All Questions
17
questions
2
votes
1
answer
75
views
geometric interpretation on covariant derivateve in curvilinear coordinates.
I'm having trouble understanding what does the covariant derivative do in a coordinate system where we have changing basis vectors. I always thought it was giving us the change in coordinates while ...
- 379
1
vote
1
answer
104
views
Exercise 5.22 in the book Geometrical Methods of Mathematical Physics (by Bernard Schutz)
I can't figure it out about the Exercise 5.22 in the book Geometrical Methods of Mathematical Physics (by Bernard Schutz).
Could any one give me a help? Thanks.
($\bar{V}$ means vector V. )
Exercise ...
- 63
0
votes
0
answers
52
views
Intuition about the divergence of a vector field in non-orthogonal basis
My textbook defines the divergence of a vector field in a non orthogonal constant basis the following way:
$$div(\vec{u})=\vec{a}^i\cdot\frac{\partial \vec{a}_ku^k}{\partial x^i}=\frac{\partial u^i}{\...
- 379
0
votes
0
answers
51
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Differential operators of tensor fields. Hamilton operator
The very first thing my textbook says is that the Hamilton operator is defined as:
$$\vec{\nabla}=\vec{a}^i\nabla_i$$
Where $\nabla_i$ is the covariant derivative and " $\vec{a}^i$ is the ...
- 379
2
votes
2
answers
93
views
Derivation or intuition on the covariant derivative for higher rank tensors
So the derivation in my textbook for the covariant derivative of a vector field $\vec{u}$ in curvilinear coordinates $\xi^k$ is the following:
$$\frac{\partial \vec{u}}{\partial\xi^j}=\frac{\partial (...
- 379
1
vote
0
answers
40
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Whats the significance of $g^{-1}$ (the inverse metric) appearing in tangential projection?
Let $M \subseteq (\mathbb{R}^n,g_E)$ be an embedded submanifold, with the embedding $F : M \to \mathbb{R}^n$. It is well known (c.f. Lee, doCarmo) that the covariant derivative on $M$ with respect to ...
- 198
6
votes
0
answers
207
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'Tensor Calculus' by J.L. Synge and A. Schild (1979 Dover publication) . Exercise 12 page 110.
I'm solving all the exercises of 'Tensor Calculus' by J.L. Synge and A. Schild (1979 Dover publication) . Till now everything went smooth, but now I'm stuck at exercise 12. page 110 of the third ...
- 116
0
votes
1
answer
210
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Vector Laplacian in Curved Spaces
The vector gradient, $\mathbb{L}$, is defined as
$$
(\mathbb{L} W)^{ij} \equiv \nabla^{i} W^{j} + \nabla^{j} W^{i} - \frac{2}{3} g^{ij} \nabla_{k} W^{k} \,,
$$
where $\nabla_{i}$ is the covariant ...
- 698
-1
votes
1
answer
111
views
Finding $\mathbf{u} \cdot \nabla \mathbf{u}$ in cylindrical coordinates
Evaluate $\mathbf{u}\cdot\nabla\mathbf{u}$ (the directional derivative of $\mathbf{u}$ in the direction of $\mathbf{u}$)in cylindrical coordinates $(r, \phi,z)$, where $\bf{u}=e_{\phi}$.
The textbook ...
- 927
1
vote
0
answers
466
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Understanding role of tangent/cotangent space changes under coordinate transformations
This is kind of a follow-up to the excellent answer to this question: https://physics.stackexchange.com/questions/445948/general-coordinate-transformations
I want a very clear understanding of what ...
- 2,569
0
votes
0
answers
1k
views
Examples of Tensor Transformation Law
Let $T_{\mu\nu}$ be a rank $(0,2)$ tensor, $V^\mu$ a vector, and $U_\mu$ a covector. Using the definition of tensors based on the tensor transformation law, determine whether each of the following is ...
- 603
0
votes
1
answer
117
views
Tensor and Vector Notation
I'm given the tensor $X^{\mu\nu}$ and vector $V^\mu$ of the form
$$X^{\mu\nu} = \begin{bmatrix}
2 & 0 & 1 & -1 \\
-1 & 0 & 3 & 2 \\
-1 & 1 & 0 & 0 \\
...
- 603
6
votes
1
answer
8k
views
How can you calculate distance in plane polar coordinates using the metric tensor
I'm trying to develop some intuition about the metric tensor and how it can be used to calculate distances/angles in curvlinear space by using the very simple example of a 2D polar surface.
The ...
- 63
2
votes
1
answer
1k
views
How to compute the divergence in polar coordinates from the Voss-Weyl formula?
The Voss-Weyl formula reads
$$\nabla_\mu V^\mu=\frac{1}{\sqrt g}\partial_\mu(\sqrt g V^\mu),$$
where $g=\mathrm{det}( g_{\mu\nu})$. In polar plane coordinates the only non-vanishing components of the ...
- 21
2
votes
1
answer
1k
views
Covariant derivative for a covector field
In the lecture we had the immersion $f: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$.
Now we said that a map $\lambda : U \rightarrow T^*f$ is a covector field. Okay, that's fine.
Also, we ...
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