This is a brief review of vector calculus.
Partial Derivatives
$$ \pd{A}{x} \equiv \lim_{\Delta x \rightarrow 0} \q{A(\ldots, x + \Delta x,\ldots) - A(\ldots, x, \ldots)}{\Delta x} $$ $$ \boxed{ dA = \sum_i \pd{A}{x_i} d x_i } $$ $$ \pdd{A}{x}{y} = \pdd{A}{y}{x} $$ $A(x, y, z)$Curvilinear Coordinates
| $u$ | $v$ | $w$ | $f$ | $g$ | $h$ | |
|---|---|---|---|---|---|---|
| R | $x$ | $y$ | $z$ | $1$ | $1$ | $1$ |
| C | $\rho$ | $\phi$ | $z$ | $1$ | $\rho$ | $1$ |
| S | $r$ | $\theta$ | $\phi$ | $1$ | $r$ | $r\sin\theta$ |
Integrals
Path Integral
$$ \dL = \uvec{u} f du + \uvec{v} g dv + \uvec{w} h dw $$ $$ \int_L \vec{F} \dot \dL \equiv \int_a^b \vec{F}(\r(t)) \dot \pd{\r}{t} dt $$ $$ \lim_{\Delta \vec{L} \rightarrow 0} \sum_i \vec{F}(\r_i) \dot \Delta \vec{L}_i = \int_L \vec{F} \dot \dL $$ $$ \oint_L \vec{F} \dot \dL = $$Surface Integral
$$ \int_S \vec{F} \dot \dS $$ $$ \oint_S \vec{F} \dot \dS = $$Volume Integral
$$ \lim_{\Delta V \rightarrow 0} \sum_i \rho(\r_i) \Delta V_i \equiv \int_V \rho dV $$ $$ dV = f du\, g dv\, h dw $$Vector Derivatives
Gradient
$A(u, v, w)$ $$ dA = \pd{A}{u} du + \pd{A}{v} dv + \pd{A}{w} dw $$ $$ dA = \grad A \dot d\r $$ $$ \boxed{ \grad A \equiv \uvec{u} \q{1}{f} \pd{A}{u} + \uvec{v} \q{1}{g} \pd{A}{v} + \uvec{w} \q{1}{h} \pd{A}{w} } $$ $$ \boxed{ \del = \ux \pd{}{x} + \uy \pd{}{y} + \uz \pd{}{z} } $$Divergence
$$ \boxed{ \div \vec{F} \equiv \lim_{\Delta V \rightarrow 0} \q{1}{\Delta V} \oint_S \vec{F} \dot \dS } $$ $$ \Delta V = f \Delta u\, g \Delta v\, h \Delta w $$ $$ \Delta \vec{S}_u = \uvec{u}\, g \Delta v\, h \Delta w; \quad \Delta \vec{S}_v = \uvec{v}\, h \Delta w\, f \Delta u; \quad \Delta \vec{S}_w = \uvec{w}\, f \Delta u\, g \Delta v $$ $$ \lim_{\Delta u \rightarrow 0} \lim_{\Delta v \rightarrow 0} \lim_{\Delta w \rightarrow 0} \q{1}{f \Delta u\,g \Delta v\,h \Delta w} [[\vec{F} \dot \Delta \vec{S}_u](u + \Delta u, v, w) - [\vec{F} \dot \Delta \vec{S}_u](u, v, w)] $$ $$ \q{1}{f g h} \lim_{\Delta u \rightarrow 0} \q{[F_u g h](u + \Delta u, v, w) - [F_u g h](u, v, w)}{\Delta u} = \q{1}{f g h} \pd{}{u}[F_u g h] $$ $$ \boxed{ \div \vec{F} = \q{1}{f g h} \bb{\pd{}{u} [F_u g h] + \pd{}{v} [f F_v h] + \pd{}{w} [f g F_w]} } $$Curl
