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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Question about calculating flux with different coordinate systems

For a question, I am asked to find the flux of $F=\langle 3x,0,2\rangle$ across the surface of $x^2+y^2+z^2=4, x>0, \ y<0,\ z<0$. I tried solving this with cylindrical and polar coordinate ...
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Show that the vector field always admits a potential function.

Let $r = ||X||$. Let $g$ be a differentiable function of one variable. Now to show that the vector field defined by $$F(X) = \frac{g'(r)}{r} X$$ in the domain $X \neq 0$ always admits a potential ...
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Do continuous conservative vector fields on $\mathbb{R}^3$ form a semigroup under composition?

Let $\mathcal{F}$ denote the set of conservative vector fields on $\mathbb{R}^3$ that are continuous. That is $$ \mathcal{F}=\{F:\mathbb{R}^3 \rightarrow \mathbb{R}^3: F \text{ is continuous and } F=\...
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4 votes
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Exercise related to vector fields, map degrees and Poincare-Hopf's Theorem

I got stuck with one exercise from Chapter 3.5 in Guillemin and Pollack's book, which I used to study differential topology by myself: Given a vector field $\overrightarrow{v}$ with isolated zeros in $...
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f-related and smooth map

This question was left as an exercise in my class of manifolds and I am not able to prove this. Question: Let $f : M \to N $ be a smooth map. Suppose that $X_1 , X_2 \in X(M)$ are f-related to $Y_1 , ...
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Is the velocity vector tangent to a dynamic path?

Imagine a particle traveling along a curve (or surface). I cannot find anything on this topic that allows the curve (or surface) to change (or deform) as the particle moves. For a static curve, it is ...
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Image of annulus under flow of vector field and differential equations

I'm studying Lee's introduction to smooth manifolds, in chapter 9 he introduces integral curves and vector flows, I'm using the following definitions. Definition: Let M be a manifold a flow domain for ...
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2 votes
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Calculating Symmetrie Groups of Differential Equations from the Infinitesimal Generators (Vectorfields)

At the moment I am working through the Book "Applications of Lie-Groups to Differential-Equations" by Peter Olver. On page 120 he calculates the symmetrie groups of the heat equation. $$ u_{...
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How to find flow equation of a complex function?

Page-199 of VDG I'm trying to figure out how to get the dipole flow equation as shown in the above picture from the complex function $f(z)=z^2$. I write it in components and get: $$ u= x^2 - y^2$$ $$ ...
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Calculating true airspeed: Horseshoe Heading Technique

This question relates to calculating the speed of an aircraft relative to the air, based on GPS measurements (i.e. groundspeed measurements). Specifically it is about David Rogers' Horseshoe Heading ...
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Derive curl of vector field in spherical coordinates

I need to calculate curl of $F$, and show that it is conservative on this region. A vector field $F$, defined on a simply-connected region $r > 0,\; \frac{\pi}{4}< \theta < \frac{3 \pi}{4},\; ...
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Evaluate the line integral $\int_{L} \frac{-y \,d x+x \,d y}{x^{2}+y^{2}}$ for a line segment $L$

Calculate the line integral $$ \int_{L} \frac{-y \,d x+x \,d y}{x^{2}+y^{2}} $$ where $L$ is the line segment from $(1,0)$ to $(0,1)$ parametrized by $$ L(t)=(1-t)(1,0)+t(0,1), \quad 0 \leq t \leq 1 $$...
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Flux through a circle in 3d space

Let $F:\mathbb{R}^3\to\mathbb{R}^3$ be the vector field $F(x,y,z)=0\,\textbf{i}+0\,\textbf{j}+2z\,\textbf{k}$. Now calculate the flux integral $\iint_S F\cdot dS$ upwards (thus facing the positive z ...
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Can I get a solution of an Algebraic Equation with Vector Fields

I would like to get a solution of an Algaibric Equations system, with the use of vector fields. Is it possible please? The idea is to go from a point in space with some directions, and to get the ...
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I want someone to explain example 1 [duplicate]

enter image description hereenter image description here I want someone to explain example 1 clearly
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Help with intuition behind invariance of ODE solution on planar curve.

Intuitively, I think I understand this, but more formally if we have a smooth vector field $$ F(x,y) = \begin{bmatrix} f(x,y) \\ g(x,y) \end{bmatrix} $$ such that for a smooth plane curve $C: x \...
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3 votes
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Kneser theorem about the Klein bottle

I know that in $1923$ H. Kneser showed that a continuous flow in a Klein bottle without singular points has a periodic trajectory. The original article is this, but does anyone know another old or new ...
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How to see that the Laplacian of a positive vector field results in a negative vector field (or negative component).

