Questions tagged [vector-fields]
Vector-fields are vector valued functions which define a vector at each point in space. Examples of the vector field include the electric field and the velocity of a fluid.
1,330 questions
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Understanding condition of Hypersurface orthogonality
Wald at page#436 under the heading B.3.2 wrote The dual formulation of Frobenius's theorem gives a useful criterion for when a vector field $\mathcal{E}^a$ is hypersurface orthogonal. Letting $T^*$ ...
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What is twisting of subspace of tangent space?
Wald in B.3 Frobenius theorem at page#435 wrote If $\dim W > 1$, it is possible for the $W$-planes to "twist around" so that integral submanifolds cannot be found This concept requires ...
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How to derive these variations of Stokes' theorem and divergence? [closed]
The book seems to expect me accept these results without building up the "why". I lack some operator formalism to understand what's going on here. Also, divergence theorem is: $$\int_{S} \...
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Fermi-Walker transport in arbitrary coordinates
I am having a hard time working with the Fermi-Walker transport equation in arbitrary coordinates. Background: The problem I was trying to analyze is that of an electron treated as a classical ...
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Fermi-Walker transport of proper acceleration vector field along timeline congruence's worldlines
If we take a generic irrotational/zero vorticity timelike congruence, do the 4-velocity and the direction of proper acceleration $($i.e. the vector in that direction at each point with norm $1$$)$ ...
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How can groundwater spread be found using gradient descent?
Suppose that a droplet of water in an aquifer flow has the position, $\gamma_t \in \mathbb{R}^3$ with a time-independent hydraulic head $h(x,y)$ (a watertable elevation map), where, \begin{align*} ...
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When is the Lamb vector the gradient of a function?
There is something I am not seeing in this derivation of advanced Bernoulli's principle: https://open.oregonstate.education/intermediate-fluid-mechanics/chapter/bernoulli-equation/ The Lamb vector is ...
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What does it means for a field to have Lorentz symmetry?
I am studying about Lorentz symmetry now, and I’m curious about what does a term ‘symmetry’ exactly means for the field. Consider a Lorentz transform $x^{\mu}\rightarrow x’^{\mu}=\Lambda^{\mu}_{\ \nu}...
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Double Null Foliation Schwarzschild Metric
Thinking two null rays coming from a timelike geodesic of , say , a star of mass $m$, one future pointing $l_a = \nabla_a u$ and one past pointing $l'_a = \nabla_a u'$ $ [l_al^a = l'_al'^a = 0] $ and ...
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How does magnetic field vary with respect to change of spatial coordinates?
I am having trouble internalizing how time independent $\mathbf{B}$ varies with respect to length, or more generally with respect to a coordinate transformation in $\mathrm{GL}(3)$ (stationary, ...
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Calculate normal vector to future light cone
Consider a future light cone in Minkowski spacetime $(-,+,+,+)$ defined by $u(t,x,y,z)=t - \sqrt{x^2+y^2+z^2}$ and $t>0$. Derivative of $u$ is $du=dt-\frac{x}{r}dx-\frac{y}{r}dy-\frac{z}{r}dz$ with ...
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Are magnetic field lines the level sets of magnetic vector potential?
I am trying to determine the shape of field lines around a set of electromagnets coils via FEM simulation and am not certain whether to use the magnetic field $\mathbf{B}$ to determine field lines a-...
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Reason why magnetic field lines never intersect [closed]
Assertion (A): Magnetic field lines around a bar magnet never intersect each other. Reason (R): Magnetic field produced by a bar magnet is a quantity that has both magnitude and direction Is the ...
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Why we need a projector in derivation of Raychaudhuri equation?
On deriving Raychaudhuri Equation (like in Wald) we define a tensor as $h_{ab}=g_{ab}+u_{a}u_{b}$ where u is tangent to timelike geodesic and then we define $h^{a}_{b}=\delta^{a}_{b}+u^{a}u_{b}$ and ...
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Prove that tangent and deviation vectors are orthogonal to each other
I am reading geodesic deviation equation from Caroll page # 144 but I have some confusion Let $T^\mu=\partial x^\mu / \partial t$ and $S^\mu=\partial x^\mu / \partial s$ are tangent vector fields and ...
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