New answers tagged vector-fields
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Accepted
Two contradictory derivations of Killing equation
As an overall comment, I stress that conservation of $Q$ is valid for the Killing vector $\xi$ if the considered curve is a geodesic.
Let us come to the issue.
First of all, generally speaking, the ...
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Two contradictory derivations of Killing equation
Both approaches are fine.
In the first approach, the analysis is done at the coordinate/component level of the equations. Simply asking the question how does $Q$ very with $\tau$ if we write ...
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Two contradictory derivations of Killing equation
I think the problem lies in the notation. I guess Tong is treating the components $\xi_\mu$ as scalars for which we have $\frac{\mathrm{d}}{\mathrm{d}\tau}=u^\alpha\partial_\alpha$. And the ...
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Cone vs. small circle parallel transport
This is in a way a Riemann-geometric analogue of the Aharonov--Bohm effect in quantum mechanics.
Consider a manifold $M$ equipped with a linear connection $\nabla$. It is well-known that the parallel ...
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Cone vs. small circle parallel transport
The path the vector you are describing takes on the cone is a closed circle while the cone is rolled, but it is no longer a closed circle once the cone is sliced and flattened!
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Accepted
Isn't the induced electric field vector always tangent to the loop?
But isn't the field $\mathbf E$ which is responsible for the induced emf also along the contour (unit tangent vector $\hat{\mathbf t}$)?
No, the equation is true for any contour you choose in a given ...
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Isn't the induced electric field vector always tangent to the loop?
Do you mean "which is responsible" or "that is responsible"?
(See Strunk and White, Which Hunts: https://clubtroppo.com.au/2008/11/30/the-great-which-hunt/).
The former is ...
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Accepted
How to compute the vector field from a potential in the complex plane?
What did you plot exactly, I rather obtained something like this:
which is compatible with the lecture. The velocity field is:
$$
\dot z = w := \sqrt{E-V(z)}
$$
with $\min V<E<0$. Since it is ...
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Killing tensor in the Kerr metric
The Killing equation is a overdetermined system of PDEs of finite type, as a result, there is a algorithm to compute all Killing tensors for a given metric. See for example arXiv 1704.02074.
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The conservative force
While @basics has the math right, I understand your question is about the physical interptetation. To answer this, we must understand what a force field is, since the definition of the rotation ...
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The conservative force
An example might be the best way to see it.
Set up some electrodes that create a horizontal electric field. Make it so the field strength is proportional to height.
This is a non-conservative field. ...
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The conservative force
Remark
the curl it's the measure of the rotation of the vector field around a specific point
Before answering your question, I'd replace (local) "rotation" with (local, or elementary) &...
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