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0
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0answers
7 views

How can I find the stability of the equilibria of this vector field?

Consider the vector field given by $y' = y - y^{3}$. This clearly has equilibria at the points $y = 0, \; y = 1, \; y = -1$. How would I find the stability of these points though? I understand that I ...
0
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0answers
25 views

Curl of a vector in the complex plane

Let there be a vector $u(z)$ in the complex plane. Are these two statements equivalent? $$\nabla\times\overline u=\overline{\nabla\times u}$$ If not, why? I think they should be equal since $\nabla$ ...
0
votes
2answers
72 views

Work = line integral over closed loop

For a velocity field $$ \textbf G(x, y) = (3x^2 − 6y^2 + 1)\textbf i + (x + 4y − 12xy)\textbf j $$ show that the work done in moving a particle on the unit circle centred at (1, 0) taking an ...
2
votes
1answer
15 views

Expression of a given vector field for the stereographic projection of the sphere

I have got stuck trying to solve the following problem. Let $X=-zx \frac{\partial}{\partial x} -zy \frac{\partial}{\partial y} + (1-z^2) \frac{\partial}{\partial z}$ be a vector field in ...
2
votes
1answer
21 views

Show that the vector field $\vec F=(yf(u),xg(u))$ has no potential

I'm trying to prove that the vector field $\vec F=(xf(u),xg(u))$ with $u=xy$ is not conservative. I suppose that there is a function $\phi$ so that $\nabla \phi= \vec F$. So I need to satisfy that: ...
3
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1answer
20 views

Show that the vector field $\vec F=(xf(u),xg(u))$ is not conservative

I'm trying to prove that the vector field $\vec F=(xf(u),xg(u))$ with $u=xy$ is not conservative. I suppose that there is a function $\phi$ so that $\nabla \phi= \vec F$. So I need to satisfy that: ...
1
vote
0answers
14 views

A clarification on an answer on residues and Polya fields

In this very informative and interesting answer about the relation between residues and representation of complex functions as vector fields the author states that the function $$f(z) = \frac{1}{z}$$ ...
0
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0answers
22 views

Helmholtz theorem (with multivariable calculus)

The Helmholtz decomposition theorem says that every smooth [I would be inclined to suppose that it is enoough for it to be of class $C^2$] vector field $\boldsymbol{F}$ on an opportune region ...
1
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1answer
43 views

Knowing a scalar field from its Laplacian and gradient

As a part of the "rabbit hole" I am descending in order to understand the meaning of the integrals of a not-so-rarely found derivation of Ampère's law, I am trying to understand how to see the ...
3
votes
1answer
21 views

Finding a domain of an integral curve of a vector field

Studying Morse theory, I am stuck on some problem. Let $M$ be a compact smooth manifold, and $f$ is a smooth real-valued function on $M$. Choose a Riemannian metric $g$ on $M$, let $X$ be the vector ...
0
votes
1answer
25 views

$L^1(0,T;(L^1_{loc}(\mathbb{R^N}))^N)$ - Confused about notations used for space and time dependent vector fields

I found this notation - $L^1(0,T;(L^1_{loc}(\mathbb{R^N}))^N)$ - in a paper of DiPerna and Lions concerning vector fields space and time dependent, "Ordinary differential equations, transport theory ...
1
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1answer
40 views

Circulation of a vector field, why the definition given in Feynman Lectures on Physics is true?

In the third chapter of Electormagnetism, R.P.Feynman teach the basics of vectoriel integral calculus and I'm stuck in the formula $(3.30)$ give as a definition, but I don't understand why if it works ...
0
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0answers
17 views

Criterion for semi-simplicity of Lie algebra generated by vector fields

Suppose I have a finite collection of smooth vector fields $V:=\{V_1,...,V_k\}$ on a smooth manifold $M$. Moreover suppose that the Lie algebra $g$, generated by $V$ (where the Lie bracket is defined ...
3
votes
1answer
40 views

A question on dynamical system's focus values on its center manifold

I have recently come across this problem involving the center focus of dynamical systems of a parameter vector field related to center manifold: We define a vector field on $ R^3 $ given by: ...
0
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1answer
15 views

Finding the flow line of a vector field that is parametrized using 3rd degree exponents

I'm trying to find the flow line $\textbf{u}(t)$ of $\textbf{F}(x,y,z)=(2,-3y,z^3)$ that passes through a point $(a,b,c)$ at time $t=0$. To solve this as I have been taught, I need to solve this ...
5
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1answer
45 views

Why curl of a vector field measures its tendency of rotation

I was trying to understand why curl measures a vector field's tendency of rotation. Two examples from physics seem to answer my question: Curl of the velocity field is twice the angular velocity ...
2
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1answer
26 views

