Questions tagged [curl]

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space.

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Curl of $u=(u_1,u_2)$

If I have $u=(u_1,u_2)$ is a vector of two components, then what is $\text{curl}\,u:=\nabla\times u$? where $\nabla=(\partial_x,\partial_y)$. I think it should give a scalar and not a vector.
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Application of Stokes' Theorem - specific vector form [closed]

I need help with this problem on Stokes' Theorem. I can't seem to find a way of solving the first part. For the second part, I got that the left-hand side equals -2A (as in this picture my work), but ...
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What is the curl of a gradient crossed with a constant vector?

Let $a$ be any scalar and $\mathbf{B}$ be a constant vector. $\nabla \times \nabla a = 0$ and $\nabla \times \mathbf{B} = 0$. But is $\nabla \times [(\nabla a) \times \mathbf{B}]=0$?
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Regularity of distributions with $L^2$ curl or divergence

Let $\Omega$ be a domain in $\mathbb R^n$. A classical result tells us that if a distribution $T \in \mathcal{D}'(\Omega)$ has partial derivatives (say gradient) in $L^2(\Omega)$ then $T \in L^2_{loc}(...
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Which operators commute with curl?

Let $X$ be the space of infinitely differentiable maps from $\mathbb{R}^3$ to $\mathbb{R}^3$. Let $C:X\rightarrow X$ denote the curl map. What are all the linear maps from $X$ to $X$ that commute with ...
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How to prove this vector identity $\nabla \cdot \Big( ({\bf u}\cdot\nabla ){\bf u}+ \dfrac{1}{2}(\nabla \cdot {\bf u}){\bf u}\Big) $?

While I was studying on analysis of the NSE and MHD problem, I encountered the following identity. \begin{eqnarray} \nabla \cdot \Big(({\bf u}\cdot\nabla ){\bf u}+ \dfrac{1}{2}(\nabla \cdot {\bf u}){\...
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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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Divergence of Cross Products of vectors in Rn.

Is there some formula for the Divergence of cross Product of (n-1) vectors in $\mathbb{R}_n$. I happen to know of such a formula for $\mathbb{R}_3$, so wanted to know. Pls add/link a proof as well. ...
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Origins of $\frac1{r^2\sin(\theta)}$ term for curl of vector field in spherical coordinates [duplicate]

The curl of a vector field $\vec A$ in spherical coordinates is given by $$\nabla \times \overrightarrow{\mathrm{A}}=\frac{1}{r^{2} \sin \theta} \left| \begin{array}{ccc} \widehat{a}_{\mathrm{r}} &...
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Curl in spherical coordinates without Del operator

Trying to get practice with differential forms and related topics I had tried to calculate curl in spherical coordinates without mystic "del" operator. Let $rot\: F = (\star d (F^\sharp))^\...
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What does Poincare's Lemma tell me about the existence of a vector potential and what restrictions have to be considered for the domain?

I am studying Vector Calculus and Poincare's lemma was mentioned in the class notes. Poincaré Lemma: If $U \subset \Bbb{R}^n$ is an open star-shaped set, then for every $k=\{0,...,n\}$ every CLOSED k-...
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Inconsistent notation fro Del-Operator

It has recently come to my attention that the operations carried out with a Del operator do not match their notation. That is, the divergence or curl are written as $\nabla \cdot \vec{v}$ and $\nabla \...
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Is Curl not a cross product?

I have a vector in cylindrical Coordinates: $$\vec{V} = \left < 0 ,V_{\theta},0 \right> $$ where $V_\theta = V(r,t)$. The Del operator in $\{r,\theta,z\}$ is: $\vec{\nabla} = \left< \frac{\...
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general definition of curl

I am studying about 2-dimensional Euler equation's fluid vorticity, and I want to know how to calculate it. $\omega = ∇\times u$ if $\omega$ is a fluid vorticity and u is the velocity vector of the ...
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Arguments to show that $\lim_{\iiint d\tau\to 0}\frac{\iint d\vec{\sigma}\times\vec{V}}{\iiint d\tau}=\nabla\times\vec{V}$

So, I was solving the problem 1.10.6 from "George B Arfken, Hans J Weber - Mathematical Methods For Physicists- Sixth edition" and the problem was to show that: $\lim_{\iiint d\tau\to 0}\...
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Clarification on stokes theorem

Stokes theorem says: $\int_C \vec{F}$•$\vec dr$ =$\iint_\sum (\nabla \mathbf x \vec{F})•\hat n$ $d\sigma$ This means $\int_C \vec{F}$•$\vec dr$ = $\iint_R (\nabla \mathbf x \vec{F})•\hat n$ $||\...
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Curl operation proof

I am new to subscript notation, and I am having a problem with the proof practice below, $$\frac{1}{2}\vec{\nabla}(\vec{f}\cdot\vec{f}) = \vec{f} \times \vec{\nabla} \times \vec{f} + (\vec{f} \cdot \...
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Is $F=(yx,x+z,yz)$ conservative?

