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Gradient is covariant. Let's consider gradient of a scalar function. The reason is that such a gradient is the difference of the function per unit distance in the direction of the basis vector. We often treat gradient as usual vector because we often transform from one orthonormal basis into another orthonormal basis. And in this case matrix transpose and ...


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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~=~...


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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 ...


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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~...


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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 ...



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