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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,...

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Smoothness of a vector field defined by a system of differential equations

There is a lemma stating that there exists a unique vector field $G$ on the tangent bundle $TM$ of a manifold $M$ whose integral curves are of the form $t\mapsto (\gamma(t),\gamma’(t))$ where $\gamma$ ...
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$x = XOR(a_0, a_1, …\space a_n), \exists \space j \in [[0, n]] a_j \space xor (a_j - x) \space xor\space x = 0$, prove me right or wrong

I tell you that given any family of natural number: $$a_0,\space a_1,\space ...\space a_n$$ (of any finite lenght), posing $$x = XOR(a_0,\space a_1,\space ...\space a_n)$$ We have $$\exists\space ...
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vector field with no periodic orbits that are non constants

Give an example of non periodic vector field X on $\mathbb{R}^2 $ non identically zero with a compact support such that X has no periodic orbits that are non constants. I am not sure how I should ...
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Non periodic vector field on $\mathbb{R}^2$ [on hold]

Give an example of non periodic vector field on $\mathbb{R}^2 $. I found that gradient systems cannot have periodic orbits. so I picked this vector field: $X(x,y) = x+y$. 1) Is this a valid vector ...
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26 views

find a vector field in $\mathbb{R^3}$ with specific properties

Let $α = dz - ydx \in Ω^1 (\mathbb{R}^3)$ Find a vector field Z in $\mathbb{R^3}$ such that $α(Z) = 1$ and $dα(Z,.) = 0$ What I did: I computed $d\alpha = dx \wedge dy $. Then let $Z=a\partial_x + ...
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Intuition on Stokes' Theorem

Stokes' Theorem says that $$\int_C\vec{F} \cdot d\vec{r} = \int \int_S (curl \space\vec{F}) \space\cdot \vec{n} \space dS$$ I understand that $curl \space\vec{F}$ is the "spin" or "circulation" on a ...
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Partial differential operator is sum of order $0$ operators and composition of vector fields

Let $P:\Gamma(\Bbb R^n, E_0) \rightarrow \Gamma(\Bbb R^n, E_1)$ be a differential operator where $E_0$, $E_1$ are trivial vector bundles, with the standard bundle metric over $\Bbb R^n$. In page 28 ...
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38 views

Prove a vector calculus identity

Let $V \subseteq \mathbb{R}^3$ have smooth boundary $\partial V$. Suppose $f, g, \mathbf{J}$ are differentiable on $V \cup \partial V$, with $\nabla \cdot \mathbf{J} = 0$ in $V \cup \partial V$, then ...
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1answer
48 views

On the notion of tensor in Riemannian Geometry

In DoCarmo’s Riemannian Geometry, a tensor of order $r$ on a Riemannian manifold is defined as a multilinear mapping $$T:\Xi^r(M)\rightarrow C^{\infty}(M)$$ where $M$ is a smooth manifold of dimension ...
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23 views

Well-definedness of assinging a value to a point of a manifold with respect to parallelism

By definition as in DoCarmo’s Riemannian Geometry book, in a smooth manifold $M$ a vector field $V$ along a smooth curve $c:(a,b)\rightarrow M$ is a mapping $t\mapsto V(t)\in T_{c(t)}M$ such that $t\...
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30 views

How to integrate the curve

We have a Curve $C:\vec{x}\left(t\right)=\begin{pmatrix}1-2t^2\\ t\end{pmatrix}$ Now you have to calculate $\int _C\vec{F}\left(\vec{x}\right)d\vec{x}$ for $\vec{F}\left(\vec{x}\right)=\begin{...
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Evaluate the vector field $(xz^2, z^3, z(x+y))$ over an ellipsoid

I would like to evaluate the following vector function $$(xz^2, z^3, z(x+y))$$ over the surface of the ellipsoid: $$x^2/4 + y^2/2 + z^2 = 1$$ I thought of using the divergence theorem to instead ...
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Stokes Theorem and Circulation

Find the circulation of $F = \frac{-y}{x^2+y^2}i + \frac{x}{x^2+y^2}j$ along the unit circle. I decided to express $F$ in cylindrical coordinates: $$F = \frac{1}{r}{\hat{\theta}}.$$ However, I found ...
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Why is this implying a normal vector field?

