Questions tagged [grad-curl-div]
For questions on the vector operators: gradient, curl and divergence.
862
questions
-5
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1
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57
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Is the gradient the equivalence class of all spanning vector bases of the tangent vector space at a manifold point? [closed]
When I try to spell out what this means the discussion becomes complicated and verbose. So I will simply ask. Is it correct to say that in finite dimensional real number differential geometry the ...
- 3,349
0
votes
1
answer
64
views
Computing flux integral in two ways what is my mistake?
So doing it this way is easy $ \int_D \nabla F dV $ which gives me $8/3$
But doing it via $ \int _{\delta D}F\cdot nds=\int _{\delta D}F\cdot n\left|r\left(t\right)\right|dt$ gives me troubles.
After ...
- 49
10
votes
1
answer
553
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Numerically compute and clear divergence of discrete vector field
I have a fluid simulation that represents velocity as a vector field in a grid of cells. The cells all have the same width and the same height, but the height is not necessarily equal to the width. I ...
- 409
3
votes
0
answers
56
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Abundance of divergence-free vector fields in noncompact manifolds
Let $(M,g)$ be a complete noncompact Riemannian manifold of dimension $n \geq 2$. How big is the space $D_b(M)$ of (pointwise) bounded divergence-free vector fields on $M$ of noncompact support? I ...
- 5,158
2
votes
1
answer
45
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Volume-preserving vector fields in noncompact manifolds
Let $(M,g)$ be a complete, connected, oriented and noncompact Riemannian manifold. If $M$ were compact, then a vector field $X$ is called volume-preserving when its associated flow $\phi : M \times \...
- 5,158
1
vote
0
answers
70
views
Curl in non-orthogonal coordinates
How can I transform the curl operator into general non-orthogonal coordinates? I have tried to transform its orthogonal expression using the determinant but to no avail. I can't get the same results ...
- 13
1
vote
0
answers
33
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If the pullback of $\phi_1(x)=\exp_x(X_x)$, preserves the volume of the Riemannian manifod $M$, then $\operatorname{div}(X)=0$?
Let $M$ be an orientable compact Rimeannian manifold whithout boundary with volume form $\operatorname{vol}_M$. Take $X\in \mathfrak{X}(M)$ such that if $\phi_t(x)$ is the flow of $X$ and $\phi_1:M\to ...
- 2,155
0
votes
1
answer
32
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Do line integrals of closed curves depend on the orientation of the curve or of the vector field?
I am trying to understand how circulation works as a line integral with the curl in green's theorem. I know that a line integral describes the relationship between a vector field and a path, i.e. how ...
- 3
0
votes
1
answer
60
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What is $\nabla$ when finding the curl/divergence of a vector field?
You are supposed to do $\nabla\times\vec{F}$ for the curl, and $\nabla\cdot\vec{F}$ for the divergence where $\nabla$ is defined as $[\frac{\partial}{\partial x}\frac{\partial}{\partial y}\frac{\...
user1318744
0
votes
0
answers
37
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Who are the divergence free vector fields of a compact Lie group?
Let $G$ be a compact Lie group and $X\in\mathfrak{X}(G)$ a divergence free vector field. Is there a characterization of such fields?
For example, if $G=S^1$, from the fact that it is parallelizable ...
- 2,155
0
votes
1
answer
31
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Calculating curl in cylindrical and cartesian coordinates
So I have this vector function $\mathbf{R}=\mathbf{i}r\cos\omega t + \mathbf{j} r\sin\omega t$, where $x=r\cos\omega t$ and $y=r\sin\omega t$.
I want to find the curl of it's time derivative, $\frac{\...
- 183
3
votes
1
answer
64
views
How do I make my formula for $\nabla \times \mathbf{F}(x,y)$ correct?
Apparently, the curl of a vector field is a function that outputs the "rotationality" of the vector field at some point, as a function of that point's coordinates. I want to go from this ...
- 1,129
1
vote
0
answers
90
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What is the Laplacian of the Gradient? Is $\boldsymbol{u}(\nabla\cdot\nabla p) = \nabla (\boldsymbol{u}\cdot\nabla p)$?
I am supposed to find out whether for a
scalar function $p$ and a
divergence-free vector function $\boldsymbol{u}$
we have that
$$\nabla\cdot\Big [\boldsymbol{u}(\nabla\cdot\nabla p) - \nabla (\...
- 93
0
votes
1
answer
41
views
Confusion about divergence theorem for flux computation
I want to compute the flux of the vector field
$$
F = \frac{\langle x,y,z\rangle}{(x^2+y^2+z^2)^{3/2}}
$$
over the unit sphere $x^2+y^2+z^2=1$.
I know this is
$$
\iint F\cdot n \, dS
$$
where $n$ is ...
- 173
0
votes
1
answer
107
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In layman's terms, what are curl and divergence? [duplicate]
As the title says, I'm wondering what $curl(\Bbb F)$ and $div(\Bbb F)$ mean, assuming $\Bbb F$ is a vector force field. Today in class I learned that if $\Bbb F$ is conservative, $curl(\Bbb F) = \vec ...
- 17