned}

\begin{aligned} \nabla (a \times b) & = (\nabla a) \times b - (\nabla b) \times a \\ \nabla \cdot (a \times b) & = (\nabla \times a) \cdot b - (\nabla \times b) \cdot a \\ \nabla \times (a \times b) & = (\nabla \cdot b + b \cdot \nabla) a - (\nabla \cdot a + a \cdot \nabla) b \end{aligned}

\begin{aligned} \nabla \cdot (φ T) & = (\nabla φ) \cdot T + φ (\nabla \cdot T) \\ \nabla \cdot (a \cdot T) & = (\nabla a) : T + a \cdot (\nabla \cdot T) \end{aligned}

\begin{aligned} \nabla \cdot (a b) & = (\nabla \cdot a) b + (a \cdot \nabla) b \\ \nabla \cdot (a b c) & = (\nabla \cdot a) b c + (a \cdot \nabla b) c + b a \cdot \nabla c \end{aligned}

\nabla \times (a \times b) = \nabla \cdot (b a - a b) = (\nabla \cdot b + b \cdot \nabla) a - (\nabla \cdot a + a \cdot \nabla) b

Taylor Series in Operator Form

φ (x + ϵ) = e^{ϵ \cdot \nabla} φ (x) = [1 + (ϵ \cdot \nabla) + \frac{1}{2!} (ϵ \cdot \nabla)^{2} + \frac{1}{3!} (ϵ \cdot \nabla)^{3} + \dots] φ (x)

here:

\frac{1}{2} (ϵ \cdot \nabla) (ϵ \cdot \nabla) φ (x) = \frac{1}{2} ϵ_{i} ϵ_{j} \frac{\partial^{2} φ}{\partial x_{i} \partial x_{j}}, \frac{1}{6} (ϵ \cdot \nabla)^{3} φ (x) = \frac{1}{6} ϵ_{i} ϵ_{j} ϵ_{k} \frac{\partial^{3} φ}{\partial x_{i} \partial x_{j} \partial x_{k}}, \dots

$T (ϵ) = e^{ϵ \cdot \nabla}$ is so-called "Translation Operator" in quantum mechanics or lie group ( $\hat{T} (ϵ) = e^{\frac{i}{ℏ} ϵ \cdot \hat{p}}$ ).

Integral operations

Coordinate-independent definition

d φ (r) = \nabla φ \cdot d l

\nabla \cdot F ≜ lim_{V \to 0} [\frac{1}{V} \oint_{\partial V} F \cdot d S]

\hat{n} \cdot (\nabla \times F) ≜ lim_{S \to 0} [\frac{1}{S} \oint_{\partial S} F \cdot d l]

Fundamental theorem of gradients

\int_{P}^{Q} d l \cdot \nabla = |_{P}^{Q}, = φ, a, T, \dots

⟹ \oint_{C} d l \cdot \nabla = 0 (conservative field)

Generalized Gauss's Theorem

∭_{V} d V \nabla = \oint_{\partial V} d σ, = φ, \times a, \cdot a, \times T \dots, with d σ = n d S

⟹ \oint_{\partial V} d σ = 0

also ⟹ \nabla = lim_{V \to 0} [\frac{1}{V} \oint_{\partial V} d σ] ⟹ {\begin{aligned} \nabla φ & = lim_{V \to 0} [\frac{1}{V} \oint_{\partial V} d σ φ] \\ \nabla \cdot F & = lim_{V \to 0} [\frac{1}{V} \oint_{\partial V} d σ \cdot F] (Classic Gauss) \\ \nabla \times F & = lim_{V \to 0} [\frac{1}{V} \oint_{\partial V} d σ \times F] \end{aligned}

In Gauss's Divergence Theorem $∭_{V} d V \nabla \cdot F = \oint_{\partial V} d σ \cdot F$ , if $F = φ \nabla ψ$ , one find the Green's first identity:

\int_{V} d V [φ \nabla^{2} ψ + \nabla φ \cdot \nabla ψ] = \oint_{\partial V} d σ \cdot φ \nabla ψ

and substituting $F = φ \nabla ψ - ψ \nabla φ$ yields the Green's second identity:

\int_{V} d V [φ \nabla^{2} ψ - ψ \nabla^{2} φ] = \oint_{\partial V} d σ \cdot [φ \nabla ψ - ψ \nabla φ]

