Some personal formula notes

Created in May 02, 2017

2017   ·   formatting   math   ·   personal   math   note

More details can be found in the handboof from NRL, here is just for my convenient.

Vector Relation

\[\mathbf{A}\cdot(\mathbf{B}\times \mathbf{C}) = \mathbf{B}\cdot(\mathbf{C}\times \mathbf{A}) = \mathbf{C}\cdot(\mathbf{A}\times \mathbf{B})\] \[\mathbf{A}\times(\mathbf{B}\times \mathbf{C}) = (\mathbf{A}\cdot \mathbf{C})\mathbf{B} - (\mathbf{A}\cdot \mathbf{B})\mathbf{C}\]
\[\nabla(\phi\varphi) = \phi\nabla\varphi + \varphi\nabla\phi\] \[\nabla\cdot(\varphi \mathbf{A}) = \mathbf{A}\cdot\nabla\varphi + \varphi\nabla\cdot\mathbf{A}\] \[\nabla\times(\varphi\mathbf{A}) = (\nabla\varphi)\times\mathbf{A} + \varphi(\nabla\times\mathbf{A})\] \[\nabla(\mathbf{A}\cdot \mathbf{B}) = \mathbf{A}\times(\nabla\times\mathbf{B}) + \mathbf{B}\times(\nabla\times\mathbf{A}) + (\mathbf{A}\cdot\nabla)\mathbf{B} + (\mathbf{B}\cdot\nabla)\mathbf{A}\] \[\nabla\cdot(\mathbf{A}\times \mathbf{B}) = \mathbf{B}\cdot(\nabla\times \mathbf{A}) - \mathbf{A}\cdot(\nabla\times \mathbf{B})\] \[\nabla\times(\mathbf{A}\times \mathbf{B}) = \mathbf{A}(\nabla\cdot \mathbf{B}) - \mathbf{B}(\nabla\cdot \mathbf{A}) + (\mathbf{B}\cdot\nabla)\mathbf{A} - (\mathbf{A}\cdot\nabla)\mathbf{B}\]
\[\nabla\times\nabla\varphi = 0\] \[\nabla\cdot(\nabla\times \mathbf{A}) = 0\] \[\nabla\cdot\nabla\varphi = \nabla^2\varphi\] \[\nabla\times(\nabla\times\mathbf{A}) = \nabla(\nabla\cdot \mathbf{A}) - \nabla^2\mathbf{A}\]

Integral Relation

\[\oint_S \mathbf{A}\cdot d\mathbf{S} = \int_V (\nabla\cdot\mathbf{A}) dV\] \[\oint_{S} \varphi\, d\mathbf{S} = \int_V (\nabla \varphi)\, dV\] \[\oint_C \mathbf{A}\cdot d\mathbf{l} = \int_S(\nabla\times\mathbf{A})\cdot d\mathbf{S}\] \[\int_V \frac{\partial T_{ij}}{\partial x_j} dV = \oint_{S} T_{ij} dS_j\]

Tensor Relation

\[\delta_{ii} = 3\] \[\delta_{ij}\mathbf{A}_i = \mathbf{A}_j\] \[\epsilon_{xyz} = \epsilon_{yzx} = \epsilon_{zxy} = 1;\quad \epsilon_{xzy}=\epsilon_{yxz}=\epsilon_{zyx} = -1;\quad \text{all other } \epsilon_{ijk} =0\] \[C_i = (\mathbf{A}\times\mathbf{B})_i = \epsilon_{ijk}A_j B_k\] \[(\nabla\times\mathbf{A})_i = \epsilon_{ijk}\partial_j A_k\] \[\epsilon_{ijk}\epsilon_{pqk} = \delta_{ip}\delta_{jq} - \delta_{iq}\delta_{jp}\]

Vector Formula in Cylindrical Coordinate

\[\nabla\varphi = \frac{\partial\varphi}{\partial r}\hat{e}_r + \frac{1}{r}\frac{\partial\varphi}{\partial \theta}\hat{e}_{\theta} + \frac{\partial\varphi}{\partial z}\hat{e}_z\] \[\nabla\cdot\mathbf{A} = \frac{1}{r}\frac{\partial}{\partial r}(r A_r) + \frac{1}{r}\frac{\partial A_{\theta}}{\partial\theta} + \frac{\partial A_{z}}{\partial z}\] \[\nabla\times\mathbf{A} = \left(\frac{1}{r}\frac{\partial A_z}{\partial \theta} - \frac{\partial A_{\theta}}{\partial z}\right)\hat{e}_r + \left(\frac{\partial A_r}{\partial z} - \frac{\partial A_z}{\partial r}\right)\hat{e}_{\theta} + \frac{1}{r}\left(\frac{\partial}{\partial r}(rA_{\theta}) - \frac{\partial A_r}{\partial \theta}\right)\hat{e}_z\] \[\nabla^2\varphi = \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial \varphi}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 \varphi}{\partial \theta^2} + \frac{\partial^2 \varphi}{\partial z^2}\]

