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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24 views

Solve the Initial Value Problems encoded with the vector field $V(x)=x^2 \frac{\partial}{\partial x^1} |_x - x^1 \frac{\partial}{\partial x^2} |_x$

Solve the Initial Value Problems encoded by the following vector fields on $\mathbb{R}^2$ Write your solutions in the form of a map $\phi_t:\mathbb{R}^2 \to \mathbb{R}^2$. $V(x)=x^2 \frac{\partial}{\...
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45 views

Differential of $0$-form

If I have a $1$-form that in local coordinates can be written as $\alpha=\alpha_i dx^i$ and a vector field $X=\xi^j \frac{\partial}{\partial x^j}$, if I apply $\alpha$ to $X$ I should obtain the ...
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How to show vector field is conservative?

Conditions for Conservative Vector Field? Is it sufficient for a vector field to be conservative, to say that it’s image is simply connected (no holes) and the partials are equivalent?? If not then ...
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Smooth lifting criteria of smooth vector fields given smooth surjective submersion whose fibers are connected.

I am working on an exercise: Suppose $F : M \to N$ is a smooth submersion, where $M$ and $N$ are positive-dimensional smooth manifolds. Given $X \in \mathfrak{X}(M)$ and $Y \in \mathfrak{X}(N)$, we ...
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Prove $\nabla \cdot (X \times Y) = (\nabla \times X) \cdot Y - X \cdot (\nabla \times Y)$ by converting to the language of differential forms

I wish to prove the identity $$\nabla \cdot (X \times Y) = (\nabla \times X) \cdot Y - X \cdot (\nabla \times Y)$$ where $X, Y$ are vector fields in $\mathbb{R}^3$ and $\times, \cdot$ are the cross, ...
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Vector field divergence on manifold

I want to proof, that a certain expression is a vector field divergence: Let $M$ be a semi-Riemannian manifold with metric $g$. Now I have an expression of the form: $$ g^{jh} \partial_j g(X, \nabla_h ...
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$X,Y,Z$ lines and directional vectors [closed]

I am wondering if anybody can show me the proof of following? Why how is it so that $u_1, u_2$ and $u_3$ are giving us a directional vector? Many thanks!
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Visualizing one-parameter subgroups

I am trying to draw "sketches" of one-parameter subgroups of subgroups of $GL_2(\mathbb{R})$, for example given $$G = \left\{\begin{bmatrix}x & y \\ 0 & 1\end{bmatrix}\in GL_2(\...
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Calculate and find the flux integral

Find the flux integral $$\iint_S \operatorname{rot} \vec{F} {N}\,dS$$ where $S$ is the half sphere $x^2+y^2+z^2=4$ and $z \ge 0$ with an aligned unit standard $\vec{N}$ (normal) and $\vec{F} =(3x-y, ...
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surface integral of vector field as the limit of a finite sum and applications of surface integrals of a vector field [closed]

what is surface integral of vector field as the limit of a finite sum and applications of surface integrals of a vector field
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Find the flux for the vector field $F =(x^2,y^2,z^2)$

Find the flux for the vector field $\vec{F}=(x^2,y^2,z^2)$ across the boundary to the ball given as $$(x-1)^2 +(y+1)^2 +(z-2)^2 \le 4$$ Edit: How much I've done: The center of the sphere is $(1,−1,2)$ ...
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Does this property of the continuity equation holds? If so, how is it possible to prove it?

A space-time dependent vector field $\mathbf{v}(t,\mathbf{x}) \in \mathbb{R}^d$ is given. Let us define the associated flow of solutions as: \begin{equation*} \begin{aligned} & \mathbf{X}(0,\...
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42 views

Determining divergence and gradient using chain rule

Suppose $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$, and $r=|r|$. Also, let $f$ be a scalar function of $r$ and let $\vec{A}$ be a vector function of $r$. We wish to determine $\nabla f$ and $\nabla.\vec{A}$....
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Vectors as differential operators act on functions.

Within my studies of differential geometry the concept of tangent space has just arisen. Now, a tangent vector $v_p$ at point $p \in M$ on a manifold $M$ is defined to be the differential operator $...
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Pushing forward vectorfields to compute flows.

So, I talked to my professor today, asking him for a way to compute flows of vectorfields. An example was given by $ v = x\partial_x + y\partial_y$ living in $\mathbb{R}^2$. Now I know that the way to ...
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What is a Lie bracket? [closed]

Given two vector fields $X$ and $Y$, what is the Lie bracket?
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Function to produce a vector field with two focal areas: one in the top right and one in the bottom left quadrant.

I'm trying to make a conceptual visualization of a system with two stable areas in a $2$-dimensional possibility space. To do this I want to produce a vector field showing the direction the system ...
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How do I imagine why divergence of curl and curl of gradient is $0$?