$$ \boxed{ [\curl \vec{F}] \dot \un \equiv \lim_{\Delta S \rightarrow 0} \q{1}{\Delta S} \oint_L \vec{F} \dot \dL } $$ $$ \Delta \vec{L}_u = \uvec{u} f \Delta u; \quad \Delta \vec{L}_v = \uvec{v} g \Delta v; \quad \Delta \vec{L}_w = \uvec{w} h \Delta w; $$ $$ \begin{split} \lim_{\Delta v \rightarrow 0} \lim_{\Delta w \rightarrow 0} \q{1}{g \Delta v h \Delta w} [& [\vec{F} \dot \Delta \vec{L}_w](u, v + \Delta v/2, w) - [\vec{F} \dot \Delta \vec{L}_v](u, v, w + \Delta w/2) \\ & - [\vec{F} \dot \Delta \vec{L}_w](u, v - \Delta v/2, w) + [\vec{F} \dot \Delta \vec{L}_v](u, v, w - \Delta w/2) ] \end{split} $$ $$ \begin{split} \q{1}{g h} \lim_{\Delta v \rightarrow 0} \lim_{\Delta w \rightarrow 0} \bigg[ &\q{[\vec{F} \dot \Delta \vec{L}_w](u, v + \Delta v/2, w) - [\vec{F} \dot \Delta %\vec{L}_w](u, v - \Delta v/2, w)}{\Delta v \Delta w} \\ &- \q{[\vec{F} \dot \Delta \vec{L}_v](u, v, w + \Delta w/2) - [\vec{F} \dot \Delta %\vec{L}_v](u, v, w - \Delta w/2)}{\Delta v \Delta w} \bigg] \end{split} $$ $$ \begin{split} \q{1}{g h} \bigg[ &\lim_{\Delta v \rightarrow 0} \q{[F_w h](u, v + \Delta v/2, w) - [F_w h](u, v - \Delta v/2, w)}{\Delta v} \\ &- \lim_{\Delta w \rightarrow 0} \q{[F_v g](u, v, w + \Delta w/2) - [F_v g](u, v, w - \Delta w/2)}{\Delta w} \bigg] \end{split} $$ $$ \q{1}{g h} \bb{\pd{}{v}[F_w h] - \pd{}{w}[F_v g]} $$ $$ \hspace{-1cm} \boxed{ \curl \vec{F} = \uvec{u}\!\q{1}{g h}\!\!\bb{\pd{}{v}\![h F_w]\!-\!\pd{}{w}\![g F_v]} \!+\!\uvec{v}\!\q{1}{f h}\!\!\bb{\pd{}{w}\![f F_u]\!-\!\pd{}{u}\![h F_w]} \!+\!\uvec{w}\!\q{1}{f g}\!\!\bb{\pd{}{u}\![h F_v]\!-\!\pd{}{v}\![f F_u]} } $$Fundamental Theorems
Gradient Theorem
applies to curl-free (conservative) fields. mention divergence-free fields somewhere $$ \odt{} A(\r(t)) = \pd{A}{x} \od{x}{t} + \pd{A}{y} \od{y}{t} + \pd{A}{z} \od{z}{t} = \grad{A} \dot \od{\r}{t} $$ $$ \int_L \grad A \dot \dL = \int_a^b \grad A \dot \od{\r}{t} dt = \int_a^b \od{}{t} A(\r(t)) dt = A(\r(b)) - A(\r(a)) $$ $$ \boxed{ \int_{\vec{a}}^{\vec{b}} \grad A \dot \dL = A(\vec{b}) - A(\vec{a}) } $$Divergence Theorem
