All Questions
19 questions
0
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69
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Is 4-velocity a Vector in the Sense of Covariant Derivative along Worldline
The definition of 4-velocity $U^{\mu} \equiv dx^{\mu}(\tau)/d\tau$, however, we've learnt that the covariant derivative for a vector along a curve parametrized by proper time is,
$$\frac{DA^{\mu}}{D\...
- 938
0
votes
1
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347
views
Isomorphism of the tangent space and the space of directional derivatives [closed]
I have already constructed the tangent space to a manifold, denoted $T_pM$, and I have a good basis for it $\{\hat e_{(\mu)}\}$. (I followed the method of equivalence classes of curves tangent at $p$....
- 713
0
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0
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103
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Conceptual confusion about the formula for parallel transport
I am examining the covariant derivative of a vector according to the formula $$\nabla_{\mu}V^{\nu} = \partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}$$ and also operating under the ...
- 1,397
0
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1
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206
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Are rates a scalar, a vector or both?
Are all rates in physics a scalar, a vector or both?
It seem to me like all rates in science are vectors.
Examples of rate that are vectors are rate of charge flow, rate of heat transfer, rate of mass ...
1
vote
2
answers
143
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Gradient of scalar field
On page183 of Rayd'inverno "An introduction to relativity" he says that the right term in parenthesis is a gradient of some scalar field i.e.
When $$\partial_a (\frac{\ X_b}{\ X^2})=\...
-2
votes
1
answer
49
views
What does the derivative of tangent means? [closed]
While studying the circular motion I had to find the derivative of a tangent so I thought what the derivative of a tangent could probably mean since the derivative of position gives velocity.
Or think ...
- 1
7
votes
2
answers
765
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What is meant when we say that a differential takes on a certain value?
As far as i understand it, total differentials are linear maps that map vectors to numbers. In thermodynamics we encounter statements that a we have reached equilibrium when a total differential of a ...
- 1,616
4
votes
1
answer
167
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What motivates defining vectors as first order differential operators?
I have read some introductions to geometrical ideas and tensors and physics and what some of them do (see, for example, Frankel's Geometry of Physics) is define a vector as a first order differential ...
- 73
1
vote
1
answer
303
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How to ("geometrically") differentiate unit vectors of spherical coordinates?
I have been trying to derive the expressions of partial derivatives of unit vectors with respect to each other in the spherical coordinate system. I was able to get all of them except $\frac{\partial \...
- 39
0
votes
1
answer
2k
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Derivative of a metric tensor
I would like to ask you a question - maybe simple - but bothering me.
We have two four-position vectors product in curvilinear coordinates given by
$(1) \quad X^{\alpha}g_{\alpha \beta}X^{\beta} = \...
- 9
1
vote
1
answer
50
views
Path Coordinates: direction problem (doubt) in derivative of tangential vector
Why is the direction of derivative of tangential vector perpendicular to the direction of the tangential vector?
- 29
3
votes
1
answer
225
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Derivative with respect to vector
How in Lagrangian and Hamiltonian mechanics we take derivatives with respect to velocity and momentum respectively if they are vectors? Can we take derivative with respect to a vector?
- 1,688
4
votes
2
answers
5k
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How is dot or cross product possible using the del operator?
Yesterday in class my teacher told me that the del operator has a direction but no value of its own (as its an operator). So it can't be called exactly a vector. But in vector calculus we see that div ...
- 1,432
5
votes
2
answers
2k
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Why the unit vector is represented as a partial derivative in GR?
Can someone give a good intuitive explanation why we represent the unit vector as a partial derivative in GR and what does it mean?
- 605
0
votes
2
answers
471
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Why exactly in general relativity are tangent vectors defined as maps from functions to $\mathbb{R}$?
I am basing this on the lectures from the hereaus international winter school on gravity and light.
If $M$ is the manifold of physical spacetime, then at any point $p \in M$, we have a tangent space ...
- 1,880
4
votes
2
answers
2k
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Confusion with partial derivatives as basis vectors
So I have seen that the directional derivative can be written as
$$ \frac{df}{d\lambda} = \frac{dx^i}{d\lambda}\frac{df}{dx^i} $$
And we can identify $ \frac{d}{dx^i} $ as basis vectors and $ \...
- 4,064
6
votes
4
answers
751
views
Do $\vec r$ and $d \vec r$ have the same direction?
One question is bugging me for a long time but I couldn't make out anything nor could my friends. Here it goes:
We know, $\vec r$ is regarded as the position vector. So we can say,
$$\vec r \cdot\vec ...
- 4,984
4
votes
1
answer
3k
views
Differentiation of a vector with respect to a vector
Does differentiation of a vector with respect to a vector make any sense? Even if it makes sense, how does it make any physical meaning? I mean what is the physical interpretation?
- 4,984
15
votes
3
answers
44k
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Derive vector gradient in spherical coordinates from first principles
Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient.
I've derived the spherical unit vectors but now I don't understand how to transform ...
- 498