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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
43 views

Curl of a vector field and orthogonality

Let $\mathbf{A}(x)$ be a scalar field defined in the 3D euclidean affine space, $\Sigma$ a plane and $\mathbf{n}$ a unit normal vector perpendicular to $\Sigma$ if: $$ \mathbf{A}(x) \times \mathbf{n}=\...
user avatar
1 vote
1 answer
41 views

divergence and curl of scalar field times a constant vector field.

Hello there I have trouble with the goal of following question about divergence and curl. More specifically I do know what these terms mean and how the maths related to them work, but the goal of the ...
user avatar
1 vote
0 answers
34 views

"Curlless" and "divergentless" fields

Is it possible for a field having zero curl to not be able to be expressed as the gradient of some scalar field and is it possible for a field having zero divergence to not be able to be expressed as ...
user avatar
0 votes
0 answers
22 views

Is the following vector expression true?

Suppose I have the following equation involving two vectors $\vec{A}$ and $\vec{B}$ that satisfy the following relation : $$\vec{\nabla}\times\frac{d\vec{A}}{dt}=k\frac{\partial \vec{B}}{\partial t}$$ ...
user avatar
0 votes
0 answers
31 views

How do I show that $\operatorname{div}(\operatorname{rot} F(x,y,z)) = 0$?

Suppose that $$ F:\mathbb{R}^{3} \to \mathbb{R}^{3} $$ is twice continuously differentiable, and $\operatorname{div}F(x,y,z)$ and $\operatorname{rot}F(x,y,z)$ are defined as shown in the picture. How ...
user avatar
  • 41
0 votes
1 answer
47 views

Does fundamental theorem of calculus apply to closed curves? If yes why the closed line integral of a function is not zero?

Why closed curve integral $$\oint A.dl$$ doesn't give us zero in many physics related examples? Closed line/loop/curve/contour have same starting and end points and according to fundamental theorem ...
user avatar
4 votes
1 answer
118 views

Why a vector field isn't determined solely by its divergence

I am an undergraduate physics student. I would prefer answers that share more information and clarification rather than try to be concise. To my understanding, Helmholtz theorem tells us that a vector ...
user avatar
0 votes
0 answers
30 views

Derive curl of vector field in spherical coordinates

I need to calculate curl of $F$, and show that it is conservative on this region. A vector field $F$, defined on a simply-connected region $r > 0,\; \frac{\pi}{4}< \theta < \frac{3 \pi}{4},\; ...
user avatar
  • 67
0 votes
0 answers
28 views

Implicit definition of stress tensor

I'm reading Giacomini, Alessandro, and Luca Lussardi. "Quasi-static evolution for a model in strain gradient plasticity." SIAM journal on mathematical analysis 40.3 (2008): 1201-1245 and I ...
user avatar
  • 701
0 votes
1 answer
39 views

Volume to surface integral of $R\times \nabla \times B$

I need to transform the following integral into a surface integral (if that's possible): $$\int\int\int_\Omega R\times (\nabla \times A) dv = \int\int_{\partial \Omega} ? . {\bf n} da, $$ where $R = (...
user avatar
  • 1,621
0 votes
1 answer
79 views

Using "Maxwell's curl equations" to get $H_y = \dfrac{j}{\omega \mu} \dfrac{\partial{E_x}}{\partial{z}} = \dfrac{1}{\eta}(E^+ e^{-jkz} - E^- e^{jkz})$

I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Chapter 1.4 THE WAVE EQUATION AND BASIC PLANE WAVE SOLUTIONS says the following: The Helmholtz Equation In ...
user avatar
  • 4,066
1 vote
0 answers
14 views

How to perturb a 2-d conservative vector field so as to maximize its curl

Given a 2-d conservative vector field $F(x,y)=(f_1(x,y),f_2(x,y))$, I would like to perturb it with another unit vector field $G(x,y)$ so as to make the curl of the new vector field $F(x,y)+G(x,y)$ ...
user avatar
4 votes
1 answer
133 views

Is there a vector identity for $\left( \nabla \times \mathbf{\vec F}\right) \times \mathbf{\vec F}$?

