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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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138 votes
10 answers
187k views

What is the Jacobian matrix?

What is the Jacobian matrix? What are its applications? What is its physical and geometrical meaning? Can someone please explain with examples?
Pratik Deoghare's user avatar
108 votes
5 answers
92k views

Difference between gradient and Jacobian

Could anyone explain in simple words (and maybe with an example) what the difference between the gradient and the Jacobian is? The gradient is a vector with the partial derivatives, right?
Math_reald's user avatar
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56 votes
6 answers
179k views

What does it mean to take the gradient of a vector field?

What does it mean to take the gradient of a vector field? $\nabla \vec{v}(x,y,z)$? I only understand what it means to take the grad of a scalar field.
fred's user avatar
  • 561
49 votes
4 answers
4k views

Is there a vector field that is equal to its own curl?

I was wondering if there is a vector field that satisfies the following condition: $$\vec F=\nabla \times \vec F$$
user avatar
46 votes
7 answers
172k views

Proof for the curl of a curl of a vector field

For a vector field $\textbf{A}$, the curl of the curl is defined by $$\nabla\times\left(\nabla\times\textbf{A}\right)=\nabla\left(\nabla\cdot\textbf{A}\right)-\nabla^2\textbf{A}$$ where $\nabla$ is ...
Demosthene's user avatar
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38 votes
2 answers
21k views

Meaning of derivatives of vector fields

I have a doubt about the real meaning of the derivative of a vector field. This question seems silly at first but the doubt came when I was studying the definition of tangent space. If I understood ...
Gold's user avatar
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35 votes
2 answers
5k views

Geometric intuition behind the Lie bracket of vector fields

I understand the definition of the Lie bracket and I know how to compute it in local coordinates. But is there a way to "guess" what is the Lie bracket of two vector fields ? What is the geometric ...
pokraka's user avatar
  • 515
26 votes
3 answers
21k views

Anti-curl operator

It is known that if a vector field $\vec{B}$ is divergence-free, and defined on $\mathbb R^3$ then it can be shown as $\vec{B} = \nabla\times\vec{A}$ for some vector field $A$. Is there a way to find ...
Max's user avatar
  • 869
24 votes
1 answer
766 views

Tricky surface integral of vector field

The following problem comes from a vector calculus exam. Let $$ S = \left\{ (x,y,z) \in \mathbb{R}: z = e^{1 - (x^2 + y^2)^2}, z > 1 \right\} $$ be an embedded surface with the orientation ...
Sam Skywalker's user avatar
19 votes
1 answer
801 views

When a vector field can be scaled to form a conservative vector field

Consider a vector field given by its components $g_i(x_1, \dots, x_n)$. It is well known that necessary and sufficient condition for a following system $$ \frac{\partial f}{\partial x_i} = g_i(x_1, \...
uranix's user avatar
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17 votes
3 answers
2k views

Does the "field" over which a vector space is defined have to be a Field?

I was reviewing the definition of a vector space recently, and it occurred to me that one could allow for only scalar multiplication by the integers and still satisfy all of the requirements of a ...
Geoffrey's user avatar
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17 votes
2 answers
2k views

Can I comb unoriented hair on a ball?

I know there is no non-vanishing vector field on $S^2$, so I cannot comb the hair on a ball. (I am treating $S^2$ as a manifold without the ambient space $\mathbb R^3$, which amounts to demanding that ...
Joonas Ilmavirta's user avatar
16 votes
2 answers
4k views

Relation between exterior derivative and Lie bracket

There is a formula connecting the exterior derivative and the Lie bracket $$d\omega (X,Y) = X \omega(Y) - Y \omega(X) - \omega([X,Y]).$$ What is a good way to remember this? By which I mean, what ...
Eric Auld's user avatar
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15 votes
2 answers
2k views

Non-vanishing vector fields on non-compact manifolds

In several papers the following result is invoked: Theorem. Every connected, non-compact, smooth manifold $M$ carries non-vanishing smooth vector fields $v$. (we are assuming $M$ is $2$nd countable ...
Pedro Teixeira's user avatar
14 votes
1 answer
5k views

extending a vector field defined on a closed submanifold

Let $M$ be a differentiable (smooth) manifold, and $S$ a closed submanifold. Let $X$ be a vector field on $S$. Prove that $X$ is the restriction of a vector field $Y$ defined on $M$. I tried this way ...
balestrav's user avatar
  • 2,091
14 votes
1 answer
2k views

