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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expression of vector fields
I am self-studying differential geometry. I am reading vector fields and their flows. I am stuck on the following basic problem.
Let $X$ be a smooth vector field on $\mathbb{R}^2$ then what is the ...
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Find the distance between a point $A$ inside a square, and the intersection point of a line starting from $A$ and intersecting the square's edge.
I was watching this video from Entagma and wanted to make the following gradient
I realised that the gradient shown in the video calculates the ratio shown in the diagram and wanted to know the math ...
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Intuition behind vector field along a local parametrization of a manifold.
In our course on several variable calculus,the following notion was defined:
Vector Field along a parametrization:
Definition:
Suppose $M$ is a $k$-manifold in $\mathbb R^n$.Let $q$ be a point on the ...
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How to integrate the newtons law of gravitation in vector form accounting for acceleration [closed]
Title.
This is the equation of newtons laws of gravitation in vector form, equated to f=ma
$$
\mathbf{a} = -\frac{G M}{|\mathbf{r}|^3} \mathbf{r}
$$
I know that we integrate this equation numerically. ...
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Is a normal vector field surjective?
Let $\Omega \subset \mathbb R^n$ be open, bounded and with smooth boundary. Then $\partial \Omega \subset \mathbb R^n$ is a closed and compact $(n - 1)$-dimensional manifold. Let $\nu : \partial \...
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If $\vec{\nabla}× \vec{A}$=0, does this qualify as the necessary and sufficient condition for being able to write $\vec{A}=\vec{\nabla}$f?
If $\vec{\nabla} × \vec{A}=\vec{0}$, does this qualify as the necessary and sufficient condition for being able to writ $\vec{A}=\vec{\nabla}$ f where f is any scalar function?
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Locally conjugate diffeomorphisms, but does not take orbits to orbits.
I'm working on the Palis's ''Geometrical Theory of Dynamical Systems''. I have the next problem:
Let $X$ and $Y$ be $C^1$ vector fields on $\mathbb{R}^m$. Suppose that $0$ is an attracting hyperbolic ...
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Vector field corresponding to linear ODE on $\mathbb{R}^n$ under coordinate change
Consider $\mathbb{R}^n$ as a differentiable manifold with the standard coordinates
$x^j: \mathbb{R}^{n} \to \mathbb{R}, x_1 e^1 + ... + p_n e^n \mapsto p_j.$
The vector field $X_A$, which in these ...
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Continuous non-vanishing vector field on annulus shape regions
Let $\Omega \subset \mathbb{R}^2$ be an open bounded region and $\Omega_0 \subset \subset \Omega$. Assume also that $\Omega_0$ is simply-connected. Suppose a continuous non-vanishing vector field $F$ ...
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Is $H(div; \Omega ) \cap H(curl; \Omega )$ compactly embedd in $L^2(\Omega)$?
I know a similar question says that $H_0^1(\Omega)=H_0(div;\Omega)\cap H_0(curl;\Omega)$, which was shown in Lemma~ 2.5 of the book "Finite Elements Methods for Navier-Stokes Equations" by ...
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how do i derive different equation from cross product of a curl [duplicate]
$\vec{a} \times (\nabla \times \vec{a}) = \frac{1}{2}\nabla{a^2}-(\vec{a} \cdot \nabla)\vec{a}$
How do I show this is true?
I attempted to use vector triple product rule however it yielded
$\vec{a} \...
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Divergence of curl is zero. Why? [closed]
I know that we can prove the statement in the question title by using the following 3D vector identity
$$
A\cdot(B\times C) = (A\times B)\cdot C.
$$ But I had a small scenario in my mind. Let's ...
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Why a vector field on a manifold defines a 1-1 mapping on the manifold?
I am reading paragraph 3.1 (Introduction: how a vector field maps a manifold into itself) of chapter 3 (LIE DERIVATIVES AND LIE GROUPS) of the book Geometrical Methods of Mathematical Physics, by B. ...
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There are smooth vector fields along a curve forming a basis of the tangent space at each point
An application of parallel transport is to prove the existence of smooth frame along a curve on the manifold. But it seems that the existence has nothing to do with the metric, so I would like to find ...
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Is there any intuitive way to understand the relationship between the definitions of divergence and curl in terms of del and in terms of geometry?
Ok so.
There are, in essence, two ways to define the two derivatives of a vector field, curl and divergence.
The first way is to define a differentiation operator, del, or ∇:
∇ = (d/dx, d/dy, d/dz)
As ...
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Combining the two polar Navier-Stokes equations into a single PDE using vorticity and streamfunction
I am starting with the following reduced form of the Navier-Stokes equations in polar coordinates.
$$u_r\frac{\partial u_r}{\partial r}+\frac{u_\theta}{r}\frac{\partial u_r}{\partial\theta}-\frac{u_\...
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An Hamiltonian diffeomorphism is also a Poisson diffeomorphism
Let $(M,\{-,-\})$ be a Poisson manifold.
