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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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Is there a counter example to disprove the following regarding vector addition in binary field?

Let $\{\mathbf{a}_1 , \mathbf{a}_2 , \mathbf{a}_3 , ...., \mathbf{a}_{30}\}\subset \mathbb{F}_2^{15}$ denote the set of binary vectors. Define the set of integers $\{p_k\}_{k=1}^{14}$ as$3 \leq p_1 &...
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Some incorrect terms in generalizing Bochner's Formula

I am interested in Bochner's Formula but for slightly more general applications. In particular, I am interested in $\Delta g(U,V)$ where $U$ and $V$ are vector fields on a manifold. This involves ...
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Visualization of skew-symmetric rank-2 tensor fields

Background I was reading Einstein's The Meaning of Relativity in which he points out that axial vectors are usually used in place of rank $2$ tensors for the sake of geometrical picturisation, as in ...
Awe Kumar Jha's user avatar
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Is it possible to show this Integral identity, by assuming the hyposeses I have made?

Assume there exists $p>2$ such that $B\in L^p_{\rm loc}(\mathbb{R}^2)$. Assume in addition that there exists $\tau >2$ such that \begin{equation} \label{B-decay-cond} |B(x)| \, = \, O(|x|^{-\...
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How to find basis of vector fields?

I'm figuring out definition of vector fields over a manifold as differentiations of algebra $C^\infty(M)$ of functions on $M$. How can we find their basis starting from this very definition? I know, ...
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extension field in the subring $K[u]$

If $F$ is an extension field of a field $K$, $u, u_i \in F$, and $X \subset F$, then \begin{enumerate}[(i)] \item the subring $K[u]$ consists of all elements of the form $f(u)$, where $f$ is a ...
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Exploring the relationship between Tensorial Techniques and Coordinate-Independence?

I am studying Differential Geometry and Riemannian Geometry.I need some help from my stack exchange community members in this regard. Recently I happened to get a lecture series on Riemannian Geometry ...
Kishalay Sarkar's user avatar
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Struggling to reconcile definition of surface integral with physical situations

I have a confusion about how only including the component of a vector field normal to a surface in surface integrals gives the intuitively correct answer in certain physical situations. I saw in an ...
Joa's user avatar
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Eigenfunction of "curl" are orthogonal

Let Ω be open, $(C^∞ (Ω))^3$=V , $v∈V$ such that $∇×v=λv$. Define $⟨u,v⟩=∫_Ω u_1 v_1+u_2 v_2+u_3 v_3 dx$. It is easy to see that $⟨∇×u,v⟩=⟨u,∇×v⟩$. I want to prove that if $u,v$ are 2 eigenvectors of ...
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Vector field of rotation around an axis in spherical coordinates

This is a qualifying exam question in Differential Geometry. I'm new to the subject and am reading up from John Lee's Introduction to Smooth Manifolds. Let $M=S^2\subset\mathbb{R}^3$ be the unit ...
giraffe's user avatar
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When defining the surface integral of a vector field, why do we use the UNIT normal vector?

I am self-studying Vector Calculus by Corral. On page 158, definition 4.3, he goes through the intuition of a surface integral for a scalar field, which leads to his definition: Makes total sense to ...
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Evaluating $\displaystyle\iint\limits_{A}\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)\cdot\overrightarrow{n}~\mathrm{d}S$, where $A$ is the unit sphere

This is from UCHICAGO (GRE Math Subject Test Preparation), Week $5$, Problem $14$. Let $A$ be the unit $2$-sphere in $\mathbb{R}^3$. Let $\overrightarrow{F}=\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)$ be ...
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Reference for wiki claim about degree of spheres

I have seen the following claim on the wikipedia page for 'vector field' about degree theory on vector fields. The index is not defined at any non-singular point (i.e., a point where the vector is ...
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Trying to use Divergence Theorem to Calculate Heat Flux of a Cylinder

Long time lurker, first time poster here. I'm new to using mathjax, so please bear with me! Given Problem in Picture Form I'm trying to solve a problem for a school project, in which we calculate ...
Say No To Decaf's user avatar
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Completeness of vector field generated by family of diffeomorphisms

Let $M$ be a smooth manifold. Let $I$ be an open interval containing zero. A smooth family of diffeomorphisms (parameterised by $I$) is a smooth map $\varphi:I \times M \rightarrow M$ such that $\...
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Norm on the space of vector fields

Let $M$ be a finite dimensional manifold. There exists the notion of the norm of a tangent vector field on M? In other words, the space of tangent vector fields is a normed space? Thank you in advance ...
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How do I find the absolute maximum and minimum values of the Lamb-Oseen Vortex?