Generalized Stokes' Theorem

\oint_{S} (d σ \times \nabla) = \oint_{\partial S} d l, = φ, \times a, \cdot a, \times T \dots

⟹ \oint_{\partial S} d l \cdot F = \oint_{S} (d σ \times \nabla) \cdot F = \oint_{S} d σ \cdot (\nabla \times F) (Classic Stokes)

Helmholtz decomposition

For any continuous differentiable vector field $F$ , if $lim_{r \to \infty} r F (x) \to 0$ ,

\begin{aligned} F & = \underset{longitudinal/irrotational part}{\underset{⏟}{- \nabla φ}} + \underset{transverse/solenoidal part}{\underset{⏟}{\nabla \times A}} \\ = - \frac{1}{4 π} \nabla \int \frac{\nabla^{'} \cdot F (x^{'})}{| x - x^{'} |} d V^{'} + \frac{1}{4 π} \nabla \times \int \frac{\nabla^{'} \times F (x^{'})}{| x - x^{'} |} d V^{'} \end{aligned}

for static magnetic field, Biot-Savart Law:

⟹ B (x) = \frac{1}{4 π} \nabla \times \int \frac{μ_{0} J (x^{'})}{| x - x^{'} |} d V^{'} = \frac{μ_{0}}{4 π} \int \frac{J (x^{'}) \times (x - x^{'})}{| x - x^{'} |^{3}} d V^{'}, x : field; x^{'} : source

for static electric field, Coulomb's Law

⟹ E (x) = - \frac{1}{4 π} \nabla \int \frac{ρ (x^{'}) / ε_{0}}{| x - x^{'} |} d V^{'} = \frac{1}{4 π ε_{0}} \int \frac{ρ (x^{'}) (x - x^{'})}{| x - x^{'} |^{3}} d V^{'}

1.1.4 Cases

Determinant

det (a b) = 0, det (I + a b) = 1 + a \cdot b

For $r = | r |, r = (x, y, z) = r_{i} {\hat{e}}_{i} = r \hat{r}$ & $\nabla$

\nabla r = \hat{r}, \nabla r = I, \nabla \cdot r = 3, \nabla \times r = 0

\nabla \times \hat{r} = 0, \nabla \cdot \hat{r} = \frac{2}{r}, \nabla \hat{r} = \frac{I - \hat{r} \hat{r}}{r}, \nabla \times [f (r) \hat{r}] = 0

\nabla^{2} (- \frac{1}{r}) = \nabla \cdot (\frac{\hat{r}}{r^{2}}) = 4 π δ^{3} (r), \nabla \frac{1}{r} = - \frac{\hat{r}}{r^{2}}, \nabla \frac{\hat{r}}{r^{2}} = \frac{I - 3 \hat{r} \hat{r}}{r^{3}} + \frac{4 π δ^{3} (x)}{3} I

Double Contraction (adopt to the proximity rule)

I : a b = a \cdot b, I : \nabla a = \nabla \cdot a

$\nabla$

\nabla \cdot φ I = \nabla φ

\nabla \cdot (T \times r) = - r \times (\nabla \cdot T), with T_{i j} = T_{j i}

1.1.5 $\nabla$ in the orthogonal curvilinear coordinates

Definition

In this section, we adopt $e_{i}$ rather than ${\hat{e}}_{i}$ to indicate that the basis vectors are orthogonal but not normalized: $e_{i} = h_{i} {\hat{e}}_{i} (no summation)$ . Replace the Cartesian coordinate values $(x_{1}, x_{2}, x_{3})$ with the Curvilinear coordinate values $(u_{1}, u_{2}, u_{3})$ .

Lamé Parameters

h_{i} = | \frac{\partial r}{\partial u_{i}} |, H = \prod_{i = 1}^{3} h_{i}, {\hat{e}}_{i} = \frac{\partial r / \partial u_{i}}{| \partial r / \partial u_{i} |} = \frac{1}{h_{i}} \frac{\partial r}{\partial u_{i}} = \frac{1}{h_{i}} e_{i} (no summation)

for Cylindrical coordinates, $r = (ρ \cos ϕ, ρ \sin ϕ, z)$ :