An Hamiltonian isotopy is a smooth family of diffeomorphisms $\{\varphi^t:M\to M\}_{t\in [0,1]}$ such that $\varphi^0=\text{id}_M$ there exists a smooth ...
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A vector field is Poisson iff its flow is a Poisson diffeomorphism
I'm studying Poisson geometry from "Lectures on Poisson Geometry" (Crainic, Fernandes, Marcut). Let $M$ be a Poisson manifold and let $X$ be a vector field on $M$. We define $X$ to be ...
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Where exactly does the integral definition of the gradient come from?
In the book "Essential mathematical methods for physicists" from Weber and Arfken, they define the integral form of the gradient,divergence and curl, althougth they give sections before an ...
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Going from discrete Poisson equation to (discrete) divergence calculation
Just to give some background: I am currently working on a fluid simulation and am trying to clear any divergence from my discretized velocity field (i.e. it's split up into grids).
To eliminate such ...
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(Stone-Goldbart Ex 6.7) Use Neumann Problem to Derive Helmholtz decomposition
In Ex 6.7 part (a) of Stone-Goldbart, we are required to use the Fredholm condition for the existence of solution for a Neumann-Poisson problem to derive the Helmholtz decomposition. Given
\begin{...
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Curl without integrals
I have a hard time following the integral derivation of a curl in a given coordinate system and it seems somewhat meandering from the entities involved. I came up with this method, I'm wondering if ...
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Determine whether this field is a gradient vector field
Let $n \in \mathbb{Z}$ and $X \colon \mathbb{R}^2\backslash\{(0,0)\} \rightarrow\mathbb{R}^2$ be the vector field
$$ X( x, y) = \begin{pmatrix}-y(x^2 + y^2)^n, x(x^2 + y^2)^n \end{pmatrix}.$$
a) For ...
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Clockwise Convention for measuring flux
The work done by a vector field $\vec F$ in moving along a trajectory C
$$\int_C \vec F \cdot d\vec r = \int_C \vec F \cdot \hat T ds$$
Where $\vec r$ is the position vector along C and $\hat T$ is ...
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Proving vector field identity with tensor index notation [closed]
How do I show, in tensor notation, that $\nabla(\frac{1}{2}v^2)= (\nabla\vec{v})\cdot\vec{v}$, where $v^2=\vec{v}\cdot\vec{v}$?
So far, the conversion to tensor index notation I have is: $\frac{\...
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Every line integral starting and ending at the boundary vanishes implies that the vector field is $0$?
Let $V\subset\mathbb{R}^3$ be compact and possess a piecewise smooth boundary $\partial V$. Let $\mathbf F$ be a continuously differentiable vector field defined on a neighborhood of $V$.
Suppose that ...
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Can one reach every state in phase-space from initial conditions on a codimension-1 submanifold?
Take the stereotypical ODE $\dot x = f(x)$ where $x \in R^n$.
Say now I take a compact, $n$-dimensional subset $P$ of my phase space in $R^n$. Can I always find a $(n-1)$-dimensional submanifold S $\...
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Problem visualizing this vector field. [closed]
Let $F(x, y) = x\hat{i} - y\hat{j}$, then the magnitude is $$|F| = \sqrt{x^2-y^2}.$$
I know that when both $x$ and $y$ have the same value the magnitude is $0$.
Is there any algorithm to visualize ...
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Shouldn't the numerator of the second term in the summation in the last result be $-3\alpha x_{j0}(x_{k}-x_{k0})$?
Shouldn't the numerator of the second term in the summation in the last result be $-3\alpha x_{j0}(x_{k}-x_{k0})$?
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Is $\frac{∂}{x_{j}} \phi(\vec{r})$ at $\vec{ r}=0$ the same as $\frac{∂}{∂x_{j}} \phi(0)$ cause this is what the author stated in the attached screen?
In the attached picture, the author evaluated first $ \frac{∂}{∂x_{j}} \phi(\vec{r})$ then to evaluate, in the line below it, $\frac{∂}{∂x_{j}} \phi(0)$, he simply substituted $\vec{r}=0$ in the ...
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eigenvalues of a first order differential operator on a manifold
I have the following, maybe naive question:
Given a smooth vector field $X$ on a smooth manifold $M$ with $\dim M \geq 1$.
Then this defines a linear operator
$$
D: C^\infty(M,\mathbb C) \to C^\infty(...
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Is line integral defined even if the curl of that vector field is not defined at certain point?
I got a question from my textbook which is as follows
A hemispherical shell is placed on the $xy$ plane centered at the origin. For a vector field $$
\vec{E}=\frac{-y\widehat{e}_{x}+x\widehat{e}_{y}}{...
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Is flux the accumulation of divergence, and circulation the accumulation of curl?
What is the simple relationship expressed verbally between flux, circulation, div, and curl, as captured by Green's, Stokes', and Gauss' Theorems? Below is what I've been able to assemble: Can you ...