I have looked through several lists of curl identities but cannot find anything for this. $\nabla \times$ denotes the curl, and $\mathbf{\vec F}$ is just some arbitrary vector field in $\Bbb R^3$
user avatar
0 votes
0 answers
16 views

When is a symmetric tensor equal to gradient times gradient?

On a ball in $\mathbb R ^n$, a vector field $v_j$ is a gradient of a function when its exterior derivative vanishes. In other words, if $$ \partial_i v_j-\partial_j v_i=0$$ then there exists a ...
user avatar
0 votes
0 answers
46 views

Calculate the Solid Angle using Stokes' theorem

The solid angle for the surface S subtended at a point P is: $$ \Omega=\iint_{S} \frac{\hat{r} \cdot \hat{n}}{r^{2}} d S $$ where $\hat{r}$ and $\hat{n}$ are unit vectors and $r =|\vec {r}|$ is the ...
user avatar
  • 21
-1 votes
1 answer
54 views

Can a unit vector be treated as a vector field?

Say you have the kth component of a vector field: $$\phi \hat{k}$$ Can we treat this as a scalar field $\phi$ multiplied by the vector field $\hat{k}$? EDIT: So, if you wanted to find the curl of $\...
user avatar
  • 325
0 votes
0 answers
28 views

Curl of a noise field on a sphere

Problem Hey people. I'm currently working on a small approximation of a fluid simulation on a sphere based on curl noise. The math behind that is based on this paper, which notes that because the curl ...
user avatar
1 vote
1 answer
146 views

Distributive property of curl?

I see via wiki that: $$\begin{aligned} \nabla \cdot(\mathbf{A}+\mathbf{B}) &=\nabla \cdot \mathbf{A}+\nabla \cdot \mathbf{B} \\ \nabla \times(\mathbf{A}+\mathbf{B}) &=\nabla \times \mathbf{A}+\...
user avatar
0 votes
0 answers
41 views

Value of curl at the origin when there is a singularity at the origin?

Say we have the following vector field: There is a singularity at the origin because we end up dividing by 0. I'm not sure what the value of curl is at the origin. On the one hand, we can work out ...
user avatar
  • 325
0 votes
0 answers
57 views

integral of vector field

I'm wondering how to prove this : if $F$ is a vector field and $g$ is a scalar function (continuously differentiable function) and $$ \nabla g$$ is not zero. assuming that $D= {x: b≥g(x)≥a}$ how can ...
user avatar
  • 11
2 votes
1 answer
112 views

Vector fields having zero curl wrt two different real inner-products?

Let $V:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ be a smooth vector field, let $e(\cdot, \cdot)$ be the standard Euclidean inner-product, and let $g(\cdot, \cdot)$ be a real inner-product that is not a ...
user avatar
0 votes
0 answers
27 views

Exact solutions to a kind of 3D wave equation

Consider the mapping ${\bf{u}}: [0,T] \times \mathbb{R}^3 \mapsto \mathbb{R}^3$, defined by ${\bf{u}} = {\bf{u}}(x,y,z,t)$. I am looking for general solutions ${\bf{u}}$ of the following PDE $$ \frac{\...
user avatar
  • 3
0 votes
1 answer
30 views

Curl of vector field must be zero or not must be zero?

I have to find is it possible for F=sinx i+cosy j+sinxyz k to be the curl of vector field? Therefore, $\begin{pmatrix}i&j&k\\ \frac{d}{dx}&\frac{d}{dy}&\frac{d}{dz}\\ sinx&cosy&...
user avatar
  • 33
2 votes
0 answers
41 views

find a funciton $f(x,y,z)$ such that $grad f=F$, where $F$ is the given vector field

So I have this vector field $$F=\langle 2xe^{x^2y}, e^{x^2}, 1\rangle$$ I am asked to find a function such that $grad f=F$ However, when I check its curl, I got $$\begin{bmatrix} \vec i & \vec j &...
user avatar
2 votes
1 answer
131 views

Calculate the circulation of the vector field alone a parameterized circle (Stoke's Theorem...?)