In a x,y,z cartesian coordinate system with the z-axis positive downwards, equations (1) and (2) are "equivalent", in that they both describe how the vector field $\mathbf{v}$ is governed by ...
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Use the divergence theorem to show that $\int _S \phi \space\vec{n} ds=\int_{V}\nabla \phi \,dv$.

Use the divergence theorem to show that if $\phi$ is a scalar field , differentiable on and inside closed surface $S$ which encloses a volume $V$ , then $\int _S \phi \space\vec{n} ds=\int_{V}\nabla \...
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Possibility of escaping from a point/within a region of a vector field by following the flow

I am not sure if there's a single answer to my questions: How can I find out if it's possible to escape from a point of a vector field by following the flow? There's a sub condition: It's not ...
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Zero Divergence - Vector Potential

I have a question regarding the implication of a vector field defined in $R^3$ having zero divergence in all of $R^3$, particularly wrt to its vector potentials. Let V be a vector field and W another ...
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1 vote
1 answer
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Parametrizing a curve through a vector field: Can I recover the full derivative from the parametric equations?

Suppose there is a differentiable vector field in 2d, $$\mathbf{F} = \mathbf{F}(x,y) \in \mathbb{R}^2.$$ I draw a curve through the $(x,y)$, parametrized in terms of $t$. So I have $(x(t), y(t))$. It ...
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2 votes
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I don't understand how vector fields on manifolds work

Sorry if this is going to be really dumb, but I really don't understand how you operate with a vector field even though I carefully read my lecture notes. I will express what I don't understand by ...
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1 vote
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Flow has no singularities

Context: Let $M$ be a compact, connected, edgeless bidimensional differentiable manifold and let $f:\mathbb R\times M\to M$ be a flow of class $C^2$ in $M$. What does it mean that a flow has no ...
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Every left invariant vector field on a Lie group is smooth. Spivak.

Hello. I'm studying A comprehensive introduction to differential geometry by Spivak's and I'm stuck on the first two equalities. Why $Xx^{i}(a)={L_a}_*X_e(x^i)=X_e(x^i\circ L_a)$? Specifically, what ...
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3 votes
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What is the relationship between the vector fields of conjugate flows?

Let $F(x,t)$ and $G(x,t)$ be two flows on $\mathbb{R}^d$ associated to the vector fields $f(x)$ and $g(x)$, respectively. Assume $F$ and $G$ are conjugate in the sense that there exists a ...
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Is it possible to write a vector form ODE to an explicit form?

Any explicit differential equation of order n, $F\left(x,y,y',y'',\ \ldots ,\ y^{(n-1)}\right)=y^{(n)}$ can be written as a system of n first-order differential equations. Conversely, can any vector ...
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1 - form and smooth vector fields

This question was asked in my mid term of smooth manifolds and I couldn't solve it in exam time. I tried it again at home and I think I need help. Question: Let w be a 1-form on $\mathbb{R}^n$. If ...
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Invariance of irrotationality and solenoidality of a vector field under pointwise rotation

Definitions of irrotational and solenoidal vector fields are usually given for vector fields in $\mathbf{R}^2$ and $\mathbf{R}^3$, but can be generalised to vector fields in $\mathbf{R}^n$ for all $n$....
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Converse of Fundamental Theorem of Line Integral

Why is this converse not called the Fundamental Theorem of Line Integral? It resembles the first part of the Fundamental Theorem of Calculus, which comes the first. Also, Wikipedia presents this ...
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Surface Integral of Vector Field $\textbf{A}=(xz,0,yz)$

I have to solve the following problem: A cylinder of radius R in $\mathbb{R}^3$ with axis along the $z$-axis of a Cartesian coordinate system $(x,y,z)$ can be parametrised as $\textbf{x}(\theta,z) = (...
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Find the Divergence of the Vector Field $A(x,y,z)=\frac{1}{(x^2+y^2+z^2)^\frac{3}{2}}(x,y,z)$

We need to find the Divergence of the Vector Field $A(x,y,z)=\frac{1}{(x^2+y^2+z^2)^\frac{3}{2}}(x,y,z)$ and show that it is $0$ apart from the origin. The part of this problem that confused me was ...
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Do the "physicist common knowledge" that "solenoidal vector fields have closed integral curves" have any mathematical foundation?