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Significance of circulation of vector field
Is there any direct significance, physical or otherwise, of the circulation of a vector field (when not connecting it to the curl)?
Every application, problem, example, or physical model of ...
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Second Fundamental Form of Hypersurface Normal to Predefined Vector Field
I am trying to compute the second fundamental form of a hypersurface that is normal to a predefined vector field. Any help would be appreciated.
More specifically, I consider a vector field $\mathbf{v}...
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Prove or disprove $span({v_0, v_1,...,v_n}) = span({v_0, v_1 - v_0, ..., v_n-v_0})$
As the title says. I am under the impression that the right hand side refers to an equivalence class (since we can show that the set ${v_0, v_1-v_0, ..., v_n - v_0}$ is a subspace and therefore the ...
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Understanding the Correspondence of Sections to Right and Left Invariant Vector Fields in Ping Xu's 'Momentum Maps and Morita Equivalence
I want to understand this passage on the articule: Ping Xu. "Momentum Maps and Morita Equivalence".
By $A → P$ we denote the Lie algebroid of $Γ\rightrightarrows P$, where the anchor map is ...
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Conservative vector field given in polar coordinates
Given a vector field $F$ in polar coordinates, for the example the field $$\vec F(r,\theta)= -r \hat r + (r^2\sin \theta ) \hat \theta $$
I am asked to check if the field is conservative.
is it right ...
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Surface integral over a divergence free field.
This question is related to the answer in this post:
https://physics.stackexchange.com/questions/776112/showing-that-the-power-is-p-ui-using-the-poynting-vector
There it is said that:
The first term ...
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Finding the tangent vector of a cubesphere using world coordinates
TLDR.:
I’m trying to find the tangent vector of a cubesphere using only world space coordinates.
Background:
A cubesphere is basically just an inflated cube. It is commonly used to project 2D textures ...
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Local coordinate proof of the Lie derivative equals the Lie bracket
I'm following Tu's proof of the fact that Lie derivative of a (smooth) vector field with respect to another is actually the Lie bracket of the two vector fields in his book An Introduction to ...
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Finding 3 linearly independent tangent vector fields on $S^3$
$S^3$ is a $3$ -surface as $f(x_1,x_2,y_1,y_2)=x_1^2+x_2^2+y_1^2+y_2^2$ gives $f^{-1}(1)= S^3$ and for every point of $S^3$ is a regular point, i.e., $\nabla f\ne 0 $ at every point of $S^3$.
Since ...
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Find vector field whose divergence is a scalar field
Say we have a Poisson problem:
$\nabla^2 \varphi = S$
As a boundary value problem, it requires the definition of boundary conditions on all surfaces of the domain. If we assume there is a vector field ...
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Stokes' Theorem Always Surface Independent?
Is Stokes' Theorem always surface independent?
In my textbook it says that if F has a vector potential A such that curl(A)=F, then the following is true:
$$\iint F \cdot dS =\int A \cdot dr$$
Excuse ...
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Functions which depend on position and time: Functional notation and derivations
In his MIT OCW course, Professor Kleitman explores the derivative of temperature when it depends on position and time:
$T(x, y, z, t)$ ... is a function of position $(x, y, z)$ and time $t$... What ...
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Electric field flux proportional to the field lines generated by (for example) a static charge
Suppose we have a stationary positive charge at a point in space that we call $+Q$. We know by definition that the flow of the electrostatic field is given by, in its simplified form,
$$\Phi_S(\vec E)=...
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Local first integral on open set
Let $M^n$ be a (compact) $n$-dimensional manifold. And we are given a set of $m$ basis functions
$$B=\{\psi_i(x): \mathbb{R}^n \rightarrow \mathbb{R}| i=1,...,m \}$$
such that all of them are ...
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Compute $\unicode{x222F}_{\partial K} |\nabla f|dS$ where $f(x,y,z)=x^2+4y^2+9z^2 , K=\{(x,y,z);f(x,y,z)\leq 1\}$
Compute $\unicode{x222F}_{\partial K} |\nabla f|dS$ where $f(x,y,z)=x^2+4y^2+9z^2 , K=\{(x,y,z);f(x,y,z)\leq 1\}$
I came across this question when studying for an upcoming exam, but it had no answer ...
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What properties does the flow of a Hamiltonian vector field have compared to the flow of a symplectic vector field?
Let $(P,\omega)$ be some real $2n$-dim symplectic manifold. A symplectic vector field, $Z\in\mathfrak{X}_{sp}(P)$, is one for which the Lie derivative satisfies $\mathcal{L}_Z \omega =0$ or, ...
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Book/articles about computing the high-order derivative of a vector field
Is there any book/article that gives a general result of this:
For any $n\in \mathbb{N}^*$, use the chain rule to compute the $n$-th order derivative of :$(f_1(x_1(t),x_2(t),\dots,x_n(t)),f_2(x_1(t),...
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