Lie derivative w.r.t. time-dependent field

Some time ago, I asked this question. CvZ answered, and with my additional answer I thought I had solved the problem. Yesterday evening, I copied those calculations into my thesis, and having $t$ and $...
MickG's user avatar
  • 8,725
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 ...
jacopoburelli's user avatar
13 votes
2 answers
2k views

Extension of Vector field along a curve always exists?

Let $c:I\to M$ be a $C^{\infty}$ curve on smooth manifold $M$ of dimension $n$ and $X:I\to TM$ be a vector field along $c$, $$\forall t\in I\qquad X(t)\in T_{c(t)}M.$$ does there exist a vector field ...
user's user avatar
  • 1,352
13 votes
1 answer
6k views

Proving The Extension Lemma For Vector Fields On Submanifolds

I need some hints to prove the following lemma (John Lee's $\textit{Introduction to Smooth Manifolds 2nd Ed}$ p.201) : EXTENSION LEMMA FOR VECTOR FIELDS ON SUBMANIFOLDS: Suppose $M$ is a smooth ...
Dubious's user avatar
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13 votes
1 answer
939 views

Is a general smooth rescaling of a complete vector field itself complete?

$\newcommand{\Ga}{\Gamma}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\til}{\tilde}$ $\newcommand{\M}{M}$ $\newcommand{\ep}{\epsilon}$ $\newcommand{\brk}[1]{\left(#1\right)} $ $\newcommand{\R}{\mathbb{...
Asaf Shachar's user avatar
  • 25.2k
13 votes
2 answers
2k views

Vector fields, line integrals and surface integrals - Why one measures flux across the boundary and the other along?

Why is it that a line integral of a vector field takes the dot product of the vector field with the tangent? This results in us taking the component of the vector field in the direction of the tangent ...
14tim4's user avatar
  • 411
12 votes
3 answers
13k views

Compute the pushforward of the vector field $\frac{\partial}{\partial x^1}$ via a given function $\phi$

We define $\phi:=(\phi^1,\phi^2):\Omega\subset\mathbb{R}^2\to\phi(\Omega)$ with $\Omega$ such that $\phi$ is a diffeomorphism by $$x^1:=\phi^1(r,\theta)=r\cos\theta\qquad\text{and}\qquad x^2:=\phi^2(r,...
Thomas Produit's user avatar
12 votes
3 answers
2k views

What is the Derivative of a Vector Field in a Manifold?

I'm studying the book "Geometric Theory of Dynamical Systems An Introduction" - Jacob Palis, Jr. Welington de Melo. On page 10, the author defines: Let $M^m\subset \mathbb{R}^k$ be a ...
Matheus Manzatto's user avatar
12 votes
4 answers
4k views

Computing the Lie bracket on the Lie group $GL(n, \mathbb{R})$

Consider the Lie group $GL(n, \mathbb{R})$. Since $GL(n, \mathbb{R})$ is an open subset of the space $M_{n,n}(\mathbb{R})$ of $n \times n$ matrices, we can identify the tangent space (Lie algebra) $T_{...
Tom Bombadil's user avatar
  • 1,706
12 votes
1 answer
104k views

What is the cross product in spherical coordinates?

The curl of an arbitrary vector, $\vec{A}$ is The curl of an arbitrary vector $\vec{A}$ in spherical coordinates \begin{align*} \nabla \times \vec{A} &= \frac{1}{r^{2}\sin{\theta}}\left| \begin{...
Joshua Burrow's user avatar
12 votes
2 answers
14k views

Find vector field given curl

I have an equation $\nabla \times \vec{B} = \mu_{0}\vec{J}$, where $\vec{J} = \left\langle f(x,y), g(x,y), 0 \right\rangle$ and need to solve for $\vec{B}$. I've looked elsewhere on here for how to "...
jackarms's user avatar
  • 245
12 votes
3 answers
406 views

Why the mapping $1/\bar{z}$ look like the field lines of an electric dipole?