I tried watching several videos on YouTube, but I failed to gain intuition. I tried to solve it by myself by imagining water flow but I was unsuccessful and got stuck. How do I imagine why divergence ...
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Confused about integral curves on differentiable manifolds

I am a little confused about integration of curves in manifolds. An element of a tangent bundle is a vector field called $X$, made of a set of vectors tangent to a point $p$. How can we find the curve ...
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What happens when C is only partly within the vector field?

So I have the following question where $$\vec{G}=(x,y)/\sqrt{x^2+y^2-1}$$ is a vector field. I have to calculate the line integral for $$\int_{\mathrm{C}}\vec{G}\, d\vec{r}\, $$ where $C$ is the ...
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Strongest increase of a gradient $f(x,y)$ at the position $x_0$

I am trying to figure out how to interpret the gradient of my scalar field at the position $x_0$. The gradient of the function $$f: \mathbb{R}^n \supset \ U \longrightarrow \mathbb {R}$$ at the point $...
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Find the line integral for the following vector field

$\vec{G} = \big(\frac{x}{\sqrt{x^2+y^2-1}}, \frac{y}{\sqrt{x^2+y^2-1}}\big)$. I need to calculate the line integral $\int_{\mathrm{CR}}\vec {G}\, d\vec{r}\, $ where $CR$ is a circle with the center in ...
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Derivation of the curl in curvilinear coordinates

I have the following question. One of the derivations for the curl of a vector field $\nabla \times \mathbf{v}$ starts with an expression $$ I_{i}[\phi] = \int ds \phi [\nabla \times \mathbf v]_{i} $$ ...
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How to calculate line integral if $c2$ has the range $0\le t \lt \infty$

I have to calculate the line integral for $\int_{\mathrm{C2}}F\, dr\,$ where $C2$ is given by $$\frac{(cost,sint)}{1+e^t}$$ and $0\le t \lt \infty$ Where $\vec{F} = \big(\frac{x}{\sqrt{1-x^2-y^2}}, \...
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Find the domain and the area where the vector field is conservative!

So I have the following question where $$F=(x,y)/\sqrt{1-x^2-y^2}$$ is a vector field. I have to define the domain for $F$, I have rewritten the function as $1/\sqrt{1-x^2-y^2} $ multiplied with $(x,y)...
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Features of a given scalar field [closed]

I have to create a visualization of a scalar field given by the formular: $$f(x,y) = x^3 - 3xy^2$$ I have to represent some features of this scalar field. I plotted the following scalar fieldbut can´t ...
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Pathological vector fields

Is it true that any smooth curve is an integral curve for some smooth vector field? More concretely, consider the curve $\gamma:\mathbb{R}\rightarrow \mathbb{R}^2$ defined by $\gamma(t)=(t,t^3sin(1/t))...
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How can I estimate a continuous 3d function from a bunch of 3d vectors?

Visualization of Problem Given: n 3d vectors at 3d positions, i.e. $[u_i, v_i, w_i] = f(x_i, y_i, z_i)$, where $i = 1, \ldots,n$ Unknown: a function $f(x,y,z)=(u,v,w)$ which best fits the given 3d ...
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Nonpositive curvature with closing geodesics.

Let $M$ be a complete Riemannian manifold, $x \in M$ a point, and $\gamma_s(t)$ be a family of geodesics starting at $x$ at time $t=0$, $s \in [-\epsilon,\epsilon]$. The Jacobi field $J(s,t)$ of this ...
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Vector Surface Integral over an Entire Sphere

This is a short, conceptual question related to the post: Surface integral of position vector over a sphere We have a vector field integrated over the the entire surface of a sphere of radius $a$, and ...
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What does capital A mean in vector field notation?

This might be a dumb question but I am stuck halfway through an assignment where I am supposed to draw a bunch of vector fields because I don't understand an element of the notation. The equation that ...
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What is an invariant torus?

Im preparing myself to read this article about an local flow on manifolds: https://www.researchgate.net/publication/303341779_An_example_of_a_local_flow_on_a_manifold In a first view, i could'nt find ...
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1answer
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How would I use Green's Theorem to evaluate this Line Integral?

If I have the closed loop $C$ (https://ibb.co/hK7xC5V) which is the union of $C_1,C_2,C_3$ and $C_4$ where $$C_1=\sin(x)-2\pi,\text{where $x$ goes from 0 to $2\pi$}$$ $$C_2=-\sin(y)+2\pi,\text{where $...
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looking for $3D$ vector field satisfying certain projection conditions

I'm searching for a $3D$ vector field $V$ in $(0,1)^3$ whose parallel projections onto the boundary of $[0,1]^3$ are the following $2D$ vector fields: The parallel projection of $V$ onto the $x-y$ ...
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If $[X,Y]=0$ then $X=0$

Let M be a manifold and $X \in \mathfrak{X}(M)$ show that if for any $Y \in \mathfrak{X}(M)$ , $[X,Y]=0$ then $X=0$. I think we can use following theorem Theorem: $[X, Y]=0$ iff the flows of $X$ and $...
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What are $C^1$ vector fields and how are they defined?

