Skip to main content

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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
11 views

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 ...
4 votes
2 answers
888 views

$X, Y$ are two complete vector fields with$[X, Y] = 0$, what is the resulting flow of $X+Y$?

If $X, Y$ are two complete vector fields with$[X, Y] = 0$, what is the resulting flow of $X+Y$? I'm kind of confused on what the flow is. I know that the respective flows for $\Phi_t^X$ and $\...
1 vote
0 answers
39 views

Could this vector field exists on a manifold?

I want to "generalize" the smooth flow $F(t,x)=e^{t}x$ on a Riemannian manifold in the following sense: Let (M,p) be a pointed positive dimensional smooth complete Riemannian manifold (that ...
5 votes
2 answers
954 views

Flowing a vector along a vector field $X$ using the pushforward of the flow of $X$

On the three-sphere $S^3$, I'm given three vector fields $X$, $Y$ and $Z$, such that at each point $p\in S^3$, the tangent vectors $X_p$, $Y_p$ and $Z_p$ form an orthogonal basis of the tangent space $...
-1 votes
1 answer
33 views

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
1 answer
65 views

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 ...
3 votes
1 answer
469 views

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 $x$ ...
0 votes
0 answers
12 views

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 ...
0 votes
1 answer
21 views

Derivation of Continuity Equation for an Incompressible flow

Good day guys, I was playing around with the following form of the continuity equation: $$ \frac{\partial \rho}{\partial t} - \nabla \cdot (\rho \vec{v}) = 0 $$ For an incompressible fluid: $\frac{D\...
3 votes
2 answers
109 views

Confusion of definition of covariant derivative along curve

I am currently studying Riemannian geometry, and have come across the following proposition: Proposition. Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$ and let $\gamma:I\...
0 votes
0 answers
30 views

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 ...
1 vote
0 answers
14 views

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 ...
0 votes
0 answers
55 views

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 ...
2 votes
3 answers
1k views

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 ...
0 votes
0 answers
23 views

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
1 answer
38 views

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 ...
4 votes
2 answers
3k views

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? ...
0 votes
1 answer
70 views

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 $\mathcal{C}$...
1 vote
1 answer
54 views

Show the relation between the derivative of a curve and Lie bracket

I'm solving this Problem: Let X and Y be vector fields over $\mathbb{R}^n$, and $\phi_t,\psi_s$ the local flow respectively. For all $p\in\mathbb{R}^n$, there is an open intervall $I_p\subseteq \...
0 votes
1 answer
32 views

Change vector's basis from 3d spherical to cartesian coordinates [duplicate]

I am writing a simulation program. I have a vector field in spherical coordinates which I need to transform into Cartesian coordinates. I understand how this works in $2D$ case - simple enough, I just ...
2 votes
0 answers
15 views

Lipschitz Vector field which acts positively on the gradient

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a Lipschitz function. Let $U_\delta=\{|f|<\delta\}$ be open. I want to show that given any $\epsilon>0$, there is a $\delta>0$ such that there is a ...
0 votes
1 answer
57 views

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{\...
0 votes
0 answers
32 views

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 ...
0 votes
1 answer
34 views

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 (...
2 votes
1 answer
657 views

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 ...
2 votes
1 answer
55 views

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 ...
4 votes
1 answer
45 views

Does anybody know this integral expression?

I stumbled upon the following integral, which is a low dimensional case of a minimization problem I currently investigate. $$F(V) = \int_{\mathbb{R}^3}|W+\nabla\times V)|^2+|\nabla\times(W+\nabla\...
13 votes
5 answers
938 views

Could exists a vector field on $\mathbb{S}^{2}$ with exactly $n$ zeroes?

I just started to learn index theory of tangent vector fields. I'm aware of two examples on the sphere $\mathbb{S}^{2}$ with exactly one zero, which, which are $F(x,y) = (1-x^2-y^2)\partial x$ thought ...
0 votes
1 answer
31 views

Oscillation of vector valued function

Assume we have $f:\mathbb{R}^n \to \mathbb{R}^m$, where $n,m\in\mathbb{N}$. If $m=1$, the notion of oscillation of $f$ is clear. However, what if $m>1$? What's the definition of the oscillation of ...
0 votes
2 answers
103 views

Find exercises to practice vector calculus

I studied vector calculus like a month ago, and now I'd like to practice to remember it. Could someone please recommend me a book or a resource to practice?
0 votes
0 answers
26 views

Can you use the interior product to calculate flux?