$X_z=\frac{d}{dt}_{|t=0} \Phi_t(z)$ has flow $\Phi_t$

Let $M$ be a manifold, $\Phi_t, t\in \mathbb R$ a one parameter group of diffeomorphisms and $X$ a vector field on $M$ definied by $$X_z:=\frac{d}{dt}_{|t=0} \Phi_t(z).$$ Show that $\Phi_t$ is the ...
8
votes
1answer
117 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)} $ ...
0
votes
0answers
20 views

Calculate flux of vector field

I want to calculate the flux of the vector field $$X(x,y)=y\partial_x-x\partial_y$$ in $\mathbb R^2$ written in polar coordinates ($\partial_x:=\frac{\partial}{\partial x}$ and so on). Step 1: ...
1
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0answers
21 views

how to visualise solenoidal vector field? [duplicate]

How to visualise solenoidal vector field;I am thinking it as a solenoid going in a circular fashion with propagation in the z direction. I was solving a problem to check that field is solenoidal or ...
1
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0answers
24 views

Matrix splitting procedures - is there equivalent of Helmholtz decomposition?

My post consists of two separate questions: I am interested in different ways to split a matrix in a form: A = B + C where both B and C would have some specific, useful properties. I am familiar ...
1
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0answers
21 views

A practical example of Helmholtz decomposition

I am familiar with the basic concept of the Helmholtz decomposition and I have read a number of materials on it (they all follow structure similar to that on Wikipedia page). However, I am not able ...
1
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0answers
26 views

Is my way of thinking correct about Green's Theorem?

Consider the vector field $$F(x,y) = F_1(x,y)\mathbf{i} + F_2(x,y)\mathbf{j} = -\frac{y}{x^2 + y^2}\mathbf{i} + \frac{x}{x^2 + y^2}\mathbf{j}$$ Show that $\frac{dF_2}{dx}=\frac{dF_1}{dy}$ for ...
1
vote
0answers
27 views

Prove that $X = \sin \phi \,\partial_\theta + \cot \theta \cos \phi\, \partial_\phi$ is a Killing vector field

I'm trying to prove that $X$ (given in the title of the question) is a Killing vector field for a metric which is spherically symmetric and static. This is what I've done so far: The vector field ...
0
votes
1answer
35 views

Why is the normal vector to the surface $f$ given by $\text{grad}(f)$? [duplicate]

According to my lecture notes, the normal vector to a surface $f$ is given by $\text{grad}(f) = \underline{\nabla} f$. However, surely the normal vector to the surface $f$ should be perpendicular to ...
4
votes
1answer
44 views

Negative divergence implies convergent flow?

Suppose we have a differentiable vector field $X:\Omega\to\mathbb{R^n}$ defined on an open, bounded and simply connected region subset $\Omega$ of $\mathbb{R^n}$, and its divergence is negative ...
1
vote
1answer
37 views

What is an invariant Simplex

Let $$\Delta=\{x_i\in\mathbb{R}^k_+: \sum_{i=1}^kx_i=1\}$$ where k is a positive integer. I've read in a book the following: The simplex $\Delta \subset \mathbb{R}^k$ can be shown to be invariant ...
0
votes
4answers
52 views

What is the difference between a linearly independent set and a spanning set? [closed]

Is a spanning set not just a linearly independent set of vectors, or is there a difference? The context of this question being the following theorem: "In any vector space, if $|I|$ is a linearly ...
1
vote
1answer
45 views

Killing fields and symmetries

In my answer to Why are Killing fields relevant in physics? I wrote: The flow of a Killing field can drag the whole manifold. Since Killing fields are divergence free and thus their flows have no ...
0
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2answers
33 views

How do we think of a field in the context of group theory?

The definition of a field (in the context of group theory) that I've been taught is as follows: "A field is defined as being a set $F$, combined with the binary operations $+$ and $\cdot$" This (to ...
-1
votes
4answers
37 views

Linear transformation $T(X_1,X_2,…,X_n)=(0,X_1,X_2,…,X_{n-1}) $

Let $T$ be a linear transformation from $F^n\to F^n$ defined by: $$T(X_1,X_2,...,X_n)=(0,X_1,X_2,...,X_{n-1}) $$ I need to calculate $T^k$ for every $k\in N$ and to find the matrix represents $T$. I ...
1
vote
2answers
27 views

Prove that divergense and curl free vector field is a constant vector field

I need to prove the fact that a vector field $\vec{B}$ that is divergence and curl free, is a constant vector field. I have attempted to prove this by referring to the divergence, but realized that ...
2
votes
1answer
33 views

Recommendation of a good source on Lyapunov theorem in dynamical systems

As part of my research I wish to read a full proof of Lyapunov's classic theorem on dynamical systems that for an analytic planar vector field where all Lyapunov/focal values are zero, the local phase ...
2
votes
0answers
31 views

A vector field with specified curl

I need a vector field $\vec{F}:\mathbb{R}^3\to\mathbb{R}^3$ such that $$\mathrm{curl}\ \vec{F}(x,y,z) \cdot \left(\frac{-x}{\sqrt{x^2+1}},\ 0,\ \frac{1}{\sqrt{x^2+1}}\right) = 1.$$ (This equality, of ...
1
vote
1answer
16 views

Find a field K where the vectors are linearly dependent in the vector space K^3

My task is to find a field $K$ where the vectors $$ \begin{pmatrix} 2 \\ 3 \\ 1 \\ \end{pmatrix} , \begin{pmatrix} 0 \\ 4 \\ 0 \\ ...
3
votes
4answers
55 views

Why isn't the fundamental theorem of line integrals applicable here?

