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There are the ${\bf k}_t,~{\bf k}_\phi$ Killing vectors. Another condition is that $${\bf k}_t\cdot{\bf k}_\phi~=~\frac{(2mr~-~Q^2)asin\phi}{\rho^2},$$ for $a~=~J/m$. This is zero for $a~=~0$ or for $Q^2~=~2mr$ or $\phi~=~0,~\pi$. There is also $${\bf k}_\phi\cdot{\bf k}_\phi~=~\frac{(r^2~+~a^2)^2sin^2\phi~-~\Delta a^2sin^4\phi}{\rho^2},$$ for $\Delta~=~... 1 In classical mechanics a system is described by a Lagrangian$\mathscr{L}\colon TQ\to \mathbb{R}$, with$Q$being the configuration space and$TQ$its tangent bundle, namely the union over$q\in Q$of all tangent spaces$T_qQ$:$TQ = \cup_q T_qQ$. A local chart on$Q$looks like$(q_1, \ldots, q_n)$, the$q_k$being the degrees of freedom of the system. The ... 6 This theorem can be used to prove Archimede's Principle in a region with a non-uniform gravitational field. The weight of the displaced fluid is $$\vec W=\int_\Omega \rho \vec g(\vec r)~\mathrm d\Omega.$$ Let us consider a body fully immersed. Then the buoyancy force is given by $$\vec B=-\oint_\Gamma p(\vec r)~\mathrm d\vec \Gamma =-\int_\Omega\vec\nabla p~... 0 The pdf J.D Callen, Fundamentals of Plasma Physics, chapter 3 defines B=\sqrt{\vec{B}\cdot \vec{B}} and \hat{b}=\vec{B}/B, and proves that as you walk along a field line (arc length segment d\ell):$$\frac{dB}{d\ell}=\hat{b}\cdot \nabla B=-B \nabla \cdot \hat{b}$$(where the second equality holds from$\nabla\cdot(B \hat{b})=0\$) If the field lines ...