All Questions
Tagged with differentiation tensor-calculus
191
questions
3
votes
1
answer
86
views
Bianchi identity in EMT [closed]
$ ∇_a∇_b F_{ab} = 0 $ ($F_{ab}$ Faraday tensor in EMT.)
proof is given by
"To see this, assume a Minkowski spacetime for simplicity and adopt
Cartesian coordinates, so that the covariant ...
- 81
1
vote
0
answers
57
views
If there is a spin-0 structure and i differentiate it with respect to a space dimension, does it become a spin-1 structure?
It might be a naive question but i was wondering what a derivative can do regarding spin. If there is a Riemann scalar it is clear that its an invariant object under tensor transformation and it does ...
- 153
1
vote
0
answers
67
views
Confusions about partial and covariant derivatives
Let, we are in 1d cartesian space with metric $g_{xx} = x^2$. Let we have a vector $v = 1/x e_x$. Since the vector is designed to shrink its components as the basis grows - its total length will ...
- 1,141
0
votes
1
answer
68
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Equating 2 sides of EFE
Can we say if the covariant derivative is zero, the partial derivative is also zero because locally covariant derivative reduces to partial derivatives (since locally spacetime is flat)? Because, in ...
- 1,141
2
votes
2
answers
337
views
Why can the dot product of two vectors be expressed as a differential?
I am reading a book by Arfken and Weber (Mathematical methods for physicists), in the section regarding rotations in $\mathbb{R}^3$. They express the elements of a rotation matrix in Cartesian ...
- 286
3
votes
2
answers
345
views
Difference and meaning of index the derivative operator
I'm a beginner in this type of math, we are just starting to study it, but I need some clarifications about the meaning and the difference of when we write
$$\partial_i \qquad \text{and}\qquad \...
- 361
1
vote
1
answer
169
views
Intuition behind parallel transport of vectors as partial derivative operators
Imagining a non-embedded manifold that forces a reformulation of the tangent space at a point as partial derivatives of any arbitrary smooth functions on the manifold along a parameterized curve is ...
- 133
1
vote
2
answers
156
views
Coordinate basis vectors on tangent bundle (extrinsic view)
Short Version: when we say that $(\pmb{q},\pmb{u}):TQ_{(q)}\to\mathbb{R}^{2n}$ are local coordinates for the tangent bundle of $Q$, which can be viewed as an embedded submanifold of a higher ...
- 477
0
votes
1
answer
170
views
Exterior derivatives Leibniz rule
I want to prove Sean Carroll's "spacetime and geometry"'s eq.(2.78):
$$
\mathrm{d}(\omega \wedge \eta)=(\mathrm{d} \omega) \wedge \eta+(-1)^p \omega \wedge(\mathrm{d} \eta) \tag{2.78}
$$
...
- 1,421
5
votes
2
answers
493
views
Lie derivative in terms of covariant derivative and the symmetry of Christoffel symbols
I want to verify that if a manifold is torsion-free with a metric compatible derivative operator $\nabla_a$, the Lie derivative of a vector $W^a$ along $V^a$ can be written as
$$L_V W^a = V^\nu\nabla_\...
- 109
0
votes
1
answer
65
views
Fraction with components of Lorentz transformation
I want to show how partial derivative transforms under a Lorentz transformation.
Since the partial derivative has a fixed definition with respect to the $x$-coordinate it stays unchanged: $\partial_\...
- 494
2
votes
2
answers
281
views
Is the contracted Christoffel symbol a tensor?
The coordinate transformation law (from coordinates x to coordinates y) for the Christoffel symbol is:
$$\Gamma^i_{kl}(y)=\frac{\partial y^i}{\partial x^m} \frac{\partial x^n}{\partial y^k} \frac{\...
- 633
3
votes
2
answers
250
views
Covariant Derivative vs Vector Differentiation
I am confused about the difference between a vector operating on another (like $u(v)$, used in Lie Brackets) and covariant derivatives ($\nabla_u v$). Can we not use Christoffel symbols in the vector ...
- 1,141
0
votes
1
answer
90
views
Are differentials of a smooth maps tensors? Connection to Jacobian matrices?
I'm trying to learn about smooth manifolds and differential geometry in order to learn the geometric view of classical mechanics. I'm confused about what kind of "object" the differential of ...
- 477
0
votes
3
answers
449
views
On varying a tensor with respect to the metric
Upon learning about the Lagrangian formulation of GR, where varying an action with respect to a metric (in order to, for instance, arrive at the Einstein field equations) is common, I can't help but ...
- 115