While I was studying Mobius Transformation in complex analysis, the complex function $\frac{1}{\bar{z}}$ caught my attention. suppose there are $n$ lines equally spaced all intersecting at $(1,0)$ as ...
shahrOZe's user avatar
  • 317
11 votes
2 answers
295 views

Which matrices can be embedded in flows?

I've been thinking about matrices, recently, and wondering about a fairly simple but maybe hard-to-answer question. A real $n\times n$ matrix $A$ yields a discrete dynamical system on $\mathbb{R}^n$, ...
Kiran Parkhe's user avatar
11 votes
1 answer
2k views

Electric field and curvature

My physic teacher said that In a conductor the electric field, which is non-zero only on the surface, is stronger where the curvature is bigger*. But he did not provide a mathematical proof for ...
CNS709's user avatar
  • 1,667
11 votes
2 answers
518 views

(Vishik's Normal Form) Behavior of a vector field near the boundary of a manifold

I'm trying to prove a special case of Vishik's Normal Form. Consider $T^2 := \frac{1}{\sqrt{2}}\cdot\mathbb{T}^2 \subset \mathbb{S}^3$, let $h: \mathbb{\mathbb{S}^3}\to \mathbb{R}$ be the function $h(...
Matheus Manzatto's user avatar
11 votes
0 answers
535 views

Flow of a Vector Field Using Sheaves

I am wondering whether the concept of the flow of a vector field can be described in the following sheaf-theoretic way: $\newcommand{\Flow}{\mathrm{Flow}} \newcommand{\pre}{\mathrm{pre}} \renewcommand{...
user avatar
11 votes
1 answer
639 views

The gradient of a function has constant Euclidean length $1$

Consider a function $f : \mathbb R^{2} \to \mathbb R$ that is defined on every point and is differentiable. Then it has a gradient $\nabla f$. Now, suppose that $|\nabla f(x,y)| = 1$ for all $x,y \in \...
calcstudent's user avatar
10 votes
2 answers
1k views

How can I prove that these definitions of curl are equivalent?

I am reading the book "Div, Grad, Curl, and All that" and I got to the section about curl. In this section, the author defines the curl to be $$ (\nabla \times \mathbf{F})\cdot \mathbf{\hat{...
Robert Lee's user avatar
  • 7,243
10 votes
1 answer
285 views

Do $S^n$ and $\mathbb{R}P^n$ admit the same number of linearly independent vector fields?

For a manifold $M$, let $i(M)$ be the maximum possible number of everywhere linearly independent vector fields on $M$. It's evident, I believe, that $i(S^n) \geq i(\mathbb{R}P^n)$. Does the equality ...
Pedro's user avatar
  • 6,538
10 votes
1 answer
6k views

Explanation for example of flow generated by vector field

The text I am reading has an example for flow in a section titled "Flows and Lie derivatives." Below is the example: Let $M = \mathbb{R}^2$, and let $X((x,y)) = -y \partial/\partial x + x \partial/\...
Alex's user avatar
  • 1,009
10 votes
2 answers
981 views

Why do we need a Lie derivative of a vector field?

Lie derivative of a smooth vector field $Y$ in the direction of a smooth vector field $X$ is defined (at least in our geometry course) as $L_X Y = \frac{d}{dt}\mid_{t=0} (\psi_\star Y)$ where $({\...
Tereza Tizkova's user avatar
10 votes
2 answers
2k views

$X$ is a Killing field $\iff \langle \nabla_YX,Z \rangle+\langle \nabla _ZX,Y\rangle=0$

I was not able to do the following exercise from Manfredo's Riemannian Geometry: Chapter 3, Exercise 5 Let $M$ be a Riemannian manifold and $X\in \mathfrak{X}(M)$. Let $p\in M$ and let $U\subset M$ ...
Derso's user avatar
  • 2,767
10 votes
2 answers
3k views

Second derivative of a vector field

I wonder how to treat the "second derivative" of a vector field. For example, imagine we have a vector field $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$. Then we evaluate the derivative at two points $...
Dan's user avatar
  • 101
9 votes
2 answers
1k views

Differential form is exact, but vector field is not conservative - what did I miss?

