Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
1answer
18 views

The image of a vector field under the differential of a diffeomorphism is a vector field

I'm learning about vector fields on manifolds, and I'm slightly confused about the following result. Let $M$ and $N$ be differentiable manifolds, $\varphi:M\to N$ a diffeomorphism, and $\mathrm d\...
0
votes
0answers
12 views

Plotting vector field lines and eigenvectors on Gnuplot

I have been trying to draw the phase portrait of a system in 3D and I need to use Gnuplot for it but I really could not find anything in the internet about drawing 3D-eigenvectors and the vector field ...
0
votes
0answers
7 views

projector onto gradient vector field

Does anyone knows what is a "projector onto gradient vector field" ? I came across with this notion in a paper for water waves with vorticity (free surface Euler equations). At some point, the ...
2
votes
2answers
24 views

Is the rank of a matrix with coefficients $\{-1,0,1\}$ the same as the rank of the matrix with coefficients in $GF(3)$?

I have a set of matrices defined over the ring of the integers, which items are using only coefficients -1, 0 and 1. For example: $$ A = \left(\begin{matrix} 1 & 0 & -1 \\ -1 & 1 &...
0
votes
0answers
35 views

Confusion about Michael Spivak's differential geometry book: vol 2, pg 260

Book: A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition, page 260. Please clarify the meanings of $X$, $X_a$ and $X_i$. Spivak says that $X_i$ an $\mathbb{R}^n$ valued ...
1
vote
1answer
19 views

Find the value of $\int_C F.dr$ for this ellipse

Let $F(x,y)=\langle -y ,x\rangle$ and $C$ be the ellipse $\frac{x^2}{16}+\frac{y^2}{9} = 1 $ oriented counter clockwise, then find the value of $\int_C F.dr$ This is how I tried it, $x=4\cos(t) \...
1
vote
1answer
51 views

Finding an explicit formula for a Hamiltonian vector field

I've been looking at this question: Existence of vector field given a smooth function That is: Given a symplectic manifold $M$ of dimension $2n$, with a symplectic form $\omega \in \Omega^2(M)$, do ...
0
votes
0answers
10 views

applying the divergence theorem in all coordinates

My Question is : is the divergence theorem same on cylindrical and cartesian coordinates? Ex: $$\iint \vec{gradA}\, \vec{dS} = \iiint \operatorname{div} \vec{gradA}\, dV $$ is this expression is ...
2
votes
0answers
12 views

How do we plot a domain wall between two strings?

Intuitively, an angular vector field $\theta(x)$ such as the following would describe a domain wall between two vortices/strings of opposite winding number. I would like to draw this on a computer ...
0
votes
1answer
48 views

Surface Integral of Vector Field

Given the scalar field $$\phi(\vec{r})=\frac{1}{|\vec{r}-\vec{a}|},$$ where $\vec{a}=(-2,0,0)$, and the corresponding vector field $$\vec{F}(\vec{r})=\operatorname{grad}\phi,$$ as well as the surface $...
0
votes
2answers
35 views

How i x j= k (vector) BUT ixj = (i)(j) (sin90) = (1)(1)(1) = 1 (Scalar) [closed]

How i x j = k (vector) , also in josiah willard Gibbs book who first given the idea of cross product did not explain the mathematical way of cross product. Also from quaternions i found no real ...
0
votes
1answer
52 views

Minimum in a non-linear system

I have the linear system: $$\begin{cases}\dot{x}=y\\ \dot{y}=-ay+x-x^3\end{cases}$$ where $a\geq 0$. I want to prove that this dynamical system has two minimum. I found the 3 equilibrium points $(...
0
votes
0answers
57 views

John Lee problem: Vector field conservative if and only if it is a gradient field

I'm looking for a push in the right direction on problem 4-5 in John Lee's Intro to Smooth Manifolds 3rd edition. The problem goes: Suppose $X$ is a smooth vector field on an open subset $U\subset \...
1
vote
0answers
21 views

Calculating Parametric Path of Particle From Force Field

This question arose when I was doing some work with line integrals over force fields. In these questions, the reader is always given a parametrization of the particle's path (in terms of time) and ...
3
votes
0answers
54 views

What is a neat way to solve $\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{\mathbf{C}}$?

Let $\mathbf{u}:\mathbb{R}^3\to\mathbb{R}^3$ be a smooth enough vector field that satisfies the following equation $$\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{C},\tag{1}$$ where $\nabla\mathbf{u}$ ...
0
votes
0answers
8 views

Is the partial gradient of a multidimensional function a continuous vector field?