Find the circulation of the following vector field $\vec{F}(x, y, z) = \langle \sin(x^2+z)-2yz, 2xz + \sin(y^2+z), \sin(x^2+y^2)\rangle$ along the circle $\vec{r}(t)=\langle\cos(t), \sin(t), 1\rangle$ ...
user avatar
0 votes
0 answers
18 views

Define vector field with known curl and div [duplicate]

I have a problem with vector analysis and can't figure out. Can vector field be determined if we know its divergence and curl? If so, what is the function of vector field? Let's say div v = f(x, y, z) ...
user avatar
2 votes
0 answers
36 views

How to call the scalar "curl" of a vector field in a plane?

I cannot find any standard name or notation for the analog of curl of a vector field in an oriented Euclidean plane. There is this operator that to a vector field $\mathbf{F}$ given in the standard ...
user avatar
  • 1,638
0 votes
0 answers
16 views

Is this method to find the conservative nature of a vector field correct?

Disclaimer: I am new to Calc-3 so I might sound stupid here. However I'd be grateful for all the help I get. A problem seeks to know if the vector field given by $[4y^2 + \frac {3x^2y}{z^2}]î + [8xy + ...
user avatar
0 votes
0 answers
48 views

Derivative by “definition” using differential

In the section Cartesian derivation, in this wiki page, they derive and define curl in an different way. I was always told that a differential is an extremely small change in a variable and that the ...
user avatar
1 vote
0 answers
86 views

How do you understand the nature of curl?

My teacher tells me the curl describes the component of rotation at a point in a vector field. When a ball is placed in a vector field with a non-zero curl, it tends to rotate. Let's consider a field ...
user avatar
  • 515
0 votes
1 answer
40 views

intuition behind differential operators

I discover these concepts in my math class, and I want to understand the intuition behind the Laplacian, the curl and the divergence operators, how do you explain this to a student who is discovering ...
user avatar
0 votes
0 answers
35 views

Curl of a vector field on all parts of space

Find $ \nabla \times \vec{F} $ of a given vector field $\vec{F}=\frac{{\hat{\phi}}}{r}$ in cylindrical coordinates on all points of space . I calculated $ \nabla \times \vec{F} $ in all points except ...
user avatar
0 votes
1 answer
40 views

Is my intuition of line integral or curl wrong?

My impression was that the curl of a vector field measures how fast a vector field turns along a closed curve around that point. Consider the vector fields $\vec{V}_1=-y\hat{i}+x\hat{j}$ and $\vec{V}...
user avatar
0 votes
1 answer
90 views

What is the determinant representation of the curl(v) index notation?

I can't visualize $\mathbf 16a$ in determinate form. I cannot read the given index notation, and convert it to the following form: $$curl \ \mathbf v = \mathbf \nabla \times \mathbf v = \begin{...
user avatar
1 vote
1 answer
53 views

Proof with div and curl

Let $$F, G:\mathbb{R}^3 \to \mathbb{R}^3$$ Prove that: $$\nabla\cdot(F\times G)=G\cdot (\nabla \times F)-F \cdot(\nabla\times G)$$ I found that: $$∇\cdot(F×G)=(∂(F_2 G_3-F_3 G_2))/∂x+∂(F_3 G_1-F_1 G_3 ...
user avatar
  • 11
0 votes
0 answers
25 views

Curl of current density over distance when deriving the Biot-Savart Law

I'm reading this Physics Exchange post on deriving the Biot-Savart Law from Maxwell's Equations. However, this step is confusing me: "Now we need only calculate B=∇×A. But" $$\nabla\times\...
user avatar
  • 121
1 vote
0 answers
78 views

Curl of vector fields in form $\begin{pmatrix}ax+by\\cx+dy\end{pmatrix}$

So I think I've got a misunderstanding about what curl is or made a calculation mistake. For the vector field $\begin{pmatrix}ax+by\\cx+dy\end{pmatrix}$ then its curl should be $\nabla \times \begin{...
user avatar
1 vote
0 answers
48 views

Time derivative of curl using tensors

I'm pretty new to tensor calculus; I would like to prove, as an exercise, the following vector calculus identity: $$ \partial_t \left( \nabla \times \vec{F} \right) = \nabla \times \partial_t \vec{F} $...
user avatar
  • 33
1 vote
1 answer
63 views

Inequality bound with $\nabla(\nabla\cdot u)$ or $\nabla \times (\nabla \times u) $