$$ \begin{split} &= \int_V \div \vec{A} dV \\ &= \int_V \bb{\pd{A_x}{x} + \pd{A_y}{y} + \pd{A_z}{z}} dV \\ &= \int_{S_x} \bb{\int \pd{A_x}{x} dx} dS_x + \int_{S_y} \bb{\int \pd{A_y}{y} dy} dS_y + \int_{S_z} \bb{\int \pd{A_z}{z} dz} dS_z \\ &= \int_{S_x} [A_x - A_x] dS_x + \int_{S_y} [A_y - A_y] dS_y + \int_{S_z} [A_z - A_z] dS_z \\ &= \oint_S A_x dS_x + \oint_S A_y dS_y + \oint_S A_z dS_z \\ &= \oint_S [A_x dS_x + A_y dS_y + A_z dS_z] \\ &= \oint_S \vec{A} \dot \dS \end{split} $$ $$ \boxed{ \int_V \div \vec{F} dV = \oint_S \vec{F} \dot \dS } $$Curl Theorem
lemma: green's theorem $$ \int_S \pd{A}{x} dS = \int \int \pd{A}{x} dx\,dy = \int [A - A] dy = \oint_L A dy $$ $$ \int_S \pd{B}{y} dS = -\oint_L B dx $$ $$ \int_S \bb{\pd{A}{x} - \pd{B}{y}} dS = \oint_L [A dx + B dy] $$ $$ \vec{S}(x, y) = \vec{S}(y, z) = \vec{S}(x, z) $$ $$ \vec{S}(x, y) = \ux x + \uy y + \uz z(x, y) $$ $$ d \vec{S}(x, y) = \pd{\vec{S}}{x} \cross \pd{\vec{S}}{y} = \bb{\ux + \uz \pd{z}{x}} \cross \bb{\uy + \uz \pd{z}{y}} = -\ux \pd{z}{x} - \uy \pd{z}{y} + \uz $$ $$ \hspace{-2cm} \begin{split} &= \int_S \curl \vec{A} \dot \dS \\ &= \int_S \bb{\bb{\pd{A_z}{y} - \pd{A_y}{z}} dS_x + \bb{\pd{A_x}{z} - \pd{A_z}{x}} dS_y + \bb{\pd{A_y}{x} - \pd{A_x}{y}} dS_z} \\ &= \int_S \bb{ \bb{\pd{A_x}{z} dS_y - \pd{A_x}{y} dS_z} +\bb{\pd{A_y}{x} dS_z - \pd{A_y}{z} dS_x} +\bb{\pd{A_z}{y} dS_x - \pd{A_z}{x} dS_y}} \\ &= \int_{S_x}\!\!\bb{\pd{A_x}{y} \pd{x}{z}\!-\!\pd{A_x}{z} \pd{x}{y}}\!dy dz +\!\!\int_{S_y}\!\!\bb{\pd{A_y}{z} \pd{y}{x}\!-\!\pd{A_y}{x} \pd{y}{z}}\!dx dz +\!\!\int_{S_z}\!\!\bb{\pd{A_z}{x} \pd{z}{y}\!-\!\pd{A_z}{y} \pd{z}{x}}\!dx dy \\ &= \oint_{L_x} A_x \bb{\pd{x}{y} dy + \pd{x}{z} dz} + \oint_{L_y} A_y \bb{\pd{y}{x} dx + \pd{y}{y} dz} + \oint_{L_z} A_z \bb{\pd{z}{x} dx + \pd{z}{y} dy} \\ &= \oint_L A_x dx + \oint_L A_y dy + \oint_L A_z dz \\ &= \oint_L [A_x dx + A_y dy + A_z dz] \\ &= \oint_L \vec{A} \dot \dL \\ \end{split} $$ $$ \boxed{ \int_S \curl \vec{A} \dot \dS = \oint_L \vec{A} \dot \dL } $$Identities
Product Rules