The question is as follows: Given vector field $$V = \left(\frac{1-y}{x^2 + (y-1)^2}, \frac{x}{x^2+(y-1)^2}\right)$$ Evaluate $$\int_{l_1}V \bullet dr\text{, }\int_{l_2}V \bullet dr$$ Where ...
0
votes
1answer
17 views

Curl F and conservative vector field.

If curl F = 0, is F conservative? I know F conservative implies curl F = 0, however, I want to know if it works the other way.
0
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0answers
8 views

curl free vector field on non-simply connected domain

I'm looking for an example of a curl free vector field on a non-simply connected region which is still a gradient.
0
votes
1answer
34 views

Green's Theorem special application

If I want to evaluate the flux of a vector field over an unfinished square (basically over three sides of a square), instead of parametrezing the 3 sides and computing the flux 3 times, can I use ...
1
vote
0answers
64 views

Vector field flux calculation

Consider the vector field: $$F=((y+1)e^{y^2}\sin(y^3),0)$$ and the curve consisting of the three line segments $A$, $B$ and $C$. Where $A$ goes between $(1,-1)$ and $(1,1)$. $B$ goes between ...
0
votes
1answer
22 views

Finding a vector field

Give a formula $F=M(x,y)i + N(x,y)j$ for the vector field in the plane that has the property that F points toward the origin with magnitude inversely proportional to the square of the distance from ...
3
votes
0answers
38 views

Moment map in general

Let the Lie group $G$ act on the smooth manifold $X$ with the map $(g,x)\to gx$. In any point $x\in X$, the differential of this map induces a linear map: $$ \mu:T_e G \to T_xX\;, $$ and globally, if ...
1
vote
2answers
32 views

For some vector field, $<P, Q>$, is it possible for $P_{y} = Q_{x}$ and yet not have $\nabla f = <P,Q>$

Even if $P_{y} = Q_{x}$, is it guaranteed that some function exists whose gradient gives $<P,Q>$?
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0answers
30 views

Differentiating a certain vector field

Let $M^n$ be a riemannian manifold and $p_0$ a point in $M$. Let $U$ be a normal neighbourhood of $p_0$, image of $B_{\delta}(0) = \{ x \in T_{p_0} M : \left\lvert x \right\rvert < \delta \}$ ...
2
votes
1answer
44 views

Is my intuition correct about vector field?

Let us take a $\mathbb{R}^2 $ coordinate system and in it let us create a vector field of acceleration/force - the vector field will be: $$\vec{r(x,y)} = 0 \boldsymbol{\hat{\textbf{i}}} -9.8 ...
1
vote
0answers
22 views

Stokes Theorem and Fundamental Theorem of Integrals

Let $S$ be an oriented surface with a normal unit vector $\vec{n}$ which is included in an equation of a vector field $\vec{F}$ where $rot\vec{F}=k\vec{n}$ ($k$ is a positive constant) ...
1
vote
1answer
21 views

How to find the potential function of this vector field?

The instructions are to use the fundamental theory of line integrals to evaluate $$\int_C \cos x \sin y dx + \sin x \cos y dy$$ Where $C:$ Line segment from $(0, -\pi)$ to $(\dfrac{3\pi}{2}, ...
0
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1answer
26 views

Set the parametric equation of an arc with two points

Like the title says, I am looking for a method to find the parametric equation expressed in the form $\vec{r}=...\vec{i}+...\vec{j}$ of the arc that connects the points (2,0) and (1,2). I am asking ...
0
votes
1answer
44 views

Find a vector field $G$ with curl ($G$) = $F$

Let $F(x, y, z) = (y, z, x^2)$ on $\mathbb{R}^3$. We know that $$y = \frac{\partial G_3}{ \partial y} - \frac{\partial G_2 }{\partial z}, \\ z = \frac{\partial G_1}{ \partial z} - \frac{\partial G_3 ...
1
vote
2answers
19 views

How would you prove that the kernel of a linear mapping $\theta\colon V \rightarrow W$ is a subspace of $V$?

In my lecture notes, it says the following: The kernel of a linear mapping $\theta\colon V \rightarrow W$ is a subspace of $V$. Proof: Straightforward I can't say that I see this as being ...