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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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Understanding exactness and finding a potential.$[e^x\cos(\pi y^2)+ay-1]dx+[by\; e^x\sin(\pi y^2)+x+y]dy$

$$\underbrace{[e^x\cos(\pi y^2)+ay-1]}_{P}dx+\underbrace{[by\; e^x\sin(\pi y^2)+x+y]}_Q dy$$ is given I want to check if the given equation is exact, has potential, conservative. Question: If I ...
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How to compute $B(X, Y) = \bar{\nabla}_\bar{X}\bar{Y} - \nabla_XY$

Let $g : \mathbb{R}^2 \rightarrow \mathbb{R}^4$ be the immersion defined by $$g(x, y) = (cos(x), sin(x), cos(y), sin(y)).$$ Let $e_1 = \frac{\partial}{\partial x}$ and $e_2 = \frac{\partial}{\partial ...
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Stable trajectories of vector fields

I'm wondering what the terminology is for trajectories (integral curves) of a vector field that are "stationary" or "stable", in the sense that in a neighborhood of the trajectory, all other ...
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Approximate commutative vector fields with convergent bound

Let $X,Y$ be two non-smooth vector field, say $C^1$-vector fields, satisfies $[X,Y]=0$ here the commutator works in $C^0$-sense. Can we find a sequence of smooth vector fields $X_k,Y_k$ such that $[...
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Product of vector fields is not a vector field

Let $M$ be a manifold and $X,Y$ be vector fields on $M$. The bracket $[X,Y]:=XY-YX$ is a vector field when $X,Y$ are smooth, but why is $XY$ not a vector field when $X,Y$ are smooth? By definition, a ...
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17 views

fields and cones

I have a euclidean cone, $X$, with some cone point $P$. I assume that a vector field on $X - P$ is parallel if an isometry that carries an open set $X - P$ to $\mathbb R^2$ carries the vector field ...
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Restriction of vector field to the circle

I'm working on a problem that asks me to describe a differentiable manifold structure for the circle $S^1$ and then to calculate the restriction of the vector field $$\xi=-y\frac{\partial}{\partial x }...
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32 views

Computation of the push forward of vectors

I am trying to understand the push forward of a vector field by going through a specific calculation. Consider $f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ given by $f(x,y,z) = (x+y+7, z-x-5)$ and ...
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32 views

ODE with discontinuous vector field

Consider the ODE $$\partial_t \Phi(t,x) = \mathbf b(\Phi(t,x)), \qquad t \in [0,T], \quad x=(x_1,x_2) \in \mathbb{R}^2$$ $$\Phi(0,x) = x, \quad x \in \mathbb R^2,$$ where $\mathbf b = (0,\chi_{\{x_1 \...
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Proving chandrasekhar-wentzel lemma (vector calculus)

We were asked to prove the Chandrasekhar-Wentzel lemma using any method different to the one given in Wikipedia. $$\oint_cr \times (dr \times n) = -\iint_s(r \times n)\nabla\cdot n ds $$ I started ...
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Understanding the idea of a pseudo-gradient vector field

I have the following definition of a pseudo-gradient vector field: Let $V$ be a Banachspace, $E\in C^1(V)$, $\tilde V = \{u\in V \mid DE(u)\neq 0\}$. Then $v: \tilde V \to V$ is called a p.g.v.f. of ...
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How to find a $f$ such that $F=grad \ f$?

I need help with this problem: For the conservative field $F$ find a function $f$ such that $F=$ grad$f$. $$F(x,y,z)=\left(\frac{x}{r},\frac{y}{r},\frac{z}{r}\right)$$, where $r=\sqrt{x^2+y^2+x^2}...
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Simplify $c_i=a_j\frac{\partial b_j}{\partial x_i}$

Consider two 3-D vector fields $\vec{a}$ and $\vec b$. Then define a new vector field $\vec c$ as $$ c_i = a_j\frac{\partial b_j}{\partial x_i}, $$ i.e., $$ \vec c = \left( \vec a \cdot \frac{\...
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Erroneous vector filtering from image based on direction (directional filter) [on hold]

I am working on the motion estimates using two images by python based algorithm. I got the attached vector field (green color). This field includes some erroneous vector. I want to filter those ...
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The existence of a smooth vector field locally

Suppose we have a smooth $k$-dim manifold $M \subset \mathbb{R}^𝑛$ and the tangent space at every point $p \in M$ (Here we translate every tangent space passing the origin point) has non-zero ...
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31 views

How to find the rotation vector by deriving the final vector with respect to the displacement?

My understanding of a rotation of a vector can be done by using a 2D rotation matrix as shown below, $R(\theta )=\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{...
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2answers
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I need to find the potential function of a vector field.