\begin{align} & \grad [f g] = [\grad f] g + f [\grad g] \\ & \grad [\vec{A} \dot \vec{B}] = \vec{B} \cross [\curl \vec{A}] + [\vec{B} \dot \del] \vec{A} + \vec{A} \cross [\curl \vec{B}] + [\vec{A} \dot \del] \vec{B} \\ & \div [f \vec{A}] = [\grad f] \dot \vec{A} + f [\div \vec{A}] \\ & \div [\vec{A} \cross \vec{B}] = \vec{B} \dot [\curl \vec{A}] - \vec{A} \dot [\curl \vec{B}] \\ & \curl [f \vec{A}] = [\grad f] \cross \vec{A} + f [\curl \vec{A}] \\ & \curl [\vec{A} \cross \vec{B}] = \vec{A} [\div \vec{B}] - \vec{B} [\div \vec{A}] - [\vec{A} \dot \del] \vec{B} + [\vec{B} \dot \del] \vec{A} \end{align}Second Derivatives
\begin{align} \del^2 \phi &\equiv \div \grad \phi \\ \div [\curl \vec{A}] &= 0 \\ \curl [\grad f] &= \vec{0} \\ \curl [\curl \vec{A}] &= \grad [\div \vec{A}] - \del^2 \vec{A} \end{align}Delta Distribution
aka Dirac Delta $$ f(0) = \int_V f(\r') \delta(\r') dV' $$ $$ f(r) = -[4 \pi r]^{-1} $$ $$ \grad \bb{-\q{1}{4 \pi r}} = \ur \pd{}{r} \bb{-\q{1}{4 \pi r}} = \ur \q{1}{4 \pi r^2} $$ $$ \del^2 \bb{-\q{1}{4 \pi r}} = \div \grad \bb{-\q{1}{4 \pi r}} = \bb{\q{1}{r^2} \pd{}{r} r^2} \bb{\q{1}{4 \pi r^2}} = 0 $$ $$ \begin{split} \int_V \del^2 \bb{-\q{1}{4 \pi r}} dV &= \int_V \div \grad \bb{-\q{1}{4 \pi r}} dV \\ &= \int_S \grad \bb{-\q{1}{4 \pi r}} \dot \dS \\ &= \int_0^{\pi} \int_0^{2 \pi} \bb{\ur \q{1}{4 \pi r^2}} \dot [\ur r d \theta r \sin \theta d \phi] \\ &= \q{1}{4 \pi} \int_0^{\pi} \sin \theta d \theta \int_0^{2 \pi} d \phi \\ &= \q{1}{4 \pi} [-\cos \pi + \cos 0] [2 \pi - 0] \\ &= \q{1}{4 \pi} [2] [2 \pi] \\ &= 1 \end{split} $$ $$ \delta(\r) = -\del^2 [4 \pi r]^{-1} $$Helmholtz Decomposition
$$ \boxed{ \R \equiv \r - \r' } $$ $$ \delta(\R) = -\del^2 [4 \pi R]^{-1} $$ $$ \vec{F}(\r) = \int_V \vec{F}(\r') \delta(\R) dV' $$ $$ \begin{split} \vec{F}(\r) &= \int_V \vec{F}(\r') \bb{-\q{1}{4 \pi} \del^2 R^{-1}} dV' \\ &= -\q{1}{4 \pi} \del^2 \int_V \vec{F} R^{-1} dV' \\ &= -\q{1}{4 \pi} [\grad \div - \curl \curl] \int_V \vec{F} R^{-1} dV' \\ &= -\q{1}{4 \pi} \bb{\grad \div \int_V \vec{F} R^{-1} dV' - \curl \curl \int_V \vec{F} R^{-1} dV'} \\ &= -\q{1}{4 \pi} \bb{\grad \int_V \vec{F} \div R^{-1} dV' - \curl \int_V \vec{F} \curl R^{-1} dV'} \\ &= \q{1}{4 \pi} \bb{\grad \int_V \vec{F} \del' \dot R^{-1} dV' - \curl \int_V \vec{F} \del' \cross R^{-1} dV'} \\ &= -\grad \phi + \curl \vec{A} \\ \end{split} $$ $$ \begin{split} \phi(\r) &\equiv -\q{1}{4 \pi} \int_V \vec{F}(\r') \del' \dot R^{-1} dV' \\ &= -\q{1}{4 \pi} \int_V [\del' \dot [\vec{F} R^{-1}] - [\del' \dot \vec{F}] R^{-1}] dV' \\ &= -\q{1}{4 \pi} \bb{\oint_S \q{\vec{F} \dot \dS}{R} - \int_V \q{\del' \dot \vec{F}}{R} dV'} \end{split} $$ $$ \begin{split} \vec{A}(\r) &\equiv -\q{1}{4 \pi} \int_V \vec{F}(\r') \del' \cross R^{-1} dV' \\ &= -\q{1}{4 \pi} \int_V [\del' \cross [\vec{F} R^{-1}] - [\del' \cross \vec{F}] R^{-1}] dV' \\ &= -\q{1}{4 \pi} \bb{-\oint_S \q{\vec{F} \cross \dS}{R} - \int_V \q{\del' \cross \vec{F}}{R} dV'} \\ \end{split} $$ $$ \int^\infty \q{X(r')}{r'} r'^2 dr' = \int^\infty X(r') r' dr' $$ $$ \vec{F} = -\grad \phi + \curl \vec{A} $$ $$ \phi(\r) = \q{1}{4 \pi} \int_V \q{\del' \dot \vec{F}}{R} dV' $$ $$ \vec{A}(\r) = \q{1}{4 \pi} \int_V \q{\del' \cross \vec{F}}{R} dV' $$Curlless Fields
$$ \curl \vec{A} = \vec{0} $$ $$ \vec{A} = -\grad f $$ $$ \oint_L \vec{A} \dot \dL = 0 $$ path integral independent of pathDivergenceless Fields
$$ \div \vec{A} = 0 $$ $$ \vec{A} = \curl \vec{F} $$ $$ \oint_S \vec{A} \dot \dS = 0 $$ surface integral independent of surfaceLemmas
$$ \begin{split} \grad R^n &= \grad [\R \dot \R]^\q{n}{2} \\ &= \grad [[\r - \r'] \dot [\r - \r']]^\q{n}{2} \\ &= \sum_j \ux_j \pd{}{x_j} \bb{\sum_i [x_i - x_i']^2}^\q{n}{2} \\ &= \sum_j \ux_j \q{n}{2} \bb{\sum_i [x_i - x_i']^2}^{\q{n}{2} - 1} \pd{}{x_j} \bb{\sum_i [x_i - x_i']^2} \\ &= \sum_j \ux_j \q{n}{2} \bb{\sum_i [x_i - x_i']^2}^{\q{n}{2} - 1} \sum_i \pd{}{x_j} [x_i - x_i']^2 \\ &= \sum_j \ux_j \q{n}{2} \bb{\sum_i [x_i - x_i']^2}^{\q{n}{2} - 1} \sum_i 2 [x_i - x_i'] \pd{}{x_j}[x_i - x_i'] \\ &= \sum_j \ux_j \q{n}{2} \bb{\sum_i [x_i - x_i']^2}^{\q{n}{2} - 1} \sum_i 2 [x_i - x_i'] \pd{x_i}{x_j} \\ &= \sum_j \ux_j \q{n}{2} \bb{\sum_i [x_i - x_i']^2}^{\q{n}{2} - 1} \sum_i 2 [x_i - x_i'] \delta_{ij} \\ &= \sum_j \ux_j \q{n}{2} \bb{\sum_i [x_i - x_i']^2}^{\q{n}{2} - 1} 2 [x_j - x_j'] \\ &= n \bb{\sum_i [x_i - x_i']^2}^{\q{n}{2} - 1} \sum_j \ux_j [x_j - x_j'] \\ &= n \bb{\R \dot \R}^{\q{n}{2} - 1} \R \\ &= n [\R \dot \R]^{\q{1}{2}[n - 2]} \R \\ &= n R^{n - 2} \R \\ &= n R^{n - 