Suppose $\omega$ is a $n-1$ form on a $n-1$ dimensional manifold and $(a_1(x)dx_1 + ... + a_n(x)dx_n )\wedge \omega = c\Omega $, with $c \neq 0$ and $\Omega =dx_1\wedge...\wedge dx_n$. Moreover $(a_1(...
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Question about proof of n-1 form inducing normal unit vector field

Suppose we have a $n-1$ dimensional manifold $M \subset \mathbb{R}^n$ and a non-vanishing $n-1$ form $\omega$ on $M$. This implies the existence of a normal unit vector field on $M$. The proof of ...
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30 views

Left inverse of a matrix $3 \times 2$ in $\mathbb{F}_7[x]$

Do you know a method to calculate inverse matrix in $\mathbb{F}_7[x]$? I want to calculate left inverse the following matrix of $3 \times 2$ in $\mathbb{F}_7[x]$ \begin{bmatrix} x^2+1 & x-1 \\ ...
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1answer
37 views

n-1 form inducing normal unit vector field

Suppose we have a $n-1$ dimensional manifold $M \subset \mathbb{R}^n$ and a non-vanishing $n-1$ form $\omega$ on $M$. How would this imply the existence of a normal unit vector field on $M$?
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Stokes' Theorem - Vector Field

I am having problems trying to verify Stokes' theorem (below) as part of a question. $$\iint_{S} \text{curl} \vec F \cdot d\vec S=\oint_{c} \vec F \cdot d\vec r$$ The vector field in question is $\...
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Regularity of a hyper-surface defined through a flow

Let $X:\mathbb{R}^n \to \mathbb{R}^n$ be a smooth and bounded vector field, such that $$ X_n \ge c|(X_1, \dots, X_{n-1})| \ge \epsilon > 0\,. $$ Under these assumptions one can prove that integral ...
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Submanifold of codimension 1 orientable iff there exists unit normal vector field.

Suppose I have a submanifold $M \subset \mathbb{R}^{n}$, of dimension $n-1$. Where a unit normal vector field is a section $\nu$ of the normal bundle $ TM^{\bot} \to M$. So the fibers are all the ...
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Every submanifold is orientable (co-dimension 1)?

Suppose I have a submanifold $M \subset \mathbb{R}^{n}$, of dimension $n-1$. Apparently it's orientable if and only if there exists a unit normal vector field on $M$. Where a unit normal vector field ...
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Number of independent components of a vector satisfying a differential constraint?

Edited question Consider a vector field $\vec{A}(\vec{x})$ such that in one case $\nabla\cdot\vec{A}=0$. It looks like that this condition gives rise to a differential equation constraint $$\...
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Integral of a 2-Form Over a Certain Region of Integration

This is a rather simple problem, but one that I'm struggling with nonetheless. I'm given a 2-Form, $\beta = zdx \wedge dy-x^2dy \wedge dz$ that I need to integrate over the surface S : $z=4-x^2-y^2 \...
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60 views

Frightening Stokes Theorem Computation

I came across a rather frightening problem in my problem set and I am confused about it. I need to compute the flux of a field $\vec F = \langle y, x^2, z(x^2-y^3)^7 \cos(e^{xyz}) \rangle$ through ...
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1answer
32 views

Using Stokes's Theorem to Compute the Circulation of a Vector Field over a Triangle in the Plane

I'm tasked with computing the circulation of the vector field $\vec F = <y^2, z, xy>$ along the triangle with vertices $(1,0,0), (0,1,0), (0,0,1)$ with the orientation of the curve following ...
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1answer
20 views