I was given F = (y+z)i + (x+z)j + (x+y)k. I found said field to be conservative, and I integrated the x partial derivative and got f(x,y,z) = xy + xz + g(y,z). The thing is that I am trying to find g(...
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How to Find a Vector Field(/Differential Form) with Div(F) = 0 but F \ne Curl(A) on R^3\(S^1x{0})

In this post, it is outlined how to find a differential $n$-form on $U_0 = \mathbb{R}^n\backslash\{\text{pt}\}$ whose exterior derivative is zero but which is not the exterior derivative of an $(n-1)$-...
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What is the integral of the 1-form field $x dy$ over the straight-line path from (2, 0) to (0, 3).

Unsure of how I should approach this... Should we parametrize the path first and then integrate?
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prove that the curve $C$ belong to a surface

$C$ is the parametrized curve given by: $$\vec{r}(t)=3\cos(t)\vec{i}+3\sin(t)\vec{j}+3\cos(2t)\vec{k },$$ $0\leq t\leq 2\pi $, clockwise oriented. How can we prove that the curve $C$ is included ...
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show that $\iint_{S}^{}{curl\vec{F}\cdot \vec{dS}}$ is proportional to the lenght of $C$

The curve $C$ is the edge of a surface S , with $\vec{T}$ unit tangent vector of the curve $C$ and $\vec{F}$ a vector field such as $\vec{F}=k\vec{T} $ for each point of $C$, where $k$ is a constant. ...
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110 views

Decompose vector fields on product manifolds

So, I know that tangent bundle of a product manifold $M \times N$ splits in a sum $$ T_{(x,y)}(M \times N) = T_xM \oplus T_yN, $$ so that is obvious that the sum $X \oplus Y$ of smooth vector fields $...
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21 views

$F(\mathbf{x})$ is tangent to $z^2=x^2+y^2$ everywhere

Show the vector field $F(\mathbf{x}) = \begin{pmatrix} x \\ y \\ z \\ \end{pmatrix}$ is tangent to the curve $z^2=x^2+y^2$ everywhere. Geometrically, this is quite straightforward: ...
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62 views

How to cross check the derivative of a vector

Can you please help in understanding how to cross verify if the derivative of a vector is correct. The problem I choose for this is as follows, There is a vector V connecting origin and point (X, Y) ...
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1answer
18 views

About the definition of divergence

The divergence is defined as: $\nabla . \mathbf{A}=\lim \limits_{V \to 0} \dfrac{ \unicode{x222F}_{\partial V} \mathbf{A}.d\mathbf{S}}{V}$ My question is of two parts: $(1)$ If we are using ...
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26 views

Cross product of unit vector in cylindrical and spherical coordinate system [closed]

For cartesian, the unit vectors are $(ax, ay, az)$ For cylindrical, the unit vectors are $(ar, a\theta, az)$ for spherical, the unit vectors are $(aR, a\theta, a\phi)$ How can one compute cross ...
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50 views

Why calculate the line integral for a vector field? *Without using work/physics*

For a while now I've been trying to find motivation and a good intuition behind the line integral for a vector field. This is the first time I'm learning this topic and I'm not interested in too much ...
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206 views

Cross check for the derivative of a unit vector $\frac{x}{|x|}$

Can you please help me in finding out the mistakes I am doing during the calculation of derivative of a vector. I am briefing the problem I am trying to solve as follows. There is a line joining two ...
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26 views

Limited Curl in Vector Fields

Is it possible for a vector field in $\mathbb{R}^2$ to have nonzero scalar curl $\bullet$ at a point $\bullet$ along a line $\bullet$ along a curve (which is not a line) $\bullet$ in a finite region ...
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1answer
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If $f$ and $\vec G$ are point functions finding the components of $\vec G$ normal and tangential to $f=0$

If $f\;and\;\vec G$ are point functions prove that the components of $\vec G$ normal and tangential to the surface $f=0$ are $\frac{(\vec G.\nabla f)\nabla f}{(\nabla f)^2}$ and $\frac{\nabla f \...
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1answer
41 views

Flux integral through tricky surface.

Let $T$ be the area lying in the first octant where $x\geq0,y\geq0,z\geq0$ limited by the surfacs $z=a^2-x^2$ and $y=a^2-x^2$. Calculate $\iint_S \vec{F}\cdot\hat{N}dS$ where $\vec{F}=(x,y,z)$ for $(...
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Evaluate $\oint_C(y^3+\cos x)dx + (\sin y+z^2)dy+xdz$

Evaluate $\oint_C(y^3+\cos x)dx + (\sin y+z^2)dy+xdz$ where C is oriented by the tangent vector given by the parametrization r. $r:[0,2\pi]\rightarrow R^3,t\rightarrow (\cos t,\sin t, \sin 2t)$ i ...
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25 views

Proving that Lie$([G,G])$ = $[\mathfrak{g},\mathfrak{g}]$

Let $G$ be a Lie-group. Let $\mathfrak{g}\equiv \text{L}(G)$ be it's Lie-algebra: i.e. the set of left-invariant vector fields on $G$: $$ \text{L}(G):= \{ X \in \Gamma(TG)\; | \; {L_g*} X = X, \forall ...
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path independence of a scalar field line integral

I had seen mathematical proofs of why it's true. But I couldn't wrap around my head with the intuition behind it. For a single variable function, $\begin{equation} \int_{a}^{b}f(x) \,dx = -\int_{b}^{...
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Why is the gradient of $f(x)= x$ the vector $(1,0,0)$?