I am researching alternative solutions to Stokes equations and I came across a problem with the Lamb-Oseen vortex I cannot solve that I hope will allow easier derivations of vortex functions with ...
Tayler Montgomery's user avatar
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Application of Chain Rule in Proof of Stokes' Theorem

In the proof of Stokes' theorem in "Vector Calculus" by Marsden and Tromba, I noticed that the chain rule is applied selectively. Specifically, the chain rule is not used when expanding the ...
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Minimum conditions in order to make this integral equal zero

What is the minimum condition to impose in $F_{\bf B}$ and $G_{\bf B}$ in order to make this integral equal to zero? \begin{equation} \int \left(\nabla\times F_{\bf B}\right)C({\bf x}) \left(\nabla\...
CristinaSardon's user avatar
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An analogue of the de Rham complex for vector fields

Let $M$ be a smooth manifold and let $\chi$ be its tangent bundle and $\Omega^1$ be its bundle of $1$-forms. Using the exterior algebra we can extend the space of sections of $\Omega^1$ to a ...
Zoltan Fleishman's user avatar
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The relation between an isotopy and a time-dependend flow (Exercise 9-21 Lee's Introduction to smooth manifolds)

The Exercise and most of the notation is from the book "Introduction to smooth manifolds" by Lee. A smooth isotopy of M is a smooth map $H:M\times J\to M$ where $J\subset\mathbb{R}$ and $H_t:...
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What are the implications of representing basis vectors as directional derivatives?

I am pretty new to differential geometry and general relativity. The notion of using directional derivative operators to represent basis vectors is starting to make sense to me, but I am still trying ...
Aidan Beecher's user avatar
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How would I calculate the flux of a given vector field through a surface?

I'm trying to work out a problem where I need to calculate the flux of the vector field $A= \langle xy, yz, zx \rangle$ through the shape pictured below. So far I've set up the following integral... $...
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"field" vs. "vector field" [duplicate]

Is the "field" in the "vector field" as the same "field" in algebra: as the commutative ring with the multiplicative inverse? If yes, then the "vector field" ...
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Why $\theta ([\xi_f, \xi_g])=\frac{1}{2}\xi_f \langle \theta ,\xi_g\rangle -\frac{1}{2}\xi_g \langle \theta ,\xi_f\rangle$?

Let $(M,\omega)$ be a symplextic manifold with $\omega =-d\theta$. For $f,g\in C^{\infty}(M)$, we can define a Poisson bracket $\{ f,g\}=\omega (\xi_{f},\xi_f)$ where $i_{\xi_f}\omega=\omega(\xi_f,\...
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Question about Vector Field Geometry

I am confused about vector fields. Let $f_i:\mathbb R^3\to \mathbb R$ be real-valued functions for $i=1,2,3$. Let $\vec{f}=(f_1,f_2,f_3):\mathbb{R}^3\to\mathbb{R}^3$ and $\vec{g}=(f_2,f_1,f_3):\mathbb{...
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Normal vector field of a hypersurface in $S^{n+1}$

Question Denote $\mathbb{R}^{n+2}=\mathbb{R}^{r+1}\times \mathbb{R}^{s+1}$, where $s+r=n$. We write a vector $\xi\in \mathbb{R}^{n+2}$ as $\xi=\xi_1+\xi_2$. Let $f:M\rightarrow S^{n+1}\subseteq \...
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Recursive approximations of inverse square law

I have a toy electrostatics simulation that consists of some number of 2D point particles that each have a real-valued "charge" $q_i$, which then exert forces on each other proportional to $...
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Model probabilistic time evolution from vector field distribution

Given an axis-aligned grid${}^1$ $x_1 , \ldots , x_n \in \mathbb{R}^2$. Let $\mu \in \mathbb{R}^{2n}$ and $\Sigma \in \mathbb{R}^{2n\times 2n}$ be the means and covariance matrix and assume we have a ...
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field lines of a $\mathbb{R}^2 \to \mathbb{R}^2$ function (ellipses) [closed]

i have the following exercise: Characterize the field lines of the following vector field $g: \mathbb{R}^2 \to \mathbb{R}^2:$ $g(x,y)=(\frac{-y}{a^2}, \frac{x}{b^2}) $ with $a,b \in \mathbb{R} \...
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Compute the pullback and d$(f\circ F)$ of a vector field in $\mathbb{ R}^3$

Let $f:\mathbb{R}^3\to \mathbb{R}$ be the function $$\tag1f(x,y,z)=x^2+y^2+z^2$$ and let $F:\mathbb{R}^2\to\mathbb{R}^3$ be the following map (the inverse of the the stereographic projection): $$\...
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Commutivity of two derivatives for exponential map

I'm reading the book Hörmander Operators by Marco Bramanti and Luca Brandolini World Scientific, 2023. I had a problem when reading the proof of the following theorem: Theorem 1.9 Let $X$ and $Y$ ...
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Electric fields and simply-connected regions

I apologize for the ignorance and the rough English in advance, I have an issue understanding how to match both what happens in physics and what I am seeing in calculus. We learned that if a vector ...
Some random guy's user avatar
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Linear map from smooth vector field to smooth functions on a manifold is determined pointwise

I'm trying to prove this theorem: If $M$ is a smooth manifold and $F\in Lin_{C_{\infty}(M)}(\mathfrak{X}(M), C_{\infty}(M))$, then $F$ is determined pointwise. Proof attempt: Let $X$ be a smooth ...
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Area of ellipse using Green’s theorem [duplicate]

Given the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, which we want to calculate the area of. Parameterization $\mathbf{r}(t) = (a \cos t, b \sin t), \ t \in [0, 2\pi]$ My book says we can ...
math.lover's user avatar
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Why don't I see "vector-valued vector fields"?

