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 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:
$$
\mathbf{A}(x) \times \mathbf{n}=\...
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divergence and curl of scalar field times a constant vector field.
Hello there I have trouble with the goal of following question about divergence and curl. More specifically I do know what these terms mean and how the maths related to them work, but the goal of the ...
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"Curlless" and "divergentless" fields
Is it possible for a field having zero curl to not be able to be expressed as the gradient of some scalar field and is it possible for a field having zero divergence to not be able to be expressed as ...
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Is the following vector expression true?
Suppose I have the following equation involving two vectors $\vec{A}$ and $\vec{B}$ that satisfy the following relation :
$$\vec{\nabla}\times\frac{d\vec{A}}{dt}=k\frac{\partial \vec{B}}{\partial t}$$
...
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How do I show that $\operatorname{div}(\operatorname{rot} F(x,y,z)) = 0$?
Suppose that
$$
F:\mathbb{R}^{3} \to \mathbb{R}^{3}
$$
is twice continuously differentiable, and
$\operatorname{div}F(x,y,z)$ and $\operatorname{rot}F(x,y,z)$
are defined as shown in the picture.
How ...
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Does fundamental theorem of calculus apply to closed curves? If yes why the closed line integral of a function is not zero?
Why closed curve integral $$\oint A.dl$$ doesn't give us zero in many physics related examples?
Closed line/loop/curve/contour have same starting and end points and according to fundamental theorem ...
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Why a vector field isn't determined solely by its divergence
I am an undergraduate physics student. I would prefer answers that share more information and clarification rather than try to be concise.
To my understanding, Helmholtz theorem tells us that a vector ...
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Derive curl of vector field in spherical coordinates
I need to calculate curl of $F$, and show that it is conservative on this region.
A vector field $F$, defined on a simply-connected region $r > 0,\; \frac{\pi}{4}< \theta < \frac{3 \pi}{4},\; ...
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Implicit definition of stress tensor
I'm reading Giacomini, Alessandro, and Luca Lussardi. "Quasi-static evolution for a model in strain gradient plasticity." SIAM journal on mathematical analysis 40.3 (2008): 1201-1245 and I ...
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Volume to surface integral of $R\times \nabla \times B$
I need to transform the following integral into a surface integral (if that's possible):
$$\int\int\int_\Omega R\times (\nabla \times A) dv = \int\int_{\partial \Omega} ? . {\bf n} da, $$
where $R = (...
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Using "Maxwell's curl equations" to get $H_y = \dfrac{j}{\omega \mu} \dfrac{\partial{E_x}}{\partial{z}} = \dfrac{1}{\eta}(E^+ e^{-jkz} - E^- e^{jkz})$
I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Chapter 1.4 THE WAVE EQUATION AND BASIC PLANE WAVE SOLUTIONS says the following:
The Helmholtz Equation
In ...
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How to perturb a 2-d conservative vector field so as to maximize its curl
Given a 2-d conservative vector field $F(x,y)=(f_1(x,y),f_2(x,y))$, I would like to perturb it with another unit vector field $G(x,y)$ so as to make the curl of the new vector field $F(x,y)+G(x,y)$ ...
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Is there a vector identity for $\left( \nabla \times \mathbf{\vec F}\right) \times \mathbf{\vec F}$?
I have looked through several lists of curl identities but cannot find anything for this.
$\nabla \times$ denotes the curl, and $\mathbf{\vec F}$ is just some arbitrary vector field in $\Bbb R^3$
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When is a symmetric tensor equal to gradient times gradient?
On a ball in $\mathbb R ^n$, a vector field $v_j$ is a gradient of a function when its exterior derivative vanishes. In other words, if $$ \partial_i v_j-\partial_j v_i=0$$ then there exists a ...
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Calculate the Solid Angle using Stokes' theorem
The solid angle for the surface S subtended at a point P is:
$$
\Omega=\iint_{S} \frac{\hat{r} \cdot \hat{n}}{r^{2}} d S
$$
where $\hat{r}$ and $\hat{n}$ are unit vectors and $r =|\vec {r}|$ is the ...
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Can a unit vector be treated as a vector field?
Say you have the kth component of a vector field:
$$\phi \hat{k}$$
Can we treat this as a scalar field $\phi$ multiplied by the vector field $\hat{k}$?
EDIT:
So, if you wanted to find the curl of $\...
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Curl of a noise field on a sphere
Problem
Hey people. I'm currently working on a small approximation of a fluid simulation on a sphere based on curl noise. The math behind that is based on this paper, which notes that because the curl ...
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Distributive property of curl?
I see via wiki that:
$$\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}+\...
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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 ...
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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 ...
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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 ...
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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
$$
\frac{\...
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Curl of vector field must be zero or not must be zero?
I have to find is it possible for F=sinx i+cosy j+sinxyz k to be the curl of vector field?
Therefore, $\begin{pmatrix}i&j&k\\ \frac{d}{dx}&\frac{d}{dy}&\frac{d}{dz}\\ sinx&cosy&...
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find a funciton $f(x,y,z)$ such that $grad f=F$, where $F$ is the given vector field
So I have this vector field $$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 &...
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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 $\vec{r}(t)=\langle\cos(t), \sin(t), 1\rangle$ ...
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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)
...
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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 ...
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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 + ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}...
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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:
$$curl \ \mathbf v = \mathbf \nabla \times \mathbf v = \begin{...
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Proof with div and curl
Let $$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 ...
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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"
$$\nabla\times\...
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Curl of vector fields in form $\begin{pmatrix}ax+by\\cx+dy\end{pmatrix}$
So I think I've got a misunderstanding about what curl is or made a calculation mistake. For the vector field $\begin{pmatrix}ax+by\\cx+dy\end{pmatrix}$ then its curl should be $\nabla \times \begin{...
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Time derivative of curl using tensors
I'm pretty new to tensor calculus; I would like to prove, as an exercise, the following vector calculus identity:
$$ \partial_t \left( \nabla \times \vec{F} \right) = \nabla \times \partial_t \vec{F} $...
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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 ...
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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 ...
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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 ...
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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:
$$
\begin{aligned}
(\phi \mathbf{e}) \times (\nabla \times ...
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Product rule identity for $\nabla \times (\mathbf{A} \cdot \mathbf{u})$, the curl of a tensor field times a vector field
I am looking to derive product rules for the curl of a 2nd-order tensor field contracted with a vector field (matrix vector multiplication),
$$\nabla \times (\mathbf{A}(\mathbf{x}) \cdot \mathbf{u}(\...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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)-\...
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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 (...
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