h_{ρ}, h_{ϕ}, h_{z} = 1, ρ, 1 and {\begin{cases} {\hat{e}}_{ρ} = \frac{1}{h_{ρ}} \frac{\partial r}{\partial ρ} = (\cos ϕ, \sin ϕ, 0) \\ {\hat{e}}_{ϕ} = \frac{1}{h_{ϕ}} \frac{\partial r}{\partial ϕ} = (- \sin ϕ, \cos ϕ, 0) \\ {\hat{e}}_{z} = \frac{1}{h_{z}} \frac{\partial r}{\partial z} = (0, 0, 1) \end{cases}

for Spherical coordinates, $r = (r \sin θ \cos ϕ, r \sin θ \sin ϕ, r \cos θ)$ :

h_{r}, h_{θ}, h_{ϕ} = 1, r, r \sin θ and {\begin{cases} {\hat{e}}_{r} = \frac{1}{h_{r}} \frac{\partial r}{\partial r} = (\sin θ \cos ϕ, \sin θ \sin ϕ, \cos θ) \\ {\hat{e}}_{θ} = \frac{1}{h_{θ}} \frac{\partial r}{\partial θ} = (\cos θ \cos ϕ, \cos θ \sin ϕ, - \sin θ) \\ {\hat{e}}_{ϕ} = \frac{1}{h_{ϕ}} \frac{\partial r}{\partial ϕ} = (- \sin ϕ, \cos ϕ, 0) \end{cases}

$\nabla$

Coordinate Component Expansion of Differential Operators

grad Φ = \sum_{i} \frac{1}{h_{i}} e_{i} \partial_{i} Φ = {\hat{e}}_{i} \partial_{i} Φ

div v = \sum_{i} \frac{1}{H} \partial_{i} (\frac{H v_{i}}{h_{i}}) = \frac{1}{h_{1} h_{2} h_{3}} [\frac{\partial}{\partial u_{1}} (h_{2} h_{3} v_{1}) + \frac{\partial}{\partial u_{2}} (h_{3} h_{1} v_{2}) + \frac{\partial}{\partial u_{3}} (h_{1} h_{2} v_{3})]

curl v = \frac{1}{h_{1} h_{2} h_{3}} | \begin{matrix} h_{1} {\hat{e}}_{1} & h_{2} {\hat{e}}_{2} & h_{3} {\hat{e}}_{3} \\ \frac{\partial}{\partial u_{1}} & \frac{\partial}{\partial u_{2}} & \frac{\partial}{\partial u_{3}} \\ h_{1} v_{1} & h_{2} v_{2} & h_{3} v_{3} \end{matrix} |

\nabla^{2} v = \nabla \cdot \nabla v = \sum_{k} \frac{1}{H} \partial_{k} (\frac{H}{h_{k}} (\nabla v)_{k}) = \sum_{k} \frac{1}{H} \partial_{k} (\frac{H}{h_{k}} \partial_{k}^{2} v)

The differential identity of the basis vector

\nabla u_{i} = {\hat{e}}_{i} = \frac{e_{i}}{h_{i}}, \nabla \times \frac{e_{i}}{h_{i}} = \nabla \times {\hat{e}}_{i} = 0, \nabla \cdot \frac{h_{i} e_{i}}{H} = 0 (no summation)

For Cylindrical coordinates, $r = (r \cos ϕ, r \sin ϕ, z)$

\begin{aligned} \nabla v = e_{r} \frac{\partial v}{\partial r} + e_{θ} \frac{1}{r} \frac{\partial v}{\partial θ} + e_{z} \frac{\partial v}{\partial z} \\ \nabla \cdot v = \frac{1}{r} \frac{\partial}{\partial r} (r v_{r}) + \frac{1}{r} \frac{\partial v_{θ}}{\partial θ} + \frac{\partial v_{z}}{\partial z} \\ \nabla \times v = \frac{1}{r} | \begin{array}{c} e_{r} & r e_{θ} & e_{z} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial θ} & \frac{\partial}{\partial z} \\ v_{r} & r v_{θ} & v_{z} \end{array} | \\ \nabla^{2} v = \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial v}{\partial r}) + \frac{1}{r^{2}} \frac{\partial^{2} v}{\partial θ^{2}} + \frac{\partial^{2} v}{\partial z^{2}} \end{aligned}