If I have a vector Field F and a 2D surface $\Sigma$ in $\mathbb{R}^3$ would $F \lrcorner dx \land dy \land dz= F_1 dy\land dz + F_2dz\land dx + F_3dx\land dy$ If this is the case, could one say that ...
0 votes
1 answer
53 views

Is divergence of a curl not 0 in this case/problem?

Q . A hemispherical shell is placed on the $x$-$y$ plane centered at the origin. For a vector field $\vec E = (-y\hat e_x +x\hat e_y)/(x²+y²),$ the value of the integral $\int_s (\vec \nabla ×\vec E). ...
2 votes
1 answer
1k views

Finding the vector field of the given potential function

Given the potential function $$f(V) = \cos (|X|^2), \quad X = (X_1, \dots, X_n)$$ find the vector field $V(X).$ I'm not really use to do this kind of question, usually they ask the other way ...
3 votes
1 answer
62 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 ...
3 votes
1 answer
694 views

Deriving Killing's Equation without Lie Derivative

I am attempting to solve this question from do Carmo's Riemannian Geometry. He begins by definining a Killing vector field in the following way: Let $M$ a Riemannian manifold, $X$ a vector field on ...
0 votes
0 answers
18 views

Finding a smooth surface with boundary as a Jordan curve in a simply connected domain

Consider the following: For any smooth Jordan curve $\gamma\subset D$, where $D$ is simply connected, we can find a smooth surface $\Sigma \subset D$ such that $\partial\Sigma = \gamma$. I am curious ...
1 vote
0 answers
64 views

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 (\...
0 votes
0 answers
47 views

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 ...
0 votes
1 answer
48 views

Inner product defined on a Vector Field on Complex Numbers

I have a homework problem that seems wrong. Previous to the problem we showed that the inner product of a vector space $V$ on the field $\mathbb{C}$ defined as $$ \langle v,w \rangle = a^i(b^j)^*\...
1 vote
1 answer
34 views

How do you know when level curves are straight?

I have a smooth function $F(x,y)$ which sends an open disc $D\subset \mathbb{R}^2$ in the plane to $\mathbb{R}$. I would like to know how to compute whether the level curves of $F$ are all each (a ...
0 votes
1 answer
68 views

Given $\gamma=\gamma(t,s):I\times I\to M$ the vector fields $\frac{\partial \gamma}{\partial s},\frac{\partial \gamma}{\partial t}$ commute

The problem is pretty much stated in the title ($I$ is the interval $[0,1]$ and $M$ is a differentiable manifold). Clearly we can't use the fact that $\gamma_*[X,Y]=[\gamma_*X,\gamma_*Y]$, because $\...
1 vote
1 answer
64 views

Lie derivative of $X^\flat$

Let $X$ be a vector field and suppose we have a Riemannian metric $g$. This allows identification of tangent and cotangent spaces, and we may consider $X^\flat := g(X, \cdot)$ as a 1-form. Does there ...
1 vote
4 answers
549 views

Confused about definition of Vector Field as map of smooth functions

I'm confused about the definition of vector field on a manifold. I've always (intuitively) understood it as a map from a point $p$ in the manifold $M$ to a vector $X(p)$ in the the tangent space $T_p$....
0 votes
0 answers
52 views

Struggling to understand vector calculus

I was working through an energy calculation for the Fokker-Planck equation: $$ \partial_t \rho + \nabla\cdot(\mathbf{v}\rho) = 0 $$where $$ \mathbf{v} = -(\frac{1}{\rho}\nabla\rho + \nabla V). $$ At ...
3 votes
1 answer
51 views

Concatenation of $f$-related vector fields/tangent vectors

This is probably a highly trivial question, but I just can't wrap my head around it. Let $M,N$ be two manifolds and $f: M \rightarrow N$ smooth. Let further $X^1, X^2$ be two smooth vector fields on $...
1 vote
0 answers
24 views

How does the del operator work exactly? [duplicate]

In Cartesian coordinates, vector operations are as simple as if we were treating the del operator as like a vector. $$\nabla = \Big(\frac{\partial}{\partial x}\hat{\textbf{i}} + \frac{\partial}{\...
1 vote
1 answer
46 views

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? ...
0 votes
1 answer
99 views

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 ...
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^...
2 votes
1 answer
1k views

When does zero divergence imply a vector potential exists?

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 ...

1
2 3 4 5
57