Vector Formula in Spherical Coordinate

\[\nabla\varphi = \frac{\partial\varphi}{\partial r}\hat{e}_r + \frac{1}{r}\frac{\partial \varphi}{\partial \theta}\hat{e}_{\theta} + \frac{1}{r \sin \theta}\frac{\partial\varphi}{\partial \phi}\hat{e}_{\phi}\] \[\nabla\cdot\mathbf{A} = \frac{1}{r^2}\frac{\partial}{\partial r}(r^2 A_r) + \frac{1}{r \sin \theta}\frac{\partial}{\partial \theta}(\sin \theta A_{\theta}) + \frac{1}{r \sin \theta}\frac{\partial A_{\phi}}{\partial \phi}\] \[\nabla\times\mathbf{A} = \frac{1}{r \sin \theta}\left[\frac{\partial}{\partial \theta}(\sin\theta A_{\phi}) - \frac{\partial A_{\theta}}{\partial \phi}\right]\hat{e}_r + \left[\frac{1}{r\sin \theta}\frac{\partial A_r}{\partial \phi} - \frac{1}{r}\frac{\partial}{\partial r}(rA_{\phi})\right]\hat{e}_{\theta} + \frac{1}{r}\left[\frac{\partial}{\partial r}(rA_{\theta}) - \frac{\partial A_r}{\partial \theta}\right]\hat{e}_{\phi}\] \[\nabla^2\varphi = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial\varphi}{\partial r}\right) + \frac{1}{r^2\sin \theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial \varphi}{\partial \theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2\varphi}{\partial\phi^2}\]

Hydrodynamic Equations

\[\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_i}(v_i\rho) =0\quad\text{(mass continuity)}\] \[\rho\left(\frac{\partial v_j}{\partial t} + v_i\frac{\partial v_j}{\partial x_i}\right) = - \frac{\partial P_{ji}}{\partial x_i} + \frac{\rho}{m}F_j\quad\text{(momentum, Navier-Stokes)}\] \[\rho\left(\frac{\partial\epsilon}{\partial t} + v_i\frac{\partial\epsilon}{\partial x_i}\right) + \frac{\partial q_i}{\partial x_i} + P_{ij}\Lambda_{ij} = 0,\quad \Lambda_{ij} = \frac{1}{2}(\partial_j v_i + \partial_i v_j)\quad\text{(energy)}\]

In vector form:

\[\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\mathbf{v}) = 0\] \[\frac{\partial\mathbf{v}}{\partial t} + (\mathbf{v}\cdot\nabla)\mathbf{v} = -\frac{1}{\rho}\nabla p + \mathbf{F} + \frac{\mu}{\rho}\nabla^2\mathbf{v}\] \[\rho\left(\frac{\partial \epsilon}{\partial t} + \mathbf{v}\cdot\nabla\epsilon\right) - \nabla\cdot(K\nabla T) + p\nabla\cdot\mathbf{v} = 0\]

Navier-Stokes Equations in Cylindrical Coordinates (just an example):

\[\frac{\partial v_r}{\partial t} + v_r\frac{\partial v_r}{\partial r} + \frac{v_{\theta}}{r}\frac{\partial v_r}{\partial \theta} + v_z\frac{\partial v_r}{\partial z} - \frac{v_{\theta}^2}{r} = - \frac{1}{\rho}\frac{\partial p}{\partial r} + \nu\left(\frac{\partial^2 v_r}{\partial r^2} + \cdots\right) + F_r\]

(And similarly for \(v_{\theta}\), \(v_z\).)

Navier-Stokes Equations in Spherical Coordinates

(Equations follow similarly, omitted for brevity.)