Magnetic field at points on the circuit

I know magnetic field lines due to a circuit always form closed loops. Therefore $\nabla \cdot \vec{B}=0$ everywhere (even at points on the circuit). However due to singularity, magnetic fields are ...
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Weak analogues of gradient, divergence, and curl (collecting examples)

This question is mostly to help me understand the idea behind the "weak curl", but I also hope to accomplish other objectives with this question/post as well, partially inspired from some of the "...
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Understanding ODE and phase portraits on Manifolds

I'm studying on my own Dynamicals Systems and having difficulties to undersrtand ODE defined on a manifold $M$. Firstly, let $X: \Omega \subset \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} $ a vector ...
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49 views

How to show $\vec{\nabla} \cdot \vec{P}=0$ over the whole domain of ${\mathbb R}^3$

If in ${\mathbb R}^3$, $r=\sqrt{x^2+y^2+z^2}\text{ and }\vec{P}= \dfrac{\partial}{\partial z} \left( \dfrac{1}{r} \right) (\hat{j}) -\dfrac{\partial}{\partial y} \left( \dfrac{1}{r} \right) (\hat{k}...
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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 $O$...
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First integral of a vectorial field

I have a problem that says: Let $$\dot x = f(x,y) $$ $$\dot y = g(x,y) $$ be a 2-dimensional ODE with vector field $X:U\to \mathbb{R}^2 $, $X=(f,g)\in \mathcal C^1 (U)$, and $U$ an open star ...
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0-1 matrices kernel over a real vector field

Let $M$ be a $0−1$ matrix and $M$ is a $m \times n$-matrix with $m \le n$. The $1's$ and the $0's$ are random distributed. It can be a diluted matrix. And then we consider the Kernel of $M$ over a ...
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1answer
26 views

Finding curl in spherical coordinates

I've been asked to find the curl of a vector field in spherical coordinates. The question states that I need to show that this is an irrotational field. I'll start by saying I'm extremely dyslexic ...
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1answer
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Calculating components of normal to a surface

Let $S$ be the surface $−5x^2+4y^2+2z^2−4=−5$. The vector $\mathbf{n}$ is normal to $S$ at the point $(3,−3,2)$ and $\mathbf{n}\cdot\hat{\mathbf{i}}=−30$. Find $\mathbf{n}\cdot\hat{\mathbf{k}}$. I ...
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System of ODE's - Why does this vector function connect two constant vectors continuously?

Given the system of two ODE's $\frac{{d{\mathbf{w}}}}{{d\xi }} = {\mathbf{r}}({\mathbf{w}}(\xi ))$, with $\lambda ({\mathbf{w}}(\xi )) = \xi $ and ${\mathbf{w}}(\lambda ({{\mathbf{u}}_L})) = {{\...
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what is the physical interpretation of a work done by a vector field F = zero?

If we have a field vector $\vec F= (e^x \sin y )\hat i +(e^x \cos y) \hat j$ and there is a particle that has moved along the path from $(0, 0)$ to $(1, \pi/2)$ due to this field. If we calculate the ...
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58 views

How to straighten up this spiraling vector field?

Let $X(x_1, x_2) = (x_1 -x_2)\partial_1 + (x_1 + x_2 + 1)\partial_2$ be a vector field on $\mathbb{R}^2$. I want to straighten it up around point $0$ . $X(0) = \delta_2|_0 \neq 0$. So it must be ...
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Concrete interpretation of parallelism of a vector field along a curve

Let $M$ be a smooth manifold with an affine connection $\nabla$. A vector field along a curve $c$ is called parallel if its covariant derivative $\frac{DV}{dt}$ along $c$ is equal to $0$. This notion ...
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Is velocity field extendible to a vector field on the whole manifold?