I have the following differential form $\omega\in\Omega_2(\mathbb{R^3})$: $$\omega = 2xzdy\wedge dz + dz\wedge dx - (z^2+e^x)dx\wedge dy$$ The question is whether it is exact, i.e whether there exists ...
clara52's user avatar
  • 93
9 votes
2 answers
3k views

BAC─CAB rule used on del operators

In my textbook, it is stated that $\nabla\times(\nabla\times A)=\nabla(\nabla\cdot A)-\nabla^2A $ So, I thought that del can be treated as if it were a vector (although it was an operator). However, ...
Danny  Han's user avatar
  • 215
9 votes
1 answer
2k views

Completion of local frames for the tangent bundle of a smooth manifold

In John M. Lee book Introductions to smooth manifolds Proposition 8.11 is left as exercise. Can anyone give me hints and suggestions to prove it? In particular, i want to show: Let $M$ be a smooth ...
Minato's user avatar
  • 1,456
9 votes
3 answers
3k views

Lie bracket of local orthonormal basis of vector fields

Let $(M, g, \nabla)$ be a Riemannian manifold (with $\nabla$ the Levi-Civita connection of $g$), and let {$E_i$} be a local orthonormal basis of vector fields ($g(E_i, E_j) = \delta_{ij} )$. What can ...
Ngicco's user avatar
  • 113
9 votes
2 answers
206 views

Weird system of PDEs defined on a sphere

Let $x$ and $y$ be functions defined on a simply connected (open or closed) portion of the surface of a (unit) sphere, and consider the following system of PDEs: \begin{align} \|\nabla x\|^2 = \|\...
DanielKatzner's user avatar
9 votes
1 answer
955 views

Translate a vector field

Imagine that you have a vector field $A = \frac{A_0}{r} e_{\theta}$ in cylindrical coordinates, where $A_0 \in \mathbb{R}$. Now you translate your coordinate system in $e_x$ direction by $x \mapsto x +...
user avatar
9 votes
3 answers
36k views

If the curl of some vector function = 0, Is it a must that this vector function is the gradient of some other scalar function?

I know of course that If the curl of a vector function is equal to zero, then the vector function is the gradient of some other scalar function, but is this a must? if so, please give mathematical ...
Mohamed Ayman's user avatar
8 votes
2 answers
2k views

What's the difference between time-dependent flow (isotopy) and time-independent flow?

Regarding the fact that both time-independent and time-dependent vector fields correspond with family of diffeomorphisms, i.e. $\{\phi_t | t\in\Re, \phi_t: M\to M\}$, what's the difference between ...
Amin's user avatar
  • 115
8 votes
3 answers
1k views

Why does it seem the inner curls within a surface always cancels in order for greens theorem to be true

Im trying to learn aerodynamics in general for my course. Every video i see to derive the concept of greens and stokes theorem shows how the inner curls within a surface area cancel to 0 and its only ...
George kirby's user avatar
8 votes
2 answers
598 views

If $\vec{\nabla} \cdot \vec{V} \neq 0$ at only one point, will this prevent us from saying that $\vec{V}=\vec{\nabla} \times \vec{U}$?

This question has an answer in the language of high level mathematics. Can somebody explain this in the language of vector calculus. Part I: Let us consider Cartesian coordinate system with origin $O$ ...
Joe's user avatar
  • 1,121
8 votes
2 answers
713 views

Show $\ker(\alpha)=\ker(\alpha)^2 \ \iff \ \ker(\alpha)\cap \mathrm{Im}(\alpha)=\{0\}$

Let $V$ be a vector space over a field $F$ and let $\alpha$ be an element of $\operatorname{End}(V)$. Show $\ker(\alpha)=\ker(\alpha^2)$ iff $\ker(\alpha)$ and $\operatorname{im}(\alpha)$ are linearly ...
cele's user avatar
  • 2,237
8 votes
1 answer
671 views

Two definitions of linearly independent vector fields

I'm studying the chapter of Lee's ISM on vector fields, and there's one thing that confuses me a lot. If $\mathfrak{X}(M)$ denotes the set of smooth vector fields on a smooth $n$-manifold, then it is ...
Boar's user avatar
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