Let $P\xrightarrow{\pi}M$ be a principal $G$-bundle with connection $\omega\in \Omega^1(P;\mathfrak{g})$. When I am studying these things, there are Lie-algebra valued differential forms all over. ...
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computaton of vector fields in KAK decomposition

I want to derive the following the left-invariant vector fields. \begin{align} {D}_t h(t) = e^{-ad \phantom{1} k_2(t)} e^{-ad \phantom{1} a(t)} {D}_t k_1(t) + e^{-ad \phantom{1} k_2(t)}{D}_t a(t) + {D}...
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Solve the system of differential equations associated with an ellipsoid.

Consider the ellipsoid, given by $$ f(u, v) = \bigl( a\sin(u)\cos(v), b\sin(u)\sin(v), c\cos(u) \bigr), \quad 0 \leq u \leq \pi, \quad 0 \leq v \leq 2\pi, $$ and consider the $1$-form in $\Bbb R^2$ ...
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2 votes
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Can the gradient of a scalar field be expressed as a surface integral per unit volume?

I am familiar with the usual equation for the gradient of a scalar field $\varphi$ $$ \nabla \phi = \frac{\partial \phi}{\partial x} \mathbf{i} + \frac{\partial \phi}{\partial y} \mathbf{j} + \frac{\...
John Keevil's user avatar
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Can I assume a horizontal vector is a horizontal lift?

I am reading a proof that the curvature 2-form is given by $(F_A)_p(v,w) = dA_p(v,w) + [A_p(v),A_p(w)]$ where $A\in \Omega^1(P;\mathfrak{g})$ is a connection 1-form on $P\xrightarrow{\pi}M$. One of ...
Wyatt Kuehster's user avatar
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Vector field with almost non-periodic orbits

Let $M$ be an n-dimensional smooth manifold (Open or compact). I want to know if it is possible to construct an smooth vector field with exactly one singularity, such that the set of periodic integral ...
Pablo Cid's user avatar
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Abundance of divergence-free vector fields in noncompact manifolds

Let $(M,g)$ be a complete noncompact Riemannian manifold of dimension $n \geq 2$. How big is the space $D_b(M)$ of (pointwise) bounded divergence-free vector fields on $M$ of noncompact support? I ...
Eduardo Longa's user avatar
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47 views

Could this vector field exists on a manifold?

I want to "generalize" the smooth flow $F(t,x)=e^{t}x$ on a Riemannian manifold in the following sense: Let (M,p) be a pointed positive dimensional smooth complete Riemannian manifold (that ...
Pablo Cid's user avatar
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1 answer
69 views

Is it really easy to derive this expression for the flow of $\dot{x} = x^2$?

I'm reading the very first chapter of Dynamical Systems I: Ordinary Differential Equations and Smooth Dynamical Systems from D. V. Anosov, et al. and found myself trying not only to understand, but ...
Ferreira H. S.'s user avatar
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1 answer
33 views

Derivation of Continuity Equation for an Incompressible flow

Good day guys, I was playing around with the following form of the continuity equation: $$ \frac{\partial \rho}{\partial t} - \nabla \cdot (\rho \vec{v}) = 0 $$ For an incompressible fluid: $\frac{D\...
STOI's user avatar
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3 votes
2 answers
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Confusion of definition of covariant derivative along curve

I am currently studying Riemannian geometry, and have come across the following proposition: Proposition. Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$ and let $\gamma:I\...
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Complete orthonormal basis of divergence free vector fields

I'm working on a problem in fluid dynamics. I need to find a complete basis of orthonormal 3D vector fields. My "inner product" between vectors $\mathbf{v}_1$, $\mathbf{v}_2$ is a dot ...
davenpi's user avatar
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Why is the lift of a foliate vector field to the transverse bundle a foliate vector field w.r.t. the lifted foliation?

Let $M$ be a manifold with a foliation $F$ of codimension $q$, and let $p: B \rightarrow M$ be the transverse frame bundle (which over each point of $M$ consists of frames in the quotient of the ...
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Is "work" a natively physical or natively mathematical concept?

I recently watched a wonderful video explaining the Cauchy and Residue theorems, and how a complex integral's components relate to work done by and flux from the Polya vector field of the function ...
HydroPage's user avatar
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1 answer
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How to exploit symmetries in vector fields

Doing electrostatics, I've found that everyone makes assumptions that nobody proves, and it's related to symmetries. If we have a system that we want to calculate the E-field, one starts with the ...
Tito Diego's user avatar

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