For Spherical coordinates, $r = (r \sin θ \cos φ, r \sin θ \sin φ, r \cos θ)$

\begin{aligned} \nabla v = e_{r} \frac{\partial v}{\partial r} + e_{θ} \frac{1}{r} \frac{\partial v}{\partial θ} + e_{φ} \frac{1}{r \sin θ} \frac{\partial v}{\partial φ} \\ \nabla \cdot v = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} v_{r}) + \frac{1}{r \sin θ} \frac{\partial}{\partial θ} (\sin θ v_{θ}) + \frac{1}{r \sin θ} \frac{\partial v_{φ}}{\partial φ} \\ \nabla \times v = \frac{1}{r^{2} \sin θ} | \begin{array}{c} e_{r} & r e_{θ} & r \sin θ e_{φ} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial θ} & \frac{\partial}{\partial φ} \\ v_{r} & r v_{θ} & r \sin θ v_{φ} \end{array} | \\ \nabla^{2} v = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial v}{\partial r}) + \frac{1}{r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial v}{\partial θ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2} v}{\partial φ^{2}} \end{aligned}

1.2 Dirac $δ$ function

1.2.1 Definition

1D definition

\int_{R} δ (x - a) φ (x) d x = φ (a) ⟹ \int_{R} δ^{(n)} (x - a) φ (x) d x = (- 1)^{n} φ^{(n)} (a)

⟹ \int_{R} δ^{(n)} (x - a) d x = {\begin{cases} 1, n = 1 \\ 0, n > 1 \end{cases}

δ (x) = \frac{d}{d x} Θ (x), Θ (x) = {\begin{cases} 0, & x < 0 \\ 1, & x > 0 \end{cases}

3D definition

δ^{3} (x - a) = δ (x_{1} - a_{1}) δ (x_{2} - a_{2}) δ (x_{3} - a_{3})

⟹ \int_{R^{3}} δ^{3} (x - a) φ (x) d^{3} x = φ (a)

\int_{R^{3}} \partial^{α} δ^{3} (x - a) φ (x) d^{3} x = (- 1)^{α} \partial^{α} φ (a), with \partial^{α} = {(\frac{\partial}{\partial x_{1}})}^{α_{1}} {(\frac{\partial}{\partial x_{2}})}^{α_{2}} {(\frac{\partial}{\partial x_{3}})}^{α_{3}}, α = α_{1} + α_{2} + α_{3}

⟹ \int_{R^{3}} [\nabla δ^{3} (x - a)] φ (x) d^{3} x = - \nabla φ (a)

⟹ \int_{R^{3}} [\nabla δ^{3} (x - a)] \cdot A (x) d^{3} x = - (\nabla \cdot A) |_{x = a}

⟹ \int_{R^{3}} [\nabla^{2} δ^{3} (x - a)] φ (x) d^{3} x = \nabla^{2} φ (a)

1.2.2 Fundamental characteristics

x δ (x) = 0, x^{n} δ^{(n)} (x) = (- 1)^{n} n! δ (x), x_{i} δ^{3} (x) = 0, x_{i} \partial_{j} δ^{3} (x) = - δ_{i j} δ^{3} (x)

for $f (x) : R \to R$ , one find

δ (f (x)) = \sum_{n} \frac{δ (x - x_{i})}{| f^{'} (x_{i}) |}, with f (x_{i}) = 0, f^{'} (x_{i}) \neq 0

for $f (x) : R^{n} \to R$

\int_{R^{n}} g (x) δ (f (x)) d^{n} x = \int_{f (x) = 0} \frac{g (x)}{| \nabla f (x) |} d S, with \nabla f (x_{i}) \neq 0

for $F (x) : R^{n} \to R^{n}$

δ (F (x)) = \sum_{i} δ (x - x_{i}) / {| \frac{\partial (F_{1}, F_{2}, \dots, F_{n})}{\partial (x_{1}, x_{2}, \dots, x_{n})} |}_{x = x_{i}}, with F (x_{i}) = 0, det (\frac{\partial F_{i}}{\partial x_{j}}) \neq 0

$⟹$ for the orthogonal curvilinear coordinates

δ^{3} (x - x^{'}) = \frac{1}{H} δ^{3} (u - u^{'})

1.2.3 Cases

the point dipole (Electric dipole or magnetic dipole)
Let the point dipole be at the origin, and its charge density distribution is:

ρ (x) = - p \cdot \nabla δ^{3} (x)