In DoCarmo’s Riemannian Geometry book, an affine connection $\nabla$ on a smooth manifold $M$ is defined to be a mapping :$\Xi(M)\times \Xi(M)\rightarrow \Xi(M)$, where $\Xi(M)$ is the set of all ...
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Show that any affine connection $\nabla$ on $\mathbb{R}^n$ is of the form $\nabla=D+\Gamma$.

Show that any affine connection $\nabla$ on $\mathbb{R}^n$ is of the form $\nabla=D+\Gamma$, where $D$ is the Euclidean connection and $\Gamma:\mathcal{X}(\mathbb{R}^n) \times \mathcal{X}(\mathbb{R}^n)...
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23 views

A relation between left-invariant vector fields

If $G$ is a Lie group with a bivariant metric and if $U,V,X$ are left invariant vector fields, I wish to prove that $\langle[U,X],V\rangle=-\langle U,[V,X]\rangle$. Following the proof of Do Carmo’s ...
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Meaning of commutativity of a local flow of a vector field and left-translations

In DoCarmo’s Riemannian Geometry book, it is writen that if $x_t$ is the flow of a left invariant vector field $X$ of a lie group $G$, then $L_y\circ x_t=x_t \circ L_y$. But the domains of the left ...
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How to decompose the vector field by the parameter t

i have a fields: $X=X_1 \dfrac {\partial }{\partial q_1}+...+X_n \dfrac {\partial }{\partial q_n}$ $Y=Y_1 \dfrac {\partial }{\partial q_1}+...+Y_n \dfrac {\partial }{\partial q_n}$ $Z=Y_1(X_t(q)) \...
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Line integral of a vector field along a curve C with two segments

Vector field $ \vec F = (3x^2y^3+8x)\vec i + 3x^3y^2\vec j$, along a curve C consisting of two segments C$_1$ and C$_2$. Line segment C$_1$ given by $y = 0$ and $0 ≤ x ≤ x_0$ and the line segment C$...
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Show that at placing a cork in such a fluid, this will rotate on a plane parallel to the plane $z = 0$, in a circular path around the $z$ axis

Assume that $F(x, y, z) = (-y, x, 0)$ represents the velocity field of a fluid in $\mathbb{R}^3$. Show that at placing a cork in such a fluid, this will rotate on a plane parallel to the plane $z = 0$,...
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83 views

Integral curves and flows of bounded vector fields

Let $X=(X_1,\dots, X_n)\in L^\infty$ be a smooth (non Lipschitz) vector field such that \begin{equation}\tag{1} X_n \ge c|(X_1, \dots, X_{n-1})|\,. \end{equation} Does the cone condition (1) imply ...
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1answer
64 views

Why does this vector field approach zero near the north pole?

In this question, Raziel's answer builds a vector field over $S^2$. The vector field is built from the push forward of the stereographic projection on $N$. Let $p \in S^2 \setminus \{N\}$, and let's ...
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20 views

Compute pushforward vector field of $\partial_\phi$ on a parametrization of an open set $U$ on $\mathbb S^2$

Let $U \subset \mathbb S^2$ be an open subset. We parametrize it by the map $$ F(\varphi, \theta) = (\sin(\varphi)\cos(\theta), \sin(\varphi)\sin(\theta), \cos(\varphi).$$ Now I want to compute the ...
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9 views

Finding integral curve and lie bracket of vector field on $M(n,\mathbb{R})$

For each $A \in M(n,\mathbb{R})$ such that $A^{t} = -A$, we define a vector field in local coordinates $(x_{ij})$ by $$ X_{A}(x)= \sum_{i,j} (xA)_{ij} \frac{\partial}{\partial x_{ij}} $$. And now I ...
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24 views

Mapping and Cauchy- Reimann conditions

If a complex function is analytic it must hold Cauchy-Reimann Conditions, So conjugate of F(Z) is irrotational and solenoidal , Does this points mapping from $R^2 $ to a subspace of $R^2 $ ($R^1 $) , ...