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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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Simple proof of $\operatorname{Curl} F = 0$ then F is conservative

Apparently this is just a simple proof with prerequisites for the domain to be a star-like domain. See picture below for proof: I don't however understand why $\operatorname{Curl} F = 0$ means ...
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Why curl of a vector field that is proportional to $1/r^2$ equal to $0$?

The curl of the vector field ${\bf F} = (-y {\bf i} + x {\bf j})/(x^2 + y^2)$ is $0$. I have an intuitive understanding of why the divergence of a radial field that is proportional to 1/r^2 is equal ...
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show that $\iint_{S}^{}{curl\vec{F}\cdot \vec{dS}}$ is proportional to the lenght of $C$

The curve $C$ is the edge of a surface S , with $\vec{T}$ unit tangent vector of the curve $C$ and $\vec{F}$ a vector field such as $\vec{F}=k\vec{T} $ for each point of $C$, where $k$ is a constant. ...
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Limited Curl in Vector Fields

Is it possible for a vector field in $\mathbb{R}^2$ to have nonzero scalar curl $\bullet$ at a point $\bullet$ along a line $\bullet$ along a curve (which is not a line) $\bullet$ in a finite region ...
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How to convert an integration of a curl to the surface integral?

$\displaystyle\int \left[ \nabla \times \dfrac {M(r')}{r} \right] d\tau =\oint \dfrac{1}{r} \left[ M(r')×da' \right]$ I came across this integral from the David Griffiths introduction to ...
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48 views

Defined and Undefined Vector Fields

Given a vector field which has circulation but no curl resulting from an undefined point, for example, $\begin{bmatrix}-\frac{y}{x^2+y^2}\\\frac{x}{x^2+y^2}\end{bmatrix}$, does there exist, or can ...
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Getting different answers for same problem on divergence and curl.

Given that $\vec{a}$ is a constant vector and $\vec{r}$ is a position vector. We are asked to prove the following: $$\nabla\times(\vec{a}\times\vec{r})=2\vec{a}$$ I tried two ways. Could prove it ...
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46 views

Is this possible in vector calculus?

$$(\boldsymbol{\nabla}\alpha)\wedge(\boldsymbol{\nabla} \wedge \boldsymbol{x} )$$ In all the examples in lecture, it has always been a $$\boldsymbol{\nabla}$$ on the left hand side. Does this give ...
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25 views

How to show this vector field is irrotational?

I have the field: $$\bar a(\bar r)=r \bar c + \frac{(\bar c\cdot \bar r)}{r}\bar r$$ where $$\bar c $$ is a constant vector. I have worked through the problem and I cant seem to easily show that: $$ \...
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Help with simplification of vector calculus problem

In my book, they simplify an expression without explanation: $$\underline{\nabla} \times \left( f\left( r\right) \underline{c}\times \underline{r}\right)=f'\left( r\right)\left( r\underline{c}-\...
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Calculate line integral in a vector field

I have the following problem and I'm not able to solve it. Given $F(x,y,z) = (1-2z, 0, 2y)$, calculate the line integral of C, where C is the contour of the surface. $S=\{(x,y,z) / x\geq 0, y \geq 0, ...
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What's the intuition behind the shear not being possible to be obtained with exterior differentiation?

Allow me to present some context, first. If we have a $K$-vector space $E$ and a non-degenerate symmetric bilinear form (a pseudo-riemannian metric) $g$, then we can project any 2nd order tensor (the ...
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I need to prove a few vector identities using Cartesian Tensor Notation, and I can't figure out how!

I have been all over the internet, but I just can't make sense of this stuff. I have done my best to learn from my textbook and different websites, but this is confusing for me. I haven't taken any ...
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Helmholtz decomposition proof

Reading derivation of Helmholtz decomposition in wikipedia, there are two points I do not understand. First one appears in the step: $$ \begin{align} \dots &=-\frac{1}{4\pi}\left[\nabla\left(\...
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Help in choosing surface for this problem on Stoke's theorem?

Let $S$ be the surface of the cone $z=\sqrt{x^2+y^