Because the total charge is zero,

Q = \int_{R^{3}} ρ (x) d^{3} x = \int_{R^{3}} [- p \cdot \nabla δ^{3} (x)] d^{3} x = - (- p \cdot \int_{R^{3}} δ^{3} (x) \nabla (1) d^{3} x) = 0

and the first-order moment (dipole moment) is $p$ ,

\begin{aligned} d & = \int x ρ (x) d^{3} x = \int_{R^{3}} x [- p \cdot \nabla δ^{3} (x)] d^{3} x \\ = \int_{R^{3}} x [- p_{i} \frac{\partial δ^{3}}{\partial x_{i}}] d^{3} x = - p_{i} \int_{R^{3}} x \frac{\partial δ^{3}}{\partial x_{i}} d^{3} x \\ = p_{i} {\frac{\partial x}{\partial x_{i}} |}_{x = 0} = p_{i} {\hat{e}}_{i} = p \end{aligned}

2 Fundamentals of Electromagnetism

2.1 Maxwell's equation (in cosmos)

In cosmos

{\begin{aligned} \nabla \cdot E & = \frac{ρ}{ϵ_{0}} & (Gauss’s Law) \\ \nabla \cdot B & = 0 & (Gauss’s Law for Magnetism) \\ \nabla \times E & = - \frac{\partial B}{\partial t} & (Faraday Law) \\ \nabla \times B & = μ_{0} J + \frac{1}{c^{2}} \frac{\partial E}{\partial t} & (Ampere-Maxwell Law) \end{aligned}

with $c = \frac{1}{\sqrt{μ_{0} ϵ_{0}}}$ .

In media

{\begin{aligned} \nabla \cdot D & = ρ_{f} & (Gauss’s Law) \\ \nabla \cdot B & = 0 & (Gauss’s Law for Magnetism) \\ \nabla \times E & = - \frac{\partial B}{\partial t} & (Faraday’s Law) \\ \nabla \times H & = J_{f} + \frac{\partial D}{\partial t} & (Ampere-Maxwell Law) \end{aligned}

\nabla \cdot E = \frac{ρ_{f} + ρ_{p}}{ϵ_{0}}, ρ_{p} = - \nabla \cdot P

\nabla \times B = μ_{0} (J_{f} + J_{M} + J_{P}) + \frac{1}{c^{2}} \frac{\partial E}{\partial t} = μ_{0} (J_{f} + \nabla \times M + \frac{\partial P}{\partial t}) + \frac{1}{c^{2}} \frac{\partial E}{\partial t}

Lorentz force

\frac{d p}{d t} = q E + q v \times B

Ohm law

j = σ (\underset{constitutive term}{\underset{⏟}{E}} + \underset{convective term}{\underset{⏟}{u \times B}} - \underset{hall term}{\underset{⏟}{\frac{1}{n e} j \times B}} + \underset{electron pressure term}{\underset{⏟}{\frac{\nabla p_{e}}{n e}}} - \underset{inertial term}{\underset{⏟}{\frac{m_{e}}{n e^{2}} \frac{\partial j}{\partial t}}} + \underset{more and more}{\underset{⏟}{\dots}})

2.2 Polarization and magnetization

Polarization and magnetization intensity

D = ϵ_{0} E + P, H = \frac{1}{μ_{0}} B - M

for linear isotropic media,

D = ϵ E, H = \frac{1}{μ} B

Charge / Current

{\begin{aligned} J_{P} & = \frac{\partial P}{\partial t} & (Polarizing Current) \\ J_{M} & = \nabla \times M, K_{M} = \hat{n} \times (M_{1} - M_{2}) & (Magnetizing Current) \\ ρ_{p} & = - \nabla \cdot P, σ_{p} = \hat{n} \cdot (P_{1} - P_{2}) & (Polarizing Charge) \end{aligned}

2.3 Boundary conditions

In cosmos ( $\hat{n} = {\hat{n}}_{1 \to 2}$ )

ϵ_{0} (E_{2} - E_{1}) \cdot \hat{n} = σ, (B_{2} - B_{1}) \cdot \hat{n} = 0, \hat{n} \times (E_{2} - E_{1}) = 0, \frac{1}{μ_{0}} \hat{n} \times (B_{2} - B_{1}) = K

or simply

ϵ_{0} (E_{2} - E_{1}) = σ \hat{n}, \frac{1}{μ_{0}} (B_{2} - B_{1}) = K \times \hat{n}

In media

(D_{2} - D_{1}) \cdot \hat{n} = σ_{f}, (B_{2} - B_{1}) \cdot \hat{n} = 0, \hat{n} \times (E_{2} - E_{1}) = 0, \hat{n} \times (H_{2} - H_{1}) = K_{f}

ϵ_{0} (E_{2} - E_{1}) \cdot \hat{n} = σ_{f} + σ_{p}, (B_{2} - B_{1}) \cdot \hat{n} = 0, \hat{n} \times (E_{2} - E_{1}) = 0, \frac{1}{μ_{0}} \hat{n} \times (B_{2} - B_{1}) = K_{f} + K_{M}

with $J_{P} = \frac{\partial P}{\partial t} = 0$ .

