7 votes

Why is the Riemann curvature tensor not zero?

I cannot see the problem. Fix $p\in M$, then $$(R^d_{cab}V^c)_p=((\nabla_a\nabla_b-\nabla_b\nabla_a)V^d)_p$$ implies that $(R^d_{cab}V)_p=0$ if the vector field $V$, defined in a neighborhood $U_p$ ...
Valter Moretti's user avatar
4 votes

What are some ways to derive $\left( \boldsymbol{E}\cdot \boldsymbol{E} \right) \nabla =\frac{1}{2}\nabla \boldsymbol{E}^2$?

Your second equation with its $({\bf E}\cdot {\bf E})\nabla$ isnot really correct as the nabla has to act on one of the ${\bf E}$'s. The correct form is best written as $({\bf E} \cdot \nabla {\bf E})...
mike stone's user avatar
  • 50.6k
3 votes

Why is the Riemann curvature tensor not zero?

I think the OP is correct to call $V^d$ as a vector field rather than a vector. Say we define covariant derivative $$ \nabla_X Y$$ Then, while evaluating the covariant derivative at a point $p$, it is ...
S.G's user avatar
  • 1,160
2 votes

How to calculate the rotation at a singularity?

Just to give a few hints: The strategy is usually to use a theorem that says that two integrals must be equal. In your case the classical Gauss' and Stokes' theorems don't seem to work but some hope I ...
Kurt G.'s user avatar
  • 1,669
1 vote

Can a non-zero curl vector force field still do a null amount of work?

Question 1: Recall from Stokes' theorem that $$\oint \vec{F} \cdot d\vec{l} = \int (\vec{\nabla} \times \vec{F}) \cdot d\vec{a}.$$So any path that can be spanned by a surface for which this latter ...
Michael Seifert's user avatar
1 vote

Is the Lorentz force a vector field or just a vector?

The force pertains to a charged body moving along a trajectory given by the equation $𝐫 = 𝐑(t)$, where I'm going to denote the function by a separate letter to remove the confusion. The velocity is $...
NinjaDarth's user avatar
  • 1,272

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