Let $U \subseteq \mathbb{R}^d$ and $V \subseteq \mathbb{R}^k$ be two open sets. Define $$f : U \times V \rightarrow \mathbb{R}$$ to be a scalar field, such that, for each fixed $v \in V$, the function ...
2
votes
1answer
35 views

Prove that the covariant derivative commutes with musical isomorphisms

Suppose I have a covector field $\omega$ and a covariant derivative $\nabla_{X}$ for some vector field $X$ on a Riemannian manifold $(M, g)$. Define $X^{\flat} \in \mathfrak{X}^{*}(M)$ as $X^{\flat}(...
1
vote
1answer
30 views

interaction of left-invariant vector fields and right-translation on a Lie-Group

Given a Lie-Group $G$ denote the set of left-invariant vector fields on $G$ by $LG$ and denote by $R_g$ the right-translation, i. e. for $g \in G$ define $$R_g \colon C^\infty (G) \to C^\infty (M) \...
1
vote
2answers
63 views

How do I show $X_{\omega(Y,Z)}=-[Y,Z]$?

How do I show $X_{\omega(Y,Z)}=-[Y,Z]$, where $\omega$ is a symplectic 2 form (in particular non-degenerate) and $Y,Z$ are vector fields and $X_f$ is the vector field correspond to the 1 form $df$ ...
1
vote
2answers
58 views

How to show that the Lie derivative $L_{Y}Z$ is equivalent to the Lie bracket $[Y, Z]$?

How can you show that, if $Y,Z \in \Gamma(TM)$ and $Y$ is complete, then: $$L_{Y}Z=\frac{\text{d}}{\text{d}t}\phi^{-1}_{t*}(Z)\bigg\rvert_{t=0}\equiv[Y, Z]$$ Where $\phi_{t}$ is in the one parameter ...
1
vote
0answers
18 views

Derivative of the flux of a vector field

Let say that $\phi$ is a vector field. By the inverse definition, we have $$\phi_t \circ \phi_t^{-1}=id.$$ I came across one line which I don't understand. $$(\forall p \in \mathbb{R})\phi_t \circ \...
4
votes
2answers
43 views

Difference between the $\nabla\cdot a$ and $a\cdot\nabla$

Hello I am new to vector calculus and I have a basic question . The del operator which is defined as $\nabla = \Bigl(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial ...
1
vote
1answer
26 views

Finding flow lines of a velocity vector field

Find and draw the flow lines of the velocity vector fields $\vec{F}(x, y) = (-2y, \frac{1}{2}x)$ Solution: $x' = -2y$ $y' = \frac{1}{2}x$ $x'' = -2y$ $x'' = -2 \frac{1}{2}x = -x$ No idea how to ...
3
votes
2answers
45 views

Non-existence of the potential function

I am wondering why this theorem is not true when $(f,g)$ are defined on a more general open set $U$ which is not necessarily the entire plane or some disc. What is an example of a vector field $$F(x,...
0
votes
1answer
57 views

Calc of potential function using line integral

I tried to calculate a potential function using the two major methods (by partial differentiation/integration, and by line integral). The two worked with all of the examples that I tried, but then I ...
7
votes
2answers
226 views

Generalized Poincaré Lemma

I'm reading the proof of an improved version of Poincaré's Lemma on Ana Cannas da Silva's Lectures on Symplectic Geometry, page 40. I am terribly confused. Here's the setup: $U_0$ is a tubular ...
6
votes
1answer
84 views

If $X,Y$ are two equivalent vector fields in two open sets $A$ and $B$, such that $A\cup B = M$. Are $X$ and $Y$ equivalent?

Let $X$ and $Y$ be smooth vector fields on $\mathbb{T}^2$. Definition 1: Let $A$ be an open subset of $\mathbb{T}^2$, we say that $X$ and $Y$ are equivalent in $A$, if there exists a homeomorphism $...
0
votes
0answers
17 views

Question of two functions which are continuous at a certain point

The electric field $$E_x$$ and its time derivative are continuous at time $t = 0$. Show that $$k = k^,$$ if for $t < 0$ $$E_x = E_0 \cos(kz-wt)$$ and if for $t > 0$ $$E_x = ...
1
vote
1answer
38 views

$1$-parameter group terminology problem.

I'm reading Kobayashi's book Transformation Groups in Differential Geometry, and I'm a bit confused about the terminology that he is using at the page 3. Here is the section that I don't get: I know ...
0
votes
2answers
22 views

Conservative field defined over a not simply-connected region

I am wondering if a field can be conservative if the region where it is defined is not simply-connected. By definition $F$ is conservative if there exists a differentiable function which satisfies $F=...
0
votes
0answers
13 views

Existence of a parallel vector field implies a splitting of the metric

Where can I find a proof of the following claim: Existence of a parallel vector field on a Riemannian manifold implies that the metric splits locally as a product of a one-dimensional manifold and $n-...
1
vote
1answer
51 views

Help setting up integral for circulation

I just need a bit of help with a problem. I'm being asked to evaluate the circulation of a velocity field. I've just finished calculating the curl of the vector field in question and I know from my ...
0
votes
1answer
19 views

Why is the circulation the line integral of the vector field over closed path bounding the region?

The circulation of vector field around some region is the line integral of the field over the closed curve surrounding it and that integral is integrating the field dotted with the vector of the ...
1
vote
0answers
50 views

Is there a missing definition for addition of vector fields?