2.4 Electromagnetic potential

Definition

E = - \nabla φ - \frac{\partial A}{\partial t}, B = \nabla \times A

Coulomb gauge

C ≜ \nabla \cdot A = 0

Lorentz gauge

L ≜ \nabla \cdot A + \frac{1}{c^{2}} \frac{\partial φ}{\partial t} = 0

Gauge invariance

φ^{'} = φ - \frac{\partial ψ}{\partial t}, A^{'} = A + \nabla ψ

Electromagnetic potential equation

\begin{aligned} ◻ φ + \partial_{t} L & = - \frac{ρ}{ϵ_{0}} \\ ◻ A - \nabla L & = - μ_{0} J \end{aligned}

with $◻ ≜ \nabla^{2} - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} (d’Alembert operator)$ .

2.5 EM Wave

2.5.1 Wave equation

Wave equation for free time-varying field

◻ E = 0, ◻ B = 0

Definition: phase of monochromatic wave

ϕ = k \cdot x - ω t

Dispersion relation

ω = k c

⟹ v_{p} = \frac{ω}{k} = c, v_{g} = \frac{d ω}{d k} = c

2.5.2 Polarization

Real description

E_{1} = A_{1} \cos ϕ, E_{2} = A_{2} \cos (ϕ + δ)

⟹ {(\frac{E_{1}}{A_{1}})}^{2} + {(\frac{E_{2}}{A_{2}})}^{2} - 2 \frac{E_{1} E_{2}}{A_{1} A_{2}} \cos δ = \sin^{2} δ

for certain $k \cdot x$ ,

E_{1} = A_{1} \cos ω t, E_{2} = A_{2} \cos (ω t - δ)

⟹ {\begin{cases} \sin δ > 0, Right-hand \\ \sin δ < 0, Left-hand \end{cases}

Complex description

{\tilde{E}}_{1} = A_{1} e^{i ϕ}, {\tilde{E}}_{2} = A_{2} e^{i (ϕ + δ)}

for certain $k \cdot x$ ,

{\tilde{E}}_{1} = A_{1} e^{i (- ω t)}, {\tilde{E}}_{2} = A_{2} e^{i (- ω t + δ)}

define the degree of polarization

\tilde{R} = \frac{{\tilde{E}}_{2}}{{\tilde{E}}_{1}} = \frac{A_{2}}{A_{1}} e^{i δ}

⟹ {\begin{cases} Im \tilde{R} > 0, Right-hand \\ Im \tilde{R} < 0, Left-hand \end{cases}

Circularly polarized basis vectors

{\begin{cases} {\hat{e}}_{+} ≜ \frac{{\hat{e}}_{1} + i {\hat{e}}_{2}}{\sqrt{2}}, Right-hand \\ {\hat{e}}_{-} ≜ \frac{{\hat{e}}_{1} - i {\hat{e}}_{2}}{\sqrt{2}}, Left-hand \end{cases}

satisfy

{\hat{e}}_{-} = {\hat{e}}_{+}^{*}, {\hat{e}}_{\pm}^{*} \cdot {\hat{e}}_{\pm} = 1, {\hat{e}}_{\pm} \cdot {\hat{e}}_{\pm} = 0

Monochromatic waves can be transformed from 2D Cartesian coordinates to circularly polarized coordinates,

E = E_{0} e^{i ϕ} = (E_{+} {\hat{e}}_{+} + E_{-} {\hat{e}}_{-}) e^{i ϕ}, E_{\pm} = {\hat{e}}_{\pm}^{*} \cdot E_{0}

2.5.3 Complex description

for ${\tilde{E}}_{1} = A_{1} e^{i ϕ}, {\tilde{E}}_{2} = A_{2} e^{i (ϕ + δ)}$ , if $A_{1, 2}$ is independent with $(t, x)$ ,

\nabla ⟶ i k, \nabla \cdot ⟶ i k \cdot, \nabla \times ⟶ i k \times, \frac{\partial}{\partial t} ⟶ - i ω