$C^1$ functions are defined as those which are differentiable and their first derivative is continuous. What are $C^1$ vector fields and how are they defined?
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Are the orbits of this discrete dynamical system bounded?

Is this a known result? If no such result (or similar result) is known, a proof or counterexample will suffice. Concise Version Conjecture: Let $f\colon \Bbb Z\to\Bbb Z$ be a function. Assume that ...
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Change of a scalar field/vector field

In my book the following is written: The change of a scalar field $du$ in an arbitrary direction, given by an infinitesimal vector with an arbitrary direction $d\vec r$ is calculated: $$du=u(\vec r + ...
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span equality $span\{v_1,v_2\}=span\{av_1,bv_2\}$

Prove for nonzero scalars a and b such that $$span\{v_1,v_2\}=span\{av_1,bv_2\}$$ My try: $$span\{v_1,v_2\}=c_1v_1+c_2v_2=c'_1(av_1)+c'_2(bv_2)=span\{av_1,bv_2\}$$ by splitting contants. Is it correct ...
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Showing that a certain vector field must be identically zero

Let $0<r<R$. Consider the region $$D=\{\mathbf x \in \mathbb R^3 \colon\ r<|\mathbf x|<R\}$$ and a continuous vector field $\mathbf E$ defined on $D$. Supposing that $$\int_{\gamma} \...
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Normal forms of differential equations and vector fields

INTRODUCTION Consider an equation $$ \dot{x} = Ax + \varphi(x), $$ where $x \in \mathbb{R}^n$, $A \neq 0$ is constant matrix, $\varphi(x)$ is a germ of smooth vector field in $0$ s.t. $$ \left\{ \...
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A question on parallel transport of a vector field along a smooth curve

I was given the following exercise: Consider the connection on $\mathbb{R}^2$ defined by $\Gamma_{1 2}^1 =1$ and all the other $\Gamma_{jk}^i$ equal to zero wrt $(x^1, x^2) = (x,y)$. Determine the ...
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What is an example and a non-example of a left-invariant vector field?

I am going through a few different books reviewing Lie Groups and Lie Algebras and came across the notion of a left-invariant vector field. Given a diffeomorphism $L_ag : G \to G, L_a(g) = ag.$ We ...
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25 views

RREF Criterion for Linear Independence

Let $v_1,v_2, \ldots , v_m$ $\in$ $K^n$ be vectors. Let $A= (v_1, \ldots, v_m)$ be a matrix with columns $v_1,v_2, \ldots , v_m$. Then the vectors are independent if and only if rref of A has a ...
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Computation of tensor elements

Suppose the matrix of linear map $T:E \to F$ in basis of $E$ and $F$ be like $T^{j}_{i}$; I want to compute the the tensorian elements of $T_{*}(X)$ and $T^{*}(W)$ for $X \in \otimes^{k} E$ and $W \...
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1answer
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Flux of a vector field that is a partial derivative

Let $X = \partial_x$ and $Y = x \partial_y$ be vectorfields on $M = \mathbb{R}^2$ I want to calculate the flux of $X$ and $Y$. I am confused on how to approach this problem. The formula to ...
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If $X,Y\in \mathfrak X^1(S^2)$ commute, then $X$ and $Y$ have a common singularity?

Original title: Dados dos campos de vectores $X, Y \in \mathfrak{X}^{r}(M)$ que conmutan Estoy aprendiendo ecuaciones diferenciales ordinarias y me piden que haga un bosquejo de la demostración de la ...
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Tangent vector of geodesic on sphere trough points $(a,b,c)$ and $(d,e,f)$

I am seeking the unit vector $v=(x,y,z)$ tangent at $p$ to the geodesic on sphere connecting $p=(a,b,c)$ and $q=(d,e,f)$. The curve is orientated from $p$ to $q$. Once the tangent vector is ortogonal ...
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1answer
25 views

Why are Line Integrals of Vector Fields independent of Magnitude of $\vec{r}'$?

If we have a line integral of a vector field $\vec{F}$ along a space curve $\vec{r}(t)$ $$\int\limits_{t=a}^{t=b}\vec{F}(\vec{r}(t))\cdot\vec{r}'(t)\,dt$$ Why doesn't it matter that $\vec{r}'$ doesn't ...
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1answer
42 views

Function defined on the tangent bundle

Suppose $F$ is a smooth real-valued function defined on the tangent bundle $TM$ of a Riemannian manifold $M$, i.e. $F:TM\to\mathbb{R}$ given by $F(p, v)\in\mathbb{R}$. Consider the function $f:M\to\...

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