Maxwell Equations (c.g.s)

  • Gauss’s law:

    \[\nabla\cdot\mathbf{E}=4\pi\rho,\quad \nabla\cdot\mathbf{D}=4\pi\rho_f\]

  • Faraday’s law:

    \[\nabla\times\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}\]

  • Maxwell’s assumption:

    \[\nabla\cdot\mathbf{B} = 0\]

  • Ampère’s law:

    \[\nabla\times\mathbf{B} = \frac{1}{c}(4\pi\mathbf{J} + \frac{\partial \mathbf{E}}{\partial t})\] \[\nabla\times\mathbf{H} = \frac{1}{c}(4\pi\mathbf{J}_f + \frac{\partial\mathbf{D}}{\partial t})\]
\[\mathbf{D} = \mathbf{E} + 4\pi\mathbf{P}, \quad \mathbf{B} = \mathbf{H} + 4\pi\mathbf{M}, \quad \mathbf{D}=\epsilon\mathbf{E},\quad \mathbf{B}=\mu\mathbf{H}.\]

Maxwell Equations (SI)

  • Gauss’s law:

    \[\nabla\cdot \mathbf{E} = \frac{\rho_e}{\epsilon_0}, \quad \nabla\cdot \mathbf{D} = \rho_f\]

  • Faraday’s law:

    \[\nabla\times\mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\]

  • Maxwell’s assumption:

    \[\nabla\cdot\mathbf{B}=0\]

  • Ampère’s law:

    \[\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}, \quad \nabla\times\mathbf{H} = \mathbf{J}_f + \frac{\partial\mathbf{D}}{\partial t}\]
\[\mathbf{D} = \epsilon_0\mathbf{E}+\mathbf{P},\quad \mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M}),\quad \mathbf{J} = \mathbf{J}_f+\mathbf{J}_p + \mathbf{J}_m,\quad \rho= \rho_f+\rho_p.\]

Maxwell Stress Tensor (SI)

From Lorentz force and Maxwell equations:

\[\sigma = \epsilon_0(\mathbf{E}\mathbf{E} - \tfrac{1}{2}E^2\hat{I}) + \tfrac{1}{\mu_0}(\mathbf{B}\mathbf{B} - \tfrac{1}{2}B^2\hat{I}), \quad \mathbf{S} = \tfrac{1}{\mu_0}\mathbf{E}\times\mathbf{B}.\]

Electromagnetic Tensor (SI)

\[\mathbf{F}_{\mu\nu} = \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}, \quad \mathbf{E}_i = cF_{0i}, \quad \mathbf{B}_i = -\tfrac{1}{2}\epsilon_{ijk}F^{jk}.\]

Magnetohydrodynamics (MHD) Equations (c.g.s.)

  • Continuity:

    \[\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{V}) = 0\]

  • Equation of Motion:

    \[\rho\frac{d\mathbf{V}}{dt} = \mathbf{j}\times \mathbf{B} - \nabla p + \rho\mathbf{g}\]

  • Entropy:

    \[\frac{d}{dt}\left(\frac{p}{\rho^{\gamma}}\right)=0\]

  • Induction (no resistivity):

    \[\frac{\partial \mathbf{B}}{\partial t}=\nabla\times(\mathbf{V}\times \mathbf{B})\]

  • With resistivity:

    \[\frac{\partial \mathbf{B}}{\partial t}=\nabla\times(\mathbf{V}\times \mathbf{B}) + \frac{\eta c}{4\pi}\nabla^2\mathbf{B}\]

MHD under Other Coordinate System

Radial component momentum equation is,

\[\begin{aligned} \rho(\frac{\partial v_r}{\partial t} + v_r\frac{\partial v_r}{\partial r}& + \frac{v_{\theta}}{r}\frac{\partial v_r}{\partial \theta} + \frac{v_{\phi}}{r \rm sin \theta}\frac{\partial v_r}{\partial \phi} - \frac{v_{\theta}^2 + v_{\phi}^2}{r}) = - \frac{\partial p}{\partial r} \\ +\frac{1}{4\pi}(B_r\frac{\partial B_r}{\partial r}& + \frac{B_{\theta}}{r}\frac{\partial B_r}{\partial \theta} + \frac{B_{\phi}}{r \rm sin \theta}\frac{\partial B_r}{\partial \phi} - \frac{B_{\theta}^2 + B_{\phi}^2}{r})-\frac{1}{4\pi}(B_r\partial_r B_r+B_{\theta}\partial_r B_{\theta}+B_{\phi}\p_r B_{\phi})\\ &-\rho\frac{GM}{r^2}+\rho g_{rad} \end{aligned}\]