I need a check on the following exercise: Let $S$ be the surface $$S= \{(x,y,z):y=x^2+z^2, y \in [0,1] \}$$ and $F(x,y,z)=(yx,x+z,yz)$ a vector field. Compute $$\oint_{\partial S} F \cdot dr$$ Is $F$ ...
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Curl identity in the sense of distributions

Let $v \in H^1(\mathbb{R}^2)^2$ such that $\operatorname{div} \ v = 0$ and $\omega = \operatorname{curl} \ v$, where $\operatorname{curl}$ is defined as $\operatorname{curl} \ v = \partial_1 v_2 - \...
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Can you compute Laplacian, divergence, curl for a function?

In my physics class, we are currently studying gradient, Laplacian, divergence, and curl, and we have a problem that states to compute all four of these (I.e., (1) gradient, (2) Laplacian, (3) ...
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are vector calculus differential equations independent of coordinate system?

What I mean by this is that, I have observed in continuum mechanics that many times we use newton's laws,etc on a differential element in suppose Cartesian coordinate system and come up with a ...
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Why is the vector Laplacian defined in a way that only makes sense for three-dimensional vector fields?

Both Wikipedia and Mathworld define the vector Laplacian of a vector field ${\bf A}$ to be $$ \nabla^2 {\bf A} := {\bf \nabla} (\nabla \cdot {\bf A}) - \nabla \times (\nabla \times {\bf A}). $$ This ...
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Differentiation in spherical coordinates

I have a vector potential as shown below. $$\vec A =I^e d\ell \dfrac{e^{-ir}}{r} \vec a_z$$ Where $d\ell$ is $$d\ell= \sqrt {dr^2+r^2d\phi^2+dz^2}$$ In order to get magnetic field, I should evaluate ...
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Is a vector field that vanishes at infinity only defined by its curl and divergence?

This subject has appears in various other questions here but I cannot seem to find a clear answer. The Helmholtz decomposition gives an answer if the field is rapidly decreasing. I understand that the ...
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Is the vector field $F(x,y)=(x+y,x^2)$ conservative?

let's consider the vector field $F(x,y)=(x+y,x^2)$. It's defined over the whole $\mathbb{R}^2$. We can see immediately that $$\partial_x F_2 = \partial_y F_1$$ iff $x = \frac{1}{2}$ Since the domain (...
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Proof that a basis vector can be written as the cross product of two dual basis vectors

I have this equation: $$\textbf{f}_i=\epsilon_{ijk}c\textbf{f}^j\times\textbf{f}^k$$ where $\textbf{f}_i^{j,k}$ are basis vectors. It showed up in my lectures about theoretical physics and I kind of ...
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vector identities in solving wave equations with different speeds of propagation

Assume that $u = (u^1, u^2, u^3)$ solves the evolution equations of linear elasticity: $$u_{tt}-µ \Delta u − (λ + µ) D (\nabla\cdot u) = 0$$ in $\mathbf{R}^3 × (0, ∞)$. Show that $w := \nabla \cdot u ...
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Vector calculus “identities”

Do these "identities" even make sense? $F$ is a 3D vector field. For the second equation, the LHS is a vector but the RHS is a scalar. $$\nabla\cdot\Delta F=\Delta(\nabla\cdot F),\,\nabla\...
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How do I calculate the work done in a vector field across a line segment?

Let C be the line segment from $(2, 3, 0)$ to $(5, 7, 9)$ followed by the line segment from $(5, 7, 9)$ to $(1, 1, 2).$ Consider a force field $F(x, y, z) = <3x^2+yz+y,xz+x+z^2,xy+2yz+2z>$ ...
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Remove curl from vector field

I have a discretized real-valued vector field on a 3D Cartesian grid as an input, which possesses some non-zero curl. I would like to find the closest possible approximation to this vector field which ...
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Compute cartesian components of the curl

Given a vector $v$, the curl of $v$ is defined as the unique vector field with the property $$(\nabla v - \nabla v^T) a = (\text{curl } v) \times a$$ for every vector $a$. (See pag. 32 of Gurtin's ...
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Curl of a vector cross the vector?

So I've seen that the expansion for (∇×A)×A is (A⋅∇)A−1/2(∇(A⋅A)) . [Given that A is a vector] Now I guess it's just really late in the night, but I can't make sense of the answer - if a vector dotted ...
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Find vector field when you know the curl $\nabla \times \mathbf{H}=\mathbf{J}$

This is a problem that arose in my electromagnetics class, but it is purely maths. Problem We have the current density vector defined in two different areas (in cylindrical coordinates) $$\mathbf{J}=\...
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How to generate analytic solutions to the systems of PDEs described by curl F = F

I'm interested in investigating the behavior of vector fields that satisfy the equation: curl(F) = F. From what I know, such vector fields have a zero divergence and the components of said "...
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Verification of Stokes’ theorem for closed path and surface