Long - term average: for $\tilde{f} = {\tilde{f}}_{0} e^{- i ω t}, \tilde{g} = {\tilde{g}}_{0} e^{- i ω t}$ ,

⟨ ℜ [\tilde{f}] \cdot ℜ [\tilde{g}] ⟩ = \frac{1}{2} ℜ [{\tilde{f}}_{0} {\tilde{g}}_{0}^{*}]

In the vector scenario,

⟨ ℜ [\tilde{f}] \cdot ℜ [\tilde{g}] ⟩ = \frac{1}{2} ℜ [\tilde{f} \cdot {\tilde{g}}^{*}], ⟨ ℜ [\tilde{f}] \times ℜ [\tilde{g}] ⟩ = \frac{1}{2} ℜ [\tilde{f} \times {\tilde{g}}^{*}]

2.6 Conservation / Continuity equation

Charge conservation

\nabla \cdot j + \frac{\partial ρ}{\partial t} = 0

Energy conservation

- \frac{\partial ε}{\partial t} = \nabla \cdot S + f \cdot v

with EMF (electromagnetic field) energy, Poynting vector and Lorentz force

ε = \frac{1}{2} ϵ_{0} E^{2} + \frac{B^{2}}{2 μ_{0}}, S = \frac{1}{μ_{0}} E \times B, f = ρ_{e} E + J \times B

Momentum conservation

- \frac{\partial g}{\partial t} = \nabla \cdot T + f

with EMF momentum and Maxwell stress tensor

g = ϵ_{0} E \times B = S / c^{2}, T_{M} = - T = - ε I + (ϵ_{0} E E + \frac{B B}{μ_{0}})

In a steady state where the electromagnetic momentum is constant over time, the total force acting on the particles within volume $V$ can be expressed as

F = ∭_{V} f d V = - \oint_{\partial V} d σ \cdot T

Angular momentum conservation

- \frac{\partial}{\partial t} (r \times g) = \underset{= r \times (\nabla \cdot T)}{\underset{⏟}{\nabla \cdot (r \times T)}} + r \times f

In linear homogeneous media
- Charge conservation

\nabla \cdot j_{f} + \frac{\partial ρ_{f}}{\partial t} = 0, \nabla \cdot (j_{P} + j_{M}) + \frac{\partial ρ_{p}}{\partial t} = 0

Energy conservation (Poynting's Theorem in Media)

\frac{\partial ε}{\partial t} = \nabla \cdot S + j_{f} \cdot E

with EMF energy and Poynting vector and Lorentz force

ε = \frac{1}{2} (E \cdot D + B \cdot H), S = E \times H

Momentum conservation

\frac{\partial g}{\partial t} = \nabla \cdot T + f_{f}

with EMF momentum (Minkowski form), Maxwell stress tensor and Lorentz force acting on free particles

g = D \times B, T_{M} = - T = - ε I + (E D + H B), f_{f} = ρ_{f} E + J_{f} \times B

In a steady state where the electromagnetic momentum is constant over time, the total force on free particles within volume V is

F_{f} = ∭_{V} f_{f} d V = - \oint_{\partial V} d σ \cdot T

Angular momentum conservation

\frac{\partial}{\partial t} (r \times g) = \nabla \cdot (r \times T) + r \times f_{f}

For inhomogeneous media

f_{t o t a l} = f_{f} - \frac{1}{2} E^{2} \nabla ϵ - \frac{1}{2} H^{2} \nabla μ

3 Special Relativity & Tensor analysis (Minkowski spacetime)

3.1 Fundamental definition

(g)_{α β} = (g)^{α β} = d i a g (- 1, + 1, + 1, + 1)_{α β}

Primary 4-vector

x^{α} = (c t, x), u^{α} = \frac{d x^{α}}{d τ} = γ (c, v), a^{α} = \frac{d u^{α}}{d τ} = (γ^{4} (β \cdot a), γ^{2} a + γ^{4} (β \cdot a) β)

u^{α} u_{α} = - c^{2}, a^{μ} u_{μ} = 0

The accelerated speed in the instantaneous co-moving frame is $a_{i n}$ ,

a^{μ} a_{μ} = | a_{i n} |^{2} = γ^{4} [| a |^{2} + γ^{2} (β \cdot a)^{2}] ⟹ | a_{i n} | = {\begin{cases} γ^{2} a, β ⊥ a \\ γ^{3} a, β ∥ a \end{cases}