In proving $\mathfrak X(U)$ is a $C^{\infty}U$-module, for an open subset $U$ of $\mathbb R^n$ my book defines scalar multiplication of smooth vector fields $U$ by smooth functions on $U$ as $$[fX]...
0
votes
1answer
25 views

Why is the curl the limiting value of circulation density?

The circulation of vector field around some region is the line integral of the field over closed path surrounding that region so why do we need the curl at all if we have a quantity that represents ...
0
votes
0answers
17 views

Why is the curl the limiting value of circulation density

Curl is measurement of the circulation around some point in a vector field so why we need the circulation density (line integral over the area) and not just the circulation (line integral of the field ...
0
votes
2answers
33 views

Finding the Closest Point to the Centre of a Rotating Vector Field

I am working on the analysis of a data set that resulted from a simulation of a Tornado. I need to be able to transform the vectors for the velocities at each measured point in the space from ...
0
votes
1answer
23 views

Given $E:N\to M$ an embedding and $V,W\in \mathfrak{X}(M)$ tangent to $N$, we claim that the commutator of $V$ and $W$ is also tangent to $N$.

I have encounter some difficulties while looking at an exercise online. It basically goes as follows: Given $E:N\to M$ an embedding and $V,W\in \mathfrak{X}(M)$ tangent to $N$, we claim that the ...
1
vote
1answer
112 views

Completion of local frames for the tangent bundle of a smooth manifold

In John M. Lee book Introductions to smooth manifolds Proposition 8.11 is left as exercise. Can anyone give me hints and suggestions to prove it? In particular, i want to show: Let $M$ be a smooth ...
2
votes
1answer
42 views

Projecting vector field from $\mathbb{R}^n$ to $T^n$

Let $T^n = S^1 \times ...\times S^1$. $T^n$ is Lie group and the map $\pi:\mathbb{R}^n \to T^n$ defined by $\pi(x_1,..x_n) = (e^{ix_1},..,e^{ix_n})$ gives a homomorphism between the additive group to ...
4
votes
1answer
58 views

How to find a global flat chart?

Let $V\subset\mathbb{R}^3$ with positive coordinates and let $T$ be an associated distribution with $T=<y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}, z\frac{\partial}{\partial x}-x\...
0
votes
0answers
16 views

Piecewise Line Integral of a Vector Field

I'm currently brushing up on some vector calc for a class on electromagnetics, and this particular practice question is giving me a bit of a headache: If: $$\mathbf H=(x - y)\mathbf a_x + (x^2 + zy)\...
0
votes
0answers
13 views

Curl and maximum rotation

I have a question, I was told that the the magnitude of the curl vector yields the maximum rotation, the questions I have are: 1) Why does the length of the curl vector give the maximum rotation. 2) ...
1
vote
1answer
37 views

How to find the critical point for this coulomb field

Two equal positive charges are at distance $d$, $-d$ from the origin on the $y$ axis. What is the distance on the $x$ axis beyond which a small perturbation in $y$ will move a particle away from the $...
0
votes
0answers
18 views

vector field generated by one parameter subgroup

How is the vector field, on a manifold, generated by a one parameter subgroup defined? I kind of got lost in the definitions that I was searching online.
2
votes
1answer
37 views

Vector Fields as Differential Operators

Given a manifold $\mathcal{M}$, the notion of a vector field $\xi$ on $\mathcal{M}$ can be interpreted as a collection of arrows on the manifold. In the book The Road to Reality by Roger Penrose, ...
0
votes
1answer
49 views

How to check if this weird vector field is conservative

I need to check if this vector field is conservative: ('$\mathrm{sgn}$' is the sign function) $$F=(\dfrac{y\cdot\mathrm{sgn}(xy)}{1+|xy|},\dfrac{x\cdot\mathrm{sgn}(xy)}{1+|xy|})$$ Do I need to check ...
1
vote
1answer
45 views

How to check if a 2 dimensional vector field is irrotational (curl=0)?

I need to check if a this vector field is irrotational and conservative: $F=\langle e^{x \cos y} \cos y,-x e^{x \cos y}\sin{y} \rangle$ curl of F should be => $\dfrac{dQ}{dx} - \dfrac{dP}{y}=0$ (...
0
votes
1answer
19 views

Breaking a path, in 2 integrals

Let $\overrightarrow V, \Gamma$, be a vectorial field and a path such that: $$\overrightarrow V=-(x^2+y^2)\overrightarrow i-(x^2-y^2)\overrightarrow j$$ and $$\Gamma=\{(x,y)\in\mathbb{...
0
votes
1answer
29 views

If $V$ is a vector field, how to interpret it when it's written in the tangent space ? For example, what are the integral curves of $V=\partial _x$?

Let $M$ a manifold and $T_{p}M$ the tangent space at $p\in M$. I'm not sure how to interpret a vector field in $M$. I know that a vector field $V$ is in $TM$ and $V(p)\in T_pM$. In $\mathbb R^n$ a ...