I remember having heard some physicist claiming that the integral curves of the magnetic field have to be closed, or "closed at $\infty$", due to the fact that the magnetic field is ...
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Commuting vector fields and product of manifolds

Given two manifolds $X$ and $Y$, the tangent space of their product is: $$T_{(x,y)}(X\times Y)=T_x(X)\times T_y(Y).$$ Let us take two vector fields $f(X,Y)\in T_x(X)\times \{0\}$ and $g(X,Y)\in \{0\}\...
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Left invariant vector fields and Lie brackets of the upper triangular matrices

This question was asked in my assignment on smooth manifolds and I am not able to solve it because vector fields has been 1 of my weak points. I have been following introduction to smooth manifolds by ...
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1 vote
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Decomposition into spherical harmonics

I'm trying to follow a text I found online. The author decomposes EM fields such $$ \mathbf{E} = \sum_{lm}\left(f_l(r) \mathbf{Y}_{lm} - i \frac{l(l+1)}{r} g_l(r) \mathbf{\Psi}_{lm} - i\left(\frac{d ...
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1 answer
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Prove this vector field on $S^2$ is smooth

I am having solving the following exercise: (iii) Fix the coordinate chart $(U_1, f_1)$ on $S^2 = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1 \} \subset \mathbb{R}^3$ where $U_1 = S^2 \cap \{z&...
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Stokes Question (non-conservative vector field but 0 flux as answer?)

I have a Stokes question. $C$ is a curve created by intersection of a cylinder $x^2+y^2=9$ and a plane $z=1+y-2x$. The curve is clockwise when viewed from the positive $z$-axis. The vector function ...
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Image of the gradient field of a smooth function is a convex set

Suppose I have a smooth function $f:\mathbb{R}^d\mapsto \mathbb{R}$. The image of the gradient filed of $f$ is defined as $$V = \{v\in \mathbb{R}^d:\exists x, s.t. \nabla f(x) = v\}.$$ Are there some ...
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Difference and intuition of derivative along a vector field and a vector field applied to a function

Let X be a vector field on a smooth manifold M and $ f \in C^{\infty}\left(M\right)$. I want to know about the difference and the intuition one should have about the following two maps: The map $Xf \...
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1 vote
1 answer
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Question about the gradient vector field of a function on a flag manifold

Let $G$ be a compact Lie group and let $T$ be a maximal torus of $G$. Consider the flag manifold $M=G/T$. Let's fix a regular element $r$ of $\mathfrak{t}^*$ and identify $M$ with the coadjoint orbit $...
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1 vote
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Vector fields, Integral curves and Lie brackets

This question is from my assignment in manifolds and I am following the book introduction to smooth manifolds. Question: Let X,Y and Z be vector fields on $\mathbb{R}^n$. Let $\sigma : I \to \mathbb{...
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How to prove the existence of $(U,\phi)$ around (0,0)

This question is from my assignment in Manifolds course. I have been following introduction to Smooth Manifolds by John Lee. Let X be a smooth vector field on $\mathbb{R}^2$ such that $X(0,0)\neq 0$....
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Every vector field on $S^1$ is i-related to some vector field on $\mathbb{R}^2$, i is the inclusion map

This question is from my quiz in Smooth manifolds and I am struck on this. Question : Let $i: S^1 \to {\mathbb{R}}^2 $ be the usual inclusion map. Prove that every vector field on $S^1$ is i-...
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Inner derivative of exterior product of forms

Can anyone briefly explain the derivation of ‘Leibniz rule’ for forms: \begin{equation} \iota_{\mathbf{v}}\left(\omega^{k_{1}}\wedge \omega^{k_{2}}\right) = \left(\iota_{\mathbf{v}}\omega^{k_{1}}\...
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1 vote
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How to perturb a 2-d conservative vector field so as to maximize its curl

Given a 2-d conservative vector field $F(x,y)=(f_1(x,y),f_2(x,y))$, I would like to perturb it with another unit vector field $G(x,y)$ so as to make the curl of the new vector field $F(x,y)+G(x,y)$ ...
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What is the trajectory of a point by the flow of a vector field?

Lemma 2.8 of this paper https://link.springer.com/article/10.1007/BF01232026?noAccess=true says For any $x \in X$ trajectory of $x$ by the gradient flow $\nabla f $ lies in the $G$-orbit $\mathcal{O}=...
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Dimension of vector space of left invariant vector fields

This question was asked in my quiz on Lie Groups and I couldn't solve it. I tried it again at home but in vain. Now, I am posting it here as I want to learn this concept. Question: Prove that $SL(2,\...
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2 votes
1 answer
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Understanding a theorem on the canonical form of a vector field

Theorem: Let $X$ be a manifold, $x \in X$ and let V be a vector field s.t. $V(x) \ne 0$. Then there exists a chart $(U,f)$ on $X$ such that $x \in U$ and for all $y \in U$, we have $\Delta_f (V(y)) = ...
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Find the additive inverse of vector space

Exercise with proposed solution So for this question, the functions I am getting for the additive inverse are the following: -u and -6-u following the process of the answer in the following question: ...
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