Why is the gradient of $f(x)= x$ the vector $(1,0,0)$? This would mean that at every point on the straight line $y = x$, I should go horizontally to experience the maximum change in slope. I was ...
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22 views

About the derivative of the Jacobian in fluid dynamics

I was studying a book on the mathematics of fluid dynamics in which there was a lemma on how to find the derivative of the Jacobian. The explanation is as follows (Sorry if it's too long): There is a ...
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1answer
94 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 $\...
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Fundamental vector field of circle bundle

I'm having some trouble getting my head around fundamental vector fields and am trying to understand them through an example. I was wondering if there is any easy examples where we can write out a ...
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1answer
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Evaluate the line integral of a vector field around a square

I am asking this question because I believe the answer sheet I was given has an incorrect solution. The task is to evaluate (by hand!) the line integral of the vector field $\mathbf{F}(x,y) = x^2y^2 \...
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Does $|(a,b)|=1 \implies (a,b) \neq 0$?

So $(a,b) \in \mathbb{R^2}$ and if $\sqrt{a^2+b^2} =1$ does this mean that the vector $(a,b) \neq (0,0)$? In general is this statement true? $(a,b) =(0,0) \iff \sqrt{a^2+b^2}=0?$ Going from left to ...
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1answer
55 views

What differential equation corresponds to this vector field?

Here is a vector field:$$ \vec F(x,y)=\{\sin(x),\sin(y)\}, $$ where $x,y \in (0,\pi).$ How do you find the differential equation, that when solved gives the integral curves for this vector field? I ...
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1answer
36 views

Finding a vector potential for a solenoidal vector field

I have to find a vector potential for $F = -y \hat{i} + x \hat{j}$ This is what I have done: We know that, if $\nabla \cdot F = 0$, we can construct the following: $$F= \nabla\times G$$ Where $G$ ...
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1answer
35 views

Adding vector fields

Consider two vector fields: $$ \vec F_1=(\sin(x),\sin(y)) $$ $$ \vec F_2=(\sin(1-x),\sin(y)), $$ where $x,y \in(0,\pi).$ Does adding the two superimposed vector fields produce a net vertical flow, ...
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46 views

Defined and Undefined Vector Fields

Given a vector field which has circulation but no curl resulting from an undefined point, for example, $\begin{bmatrix}-\frac{y}{x^2+y^2}\\\frac{x}{x^2+y^2}\end{bmatrix}$, does there exist, or can ...
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92 views

Vector field over $\Bbb Q^2$

Consider the vector field restricted to the rationals, $\vec F_\Bbb {Q^2}=(x,y).$ This is a vector field $\vec F:\Bbb Q^2\to \Bbb Q^2.$ Is this vector field weak with respect to the same vector field ...
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42 views

Integrating Factor Techniques for Exact ODE

As explained in this answer, an inexact ODE in the form $M(x,\ y)\ dx + N(x,\ y)\ dy = 0$ can be transformed into an exact ODE via multiple integrating factors which vary from each other non-trivially ...
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1answer
30 views

What is the vector field on this sphere?

What is the vector field shown on this sphere? The source should be at $P(0,0,0)$ and the sink should be at $P(1,1,1).$ How would I derive this vector field? It looks like the vectors are all ...
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141 views

Do Carmo 3.4. exercise 8: Vector Field on a Surface

I'm having trouble trying to start this. Here is the problem statement: Show that if $w : S \to \mathbb{R^3}$ is a differentiable vector field on a regular surface $S \subset \mathbb{R}^3$, and $w(...
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1answer
48 views

Euler vector field is a $C^{\infty}$ vector field.

Let $A$ be a finite dimensional vector field over $\mathbb{R}$, we defined the Euler vector field $\chi$ such that $\chi(f)(a) = \frac{d}{d\lambda}\mid_{\lambda=1} f(\lambda a) $ for $a\in A$ and $f \...
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1answer
80 views

The gravitational potential experienced by a point on the $z$-axis due to a hemisphere

A solid hemisphere of uniform density $\rho$ occupies the region $$x^2+y^2+z^2\le a^2,\qquad z\le0.$$ Find the gravitational potential due to the hemisphere at the point $(0,0,s)$ where $s\gt0$. A ...