I'm studying regularity for linear elasticity problem where I encountered $\nabla (\nabla \cdot u) = \nabla \times (\nabla \times u) + \Delta u$ I am hoping to find a bound of $\|\nabla (\nabla \cdot ...
user avatar
  • 55
0 votes
1 answer
126 views

Vorticity Equation in two dimensions, the vector stream function and curl

So I am working through Peter Olver's book Application of Lie Groups to Differential Equations and there is this example on page 445 with the Euler Equations for inviscid ideal fluid flow in two ...
user avatar
  • 657
1 vote
0 answers
35 views

Deriving 2-D curl from the formal definition, the hard way

The curl of $\mathbf{F}$ at some point $(x_0,y_0)$ is defined as $$\lim_{A\to0}\frac{1}{|A|}\oint_\gamma\mathbf{F}\cdot ds$$ I was curious to see what happens when we expand this, in the simple two ...
user avatar
  • 10.5k
0 votes
1 answer
34 views

Simplifying Expression involving cross product and curl. [duplicate]

I have the following expression that i would like to simplify further. $\mathbf{e}$ is a unit vector. I have used the BAC-CAB Rule to obtain: $$ \begin{aligned} (\phi \mathbf{e}) \times (\nabla \times ...
user avatar
  • 1
2 votes
0 answers
141 views

Product rule identity for $\nabla \times (\mathbf{A} \cdot \mathbf{u})$, the curl of a tensor field times a vector field

I am looking to derive product rules for the curl of a 2nd-order tensor field contracted with a vector field (matrix vector multiplication), $$\nabla \times (\mathbf{A}(\mathbf{x}) \cdot \mathbf{u}(\...
user avatar
2 votes
0 answers
99 views

Given that we have to find the $ Curl (V)$. How could we find $\phi$?

Question : By calculating $\nabla \times V$, show that $$V~(x, y , z )=\begin{pmatrix} ~yz^2+3\\ xz^2+2z+1\\ 2xyz+2y\end{pmatrix}$$ can be expressed as V $=\nabla \phi$ for some function $\phi$, and ...
user avatar
  • 418
1 vote
1 answer
71 views

General line integral when curl of vector field is nonzero

Given $\mathbf{F}(x,y,z) = (y-z,z-x,x-y)$, calculate the line integral $\int_C\mathbf{F}\cdot d\mathbf{l}$ where $C$ is any curve on the plane $x-z=1$. My initial instincts to tackling this problem ...
user avatar
  • 452
1 vote
1 answer
73 views

Irrotational vector Field in Spherical coordinates

If the vector field $ \overrightarrow{A} = A_{\varphi} (r, \theta) \widehat{e}_{\varphi} $ is irrotational, then what will be the value of $ A_{\varphi} (r, \theta) $ My work: For irrotational case, ...
user avatar
  • 347
3 votes
3 answers
350 views

Using Stokes theorem to find the line integral over the boundary of a paraboloid in the first octant opening downward the z-axis

I've been trying at this problem on my homework, but I think I am going about it the wrong way. I tried breaking it down into the line integrals of the boundaries of the surface, but I think I might ...
user avatar
3 votes
1 answer
84 views

Proving $(\nabla \times \mathbf{v}) \cdot \mathbf{c} = \nabla \cdot (\mathbf{v} \times \mathbf{c})$ using cylindrical coordinates

Assuming the form of divergence in polar coordinates is known, I am attempting to use the following definition of the curl of a vector field to determine the form of the curl in cylindrical ...
user avatar
2 votes
1 answer
43 views

Does the relation $\nabla\times\vec{A}\approx\nabla\vec{A}-\left(\nabla\vec{A}\right)^T$ have a proper name?

I discovered that curl seems to have an analog which could be used in dimensions other than n=3. $$\nabla\times\vec{A}\approx\left[\begin{matrix}0&\left(\frac{\partial A_x}{\partial y}\right)-\...
user avatar
  • 182
0 votes
1 answer
108 views

Does Stokes law on a non-conservative field?

I was trying to check if Stokes would hold in a field $(3x^2y^3,-x^3y^2)$ (non conservative as curl is non zero) for the figure below but I got that LHS (the surface integral) was not equal to RHS (...
user avatar
  • 279

1
2 3 4 5