I am having trouble with the follow problem about Stoke's theorem: Assume the vector field $A= <3x^2y^3,−x^3y^2>$ Verify Stoke's theorem for the closed path around the triangle with vertices (1,...
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Find $\nabla \cdot (F \times G)$ in terms of $\nabla \times F$ and $\nabla \times G$ [closed]

If $F,G$ are differentiable vector fields, Find $\nabla \cdot (F \times G)$ in terms of $\nabla \times F$ and $\nabla \times G$. These are the types of questions I am stuck on and for past 3 hours ...
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Conservative vector field partial derivatives

Given a 3D vector field $\vec{F}=(P,Q,R)$, I know that if $\boldsymbol\nabla \times \vec{F}=\mathbf{0}$, then it is a conservative field. Moreover, I made an observation that if a vector field is ...
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Integral definition of curl

My book, Mathematical Methods for Physics and Engineering - K. F. Riley, explains that the 'Integral definition of curl' is:- $$\nabla\times a = \lim_{V\to0}(\frac{1}{V}\int_S dS\times a) $$ and then, ...
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How can I prove that these definitions of curl are equivalent?

I am reading the book "Div, Grad, Curl, and All that" and I got to the section about curl. In this section, the author defines the curl to be $$ (\nabla \times \mathbf{F})\cdot \mathbf{\hat{...
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What's the dimension of the image of Curl in ${\rm I\!R}^2$?

When working in ${\rm I\!R}^2$, if I apply the Curl operator to a vector field, I obtain a scalar function. Now, the value of this scalar function on a certain point is the Curl on that point. I ...
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Is the curl of matrix times gradient equal to zero?

We know that if $u$ is a smooth function, $\nabla \times (\nabla u) = 0$. If $M$ is a positive-definite, invertible matrix, is there a chance that $\nabla \times (M\nabla u)$ is also zero? The vector ...
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If $\nabla\cdot u=0$ and $w=\operatorname{curl}u$, then $\int w=0$

Let $\Lambda\subseteq\mathbb R^2$ be open, $u\in C^1(\Lambda,\mathbb R^2)$ with $\nabla\cdot u=0$ and $$w:=\frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2}.$$ How can we show that $...
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If $\nabla\cdot v=0$ and $w=\nabla^\perp\cdot v$, then $v=\nabla^\perp g\ast w$, where $g$ is the fundamental solution of the Poission equation

Let $\Omega\subseteq\mathbb R^2$ be open, $v:\Omega\to\mathbb R^2$ with $\nabla\cdot v=0$ (in a sense to be specified later), $$\nabla^\perp:=\left(-\frac\partial{\partial x_2},\frac\partial{\partial ...
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1answer
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Expression for potential vector of a central field

I know that for the central field $$ {\bf F(x)}=\alpha\cdot\frac{\bf x}{|{\bf x}|^{3}}=\alpha\cdot\left(\frac{x_{1}}{|{\bf x}|^{3}},\frac{x_{2}}{|{\bf x}|^{3}},\frac{x_{3}}{|{\bf x}|^{3}}\right) $$ ...
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Show that for all $w$, there is a unique $v$ with $\nabla\cdot v=0$ and $w=\nabla\wedge v$

Let $\Omega\subseteq\mathbb R^2$ be open and $w:\Omega\to\mathbb R$. I've read that there is a unique $v:\Omega\to\mathbb R^2$ with \begin{align}\nabla\cdot v&=0,\tag1\\\int v(x)\:{\rm d}x&=0,\...
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Analogue of $(f\cdot\nabla)f=\frac12\nabla|f|^2-f\times\operatorname{curl}f$ in the two-dimensional case

If $\Omega\subseteq\mathbb R^3$ is open and $f:\Omega\to\mathbb R^3$ is differentiable, there is the identity $$(f\cdot\nabla)f=\frac12\nabla|f|^2-f\times\operatorname{curl}f\tag1,$$ where $$\...
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48 views

What is $\nabla\wedge\vec{\mathbf F}(x,y,z)$

What is $\nabla\wedge\vec{\mathbf F}(x,y,z) \quad$ where F is a vector field? I assume its the dual of the curl but I need an equation. And what is it called? Google is getting me nowhere.
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1answer
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Can a vector and its curl be collinear?

While I was studying fluid mechanics and doing some vector calculus. I wondered if the following statement is true or false. Given that $A$ is a smooth vector field and given that $V\times ( \nabla \...
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2answers
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$\nabla \times \bf{u} \neq 0$ but $\oint_{c} \bf{u} \cdot \textit{d}r \textit{=0}$?

Consider the vector field $\vec{u}=(xy^2,x^2y,xyz^2)$ The curl of the vector field is $$\nabla \times\vec{u}=(xz^2,-yz^2,0)$$ Consider the line integral of $\vec{u}$ around the ellipse $C$ $x^2+4y^2=...
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66 views

Surface-Curl integral questions

Good morning, I have some questions about a surface integral with curl. The exercise is the following: Be $(\Sigma, \omega)$ an oriented surface with boundary where $$\Sigma = \{(x, y, z): x^2 + ...