# Questions tagged [vector-fields]

In vector calculus, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space.

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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 ...
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### Is this non-linear system: $\begin{array}{cc} & \dot{x}_1 = {x_2}^3 \\ & \dot{x}_2 = u \end{array}$, solvable using backstepping? [closed]

This problem isn't in the general form to perform backstepping control, I'm wondering is it even doable.
1 vote
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### Is it really easy to derive this expression for the flow of $\dot{x} = x^2$?

I'm reading the very first chapter of Dynamical Systems I: Ordinary Differential Equations and Smooth Dynamical Systems from D. V. Anosov, et al. and found myself trying not only to understand, but ...
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### Jacobi field J along geodesic, satisfying additional condition $[J, \frac{\partial}{\partial r}] = 0.$

Following came up reading different articles and books: Let $p$ be a point on a Riemannian manifold. For $x \in M \backslash Cut(p)$ let $\gamma$ be the minimal geodesic joining $p$ and $x$ ...
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### Decompose a vectorial flow field into parts which form closed/open paths within a finite domain [closed]

I get this question from a physical problem. Suppose I have found a current density distribution $\mathbf{J}(\mathbf{r})$ in a finite domain. The current may flow into or out of the domain. I would ...
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### Complete orthonormal basis of divergence free vector fields

I'm working on a problem in fluid dynamics. I need to find a complete basis of orthonormal 3D vector fields. My "inner product" between vectors $\mathbf{v}_1$, $\mathbf{v}_2$ is a dot ...
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### Why is the lift of a foliate vector field to the transverse bundle a foliate vector field w.r.t. the lifted foliation?

Let $M$ be a manifold with a foliation $F$ of codimension $q$, and let $p: B \rightarrow M$ be the transverse frame bundle (which over each point of $M$ consists of frames in the quotient of the ...
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### Is "work" a natively physical or natively mathematical concept?

I recently watched a wonderful video explaining the Cauchy and Residue theorems, and how a complex integral's components relate to work done by and flux from the Polya vector field of the function ...
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### Line Integrals: Center of Mass

A thin wire of constant linear mass density $k$ takes the shape of an arch of the cycloid $$x = a(t − \sin t),\quad y = a(1 − \cos t), \quad 0 ≤ t ≤ 2π.$$ Determine the mass $m$ of the wire, and ...
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### How to find the parameterization of a Curve for a conservative field? [closed]

So I have the following question with the following facts: $F = ∇f$ so the field is conservative $f(x,y) = x^2y - x + xy^2$ $\int_c F dr = 0$ How do I find a parameterization of the Curve? Since the ...
1 vote
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### How to exploit symmetries in vector fields

Doing electrostatics, I've found that everyone makes assumptions that nobody proves, and it's related to symmetries. If we have a system that we want to calculate the E-field, one starts with the ...
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### Nonvanishing vector field on an odd sphere

This is an exercise I am somewhat confused about. Here $X$ looks like a vector field on $\mathbb{R}^{2n}$, not $S^{2n-1}$. Then how should I interpret $X$ to make it a vector field on the sphere? ...
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### Evaluate the line integral of $\textbf{F}(x,y,z) = (f(x),g(y),h(z))$ along the curve $x^{2/3} + y^{2/3} = 1$

Consider the vector field $\vec{F}(x,y,z)=f(x) \vec{i} + g(y) \vec{j}+ h(z) \vec{k}$ where $f,g,h$ are continuous functions. Evaluate $$\int_{\mathcal{C}}\vec{F} \cdot \vec{dr},$$ where $\mathcal{C}$...
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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 ...
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### Volume integral of vector field in spherical and cartesian coordinates

I am trying to reconcile the different results obtained when integrating a vector field in either spherical or cartesian coordinates. Take for example the vector field in spherical coordinates (...
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### How does a divergence-free field $\vec F$ imply $\vec F=\vec\nabla\times\vec G$?

I am experiencing a slight dilemma here. Starting from the Divergence theorem, I am trying to show how a divergence-free field $\vec F$ implies that $\vec F$ can be written as the curl of another ...
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### Line integral of a vector field over a triangle [closed]

I am trying to solve this line integral, but I get different answers when doing a normal line integral and using the vector form of green's theorem and am wondering why this is the case? I got 4 as my ...
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### Pullback of vector field under inclusion map

Suppose we have a manifold $X$ embedded in ambient space (this is $\mathbb{P}^n$), $$\iota: X \hookrightarrow \mathbb{P}^n.$$ Given a vector field on $\mathbb{P}^n$, is there any way, canonical or ...
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### Domain $U$ for "if $\nabla\times \mathbf{F}=\mathbf{0}$, then $\mathbf{F}$ is conservative"

$U$ is the union of two disjoint open simply connected sets. $\mathbf{F}:U\to\mathbb{R}^3$ is $C^1$. Then is it true that if $\nabla\times \mathbf{F}=\mathbf{0}$, then $\mathbf{F}$ is conservative? ...
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, $curl(\Bbb F) = \vec ... 1 vote 1 answer 32 views ### Lipschitz continuous pseudo gradient flow Im currently looking at a proof of a deformation lemma for functionals on Banach spaces from "Linking Methods in Critical Point Theory" from Schechter. The situation is as follows,$G \in C^...
From electrodynamics we know that $\boldsymbol{\nabla}\mathbf{B}=\mathbf 0$ hence we can introduce a vector potential such that $\mathbf{B}=[\boldsymbol \nabla\times \mathbf{A}]$. What is the general ...