1} \uR \\ \end{split} $$ $$ \boxed{ \grad R^n = n R^{n - 1} \uR } $$ $\div [\uR / R^2]$ $$ \div \bb{\q{\uR}{R^2}} = [\grad R^{-3}] \dot \R + R^{-3} \div \R = [-3 R^{-4} \uR] \dot \R + R^{-3} [3] = 0 $$ $\r' = \vec{0}$ $$ \int_V \div \bb{\q{\ur}{r^2}} dV' = \oint_S \bb{\q{\ur}{r^2}} \dot \dS' $$ $\dS' = r d\theta r \sin\theta d\phi \ur$ $$ \int_0^{2 \pi} \int_0^\pi \bb{\q{\ur}{r^2}} \dot [r d\theta r \sin\theta d\phi \ur] = \int_0^{2 \pi} d\phi \int_0^\pi \sin \theta d\theta = 2 \pi [-\cos\pi + \cos 0] = 4 \pi $$ $$ \div \bb{\q{\ur}{r^2}} = 4 \pi \delta(\r) $$ $$ \boxed{ \div \bb{\q{\uR}{R^2}} = 4 \pi \delta(\R) } $$ $\curl [\uR / R^2]$ $$ \curl \bb{\q{\R}{R^3}} = \grad R^{-3} \cross \R + R^{-3} \curl \R = -3 R^{-4} \uR \cross \R + R^{-3} \vec{0} = \vec{0} $$ $$ \boxed{ \curl \bb{\q{\uR}{R^2}} = \vec{0} } $$ Multipole expansion $$ R^2 = r^2 + r'^2 - 2 r r' \cos \theta = r^2 \bb{1 + \bb{\q{r'}{r}}^2 - 2 \bb{\q{r'}{r}} \cos \theta} = r^2 [1 + \epsilon] $$ $$ \epsilon \equiv \bb{\q{r'}{r}} \bb{\q{r'}{r} - 2 \cos \theta} $$ Taylor expand (binomial theorem), see Calculus review $$ [1 + x]^p = \sum_{n=0}^{\infty} \begin{pmatrix} p \\ n \end{pmatrix} x^n \quad\text{where}\quad \begin{pmatrix} p \\ n \end{pmatrix} \equiv \q{1}{n!} \prod_{m=1}^n [p - m + 1] $$ $$ \q{1}{R} = \q{1}{r} [1 + \epsilon]^{-\q{1}{2}} = \q{1}{r} \bb{1 - \q{1}{2} \epsilon + \q{3}{8} \epsilon^2 - \q{5}{16} \epsilon^3 + \ldots} $$ $$ \q{1}{R} = \q{1}{r} \bb{ 1 - \q{1}{2} \bb{\q{r'}{r}} \bb{\q{r'}{r} - 2 \cos \theta} + \q{3}{8} \bb{\q{r'}{r}}^2 \bb{\q{r'}{r} - 2 \cos \theta}^2 - \q{5}{16} \bb{\q{r'}{r}}^3 \bb{\q{r'}{r} - 2 \cos \theta}^3 + \ldots} $$ $$ \q{1}{R} = \q{1}{r} \bb{ 1 + \bb{\q{r'}{r}} \cos \theta + \bb{\q{r'}{r}}^2 \q{1}{2} \bb{-1 + 3 \cos^2 \theta} + \bb{\q{r'}{r}}^3 \q{1}{2} \bb{-3 \cos \theta + 5 \cos^3 \theta} + \ldots} $$ $$ \boxed{ \q{1}{R} = \q{1}{r} \sum_{n=0}^{\infty} \bb{\q{r'}{r}}^n P_n(\cos \theta) } $$| $P_0(x) = 1$ |
| $P_1(x) = x$ |
| $P_2(x) = \q{1}{2} [-1 + 3x^2]$ |
| $P_3(x) = \q{1}{2} [-3 x + 5 x^3]$ |
| $P_4(x) = \q{1}{8} [3 - 30 x^2 + 35 x^4]$ |
| $P_5(x) = \q{1}{8} [15 x - 70 x^3 + 63^5]$ |
References
- link some references