Primary characteristics of metric

g_{α β} g^{β γ} = δ_{α}^{γ}

Lorentz transformation

d s^{2} = g_{α β} d x^{α} d x^{β} = d x_{α} d x^{β} = - c^{2} d t^{2} + | d x |^{2} = - c^{2} d τ^{2}

Christoffel symbols (the first kind)*

Γ_{ρ μ ν} = \frac{1}{2} (\frac{\partial g_{ρ μ}}{\partial x^{ν}} + \frac{\partial g_{ρ ν}}{\partial x^{μ}} - \frac{\partial g_{μ ν}}{\partial x^{ρ}}) = g_{α β} \frac{\partial^{2} x^{' α}}{\partial x^{ν} \partial x^{μ}} \frac{\partial x^{' β}}{\partial x^{ρ}} = 0

⟹ x^{' α} = Λ_{β}^{α} x^{β} + a^{α}

the inverse of the matrix $Λ_{β}^{α}$ is $Λ_{β}^{α}$ :

(Λ^{- 1})_{α}^{μ} = g_{α β} Λ_{ν}^{β} g^{ν μ} = Λ_{α}^{μ}

Λ_{β}^{α} Λ_{α}^{γ} = δ_{β}^{γ} or Λ_{γ}^{α} Λ_{β}^{γ} = δ_{β}^{α}

for $x$ -axis boost:

Λ_{β}^{α} = [\begin{matrix} γ & γ β \\ γ β & γ \\ 1 \\ 1 \end{matrix}], Λ_{β}^{α} = [\begin{matrix} γ & - γ β \\ - γ β & γ \\ 1 \\ 1 \end{matrix}]

for general boost

Metric invariance condition

g_{α β} Λ_{μ}^{α} Λ_{ν}^{β} = g_{μ ν}

⟹ g^{μ ν} Λ_{μ}^{α} Λ_{ν}^{β} = g^{α β}, g_{α β} Λ_{μ}^{α} Λ_{ν}^{β} = g_{μ ν}, g^{μ ν} Λ_{μ}^{α} Λ_{ν}^{β} = g^{α β}

The Proper Orthochronous Lorentz Group*
The Full Lorentz Group $O (1, 3)$ consists of all transformations that preserve the Minkowski metric. From the property $Λ^{T} η Λ = η$ above, it follows that $| det (Λ) | = 1, | Λ_{0}^{0} | \geq 1$ . $O (1, 3)$ is a Lie group that possesses four disjoint, connected components. It can be expressed as the union of the Proper Orthochronous Lorentz Group $S O (1, 3)^{↑}$ (the identity component) and its cosets:

P = d i a g (1, - 1, - 1, - 1), T = d i a g (- 1, 1, 1, 1)

O (1, 3) = {S O (1, 3)^{↑}, P S O (1, 3)^{↑}, T S O (1, 3)^{↑}, P T S O (1, 3)^{↑}}

Infinitesimal Lorentz transformation

Λ_{ν}^{μ} = δ_{ν}^{μ} + ω_{ν}^{μ}, with ω_{ν}^{μ} = (\begin{matrix} 0 & β_{x} & β_{y} & β_{z} \\ β_{x} & 0 & θ_{z} & - θ_{y} \\ β_{y} & - θ_{z} & 0 & θ_{x} \\ β_{z} & θ_{y} & - θ_{x} & 0 \end{matrix})

K^{'} \to K : x^{α} = Λ_{β}^{α} x^{' β} = [\begin{matrix} c t^{'} + β \cdot x^{'} \\ x^{'} + β c t^{'} - θ \times x^{'} \end{matrix}]

For infinite transformation $Ω = lim_{n \to \infty} n ω$ ,

Λ = l i m_{n \to \infty} {(1 + \frac{Ω}{n})}^{n} = e^{Ω} = \sum_{k = 0}^{\infty} \frac{Ω^{k}}{k!} = {\begin{cases} [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & \cos θ & \sin θ & 0 \\ 0 & - \sin θ & \cos θ & 0 \\ 0 & 0 & 0 & 1 \end{array}], & z-axis rotation \\ [\begin{array}{c} ch ϕ & sh ϕ \\ sh ϕ & ch ϕ \\ 1 \\ 1 \end{array}], & x-axis boost, with \tanh ϕ = β \end{cases}