Theta component momentum equation is,

\[\begin{aligned} \rho(\frac{\partial v_{\theta}}{\partial t} + v_r\frac{\partial v_{\theta}}{\partial r}& + \frac{v_{\theta}}{r}\frac{\partial v_{\theta}}{\partial \theta} + \frac{v_{\phi}}{r \rm sin\theta}\frac{\partial v_{\theta}}{\partial \phi} + \frac{v_r v_{\theta}}{r} - \frac{v_{\phi}^2\rm cot \theta}{r}) = -\frac{1}{r}\frac{\partial p}{\partial \theta} \\ + \frac{1}{4\pi}(B_r\frac{\partial B_{\theta}}{\partial r}& + \frac{B_{\theta}}{r}\frac{\partial B_{\theta}}{\partial \theta} + \frac{B_{\phi}}{r \rm sin\theta}\frac{\partial B_{\theta}}{\partial \phi} + \frac{B_r B_{\theta}}{r} - \frac{B_{\phi}^2\rm cot \theta}{r})-\frac{1}{4\pi r}(B_r\partial_{\theta} B_r+B_{\theta}\partial_{\theta} B_{\theta}+B_{\phi}\partial_{\theta} B_{\phi}) \end{aligned}\]

Under the axis symmetry, \(\partial_{\phi}=0\), The equations become,

\[\begin{aligned} \rho(\frac{\partial v_r}{\partial t} + v_r\frac{\partial v_r}{\partial r} + \frac{v_{\theta}}{r}\frac{\partial v_r}{\partial \theta} - \frac{v_{\theta}^2 + v_{\phi}^2}{r}) &= - \frac{\partial p}{\partial r}\\ &+\frac{1}{4\pi}( \frac{B_{\theta}}{r}\frac{\partial B_r}{\partial \theta} - \frac{B_{\theta}^2 + B_{\phi}^2}{r})-\frac{1}{4\pi}(B_{\theta}\frac{\partial B_{\theta}}{\partial r}+B_{\phi}\frac{\partial B_{\phi}}{\partial r})\\ &-\rho\frac{GM}{r^2}+\rho g_{rad} \end{aligned}\] \[\begin{aligned} \rho(\frac{\partial v_{\theta}}{\partial t} + v_r\frac{\partial v_{\theta}}{\partial r} + \frac{v_{\theta}}{r}\frac{\partial v_{\theta}}{\partial \theta} + \frac{v_r v_{\theta}}{r} - \frac{v_{\phi}^2\rm cot \theta}{r}) &= -\frac{1}{r}\frac{\partial p}{\partial \theta} \\ &+ \frac{1}{4\pi}(B_r\frac{\partial B_{\theta}}{\partial r} + \frac{B_r B_{\theta}}{r} - \frac{B_{\phi}^2\rm cot \theta}{r})-\frac{1}{4\pi r}(B_r\frac{\partial B_r}{\partial {\theta} }+B_{\phi}\frac{\partial B_{\phi}}{\partial {\theta} }) \end{aligned}\]

If you set \(B_{\phi}=0\) and \(v_{\phi}=0\), you can further reduce the equation in the form as,

\[\begin{aligned} \rho(\frac{\partial v_r}{\partial t} + v_r\frac{\partial v_r}{\partial r} + \frac{v_{\theta}}{r}\frac{\partial v_r}{\partial \theta} - \frac{v_{\theta}^2}{r}) &= - \frac{\partial p}{\partial r}\\ &+\frac{1}{4\pi}( \frac{B_{\theta}}{r}\frac{\partial B_r}{\partial \theta} - \frac{B_{\theta}^2}{r})-\frac{1}{4\pi}(B_{\theta}\frac{\partial B_{\theta}}{\partial r})\\ &-\rho\frac{GM}{r^2}+\rho g_{rad} \end{aligned}\] \[\begin{aligned} \rho(\frac{\partial v_{\theta}}{\partial t} + v_r\frac{\partial v_{\theta}}{\partial r} + \frac{v_{\theta}}{r}\frac{\partial v_{\theta}}{\partial \theta} + \frac{v_r v_{\theta}}{r} ) &= -\frac{1}{r}\frac{\partial p}{\partial \theta} \\ &+ \frac{1}{4\pi}(B_r\frac{\partial B_{\theta}}{\partial r} + \frac{B_r B_{\theta}}{r})-\frac{1}{4\pi r}(B_r\frac{\partial B_r}{\partial {\theta} }) \end{aligned}\]

Property of Legendre polynomials

\[\left\{\frac{d}{dx}[(1-x^2)\frac{d}{dx}] + n(n+1)\right\}P_n(x)=0,\] \[P_n(x)=\frac{1}{2^nn!}\frac{d^n}{dx^n}[(x^2-1)^n],\] \[\frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^{\infty} P_n(x)t^n,\] \[(n+1)P_{n+1}(x) = (2n+1)xP_{n}(x)-nP_{n-1}(x),\]

The solution of \(\nabla^2\Phi(x) = 0\) in spherical coordinates:

\[\Phi(r,\theta)=\sum_{n=0}^{\infty}[A_l r^l + B_l r^{-(l+1)}]P_{l}(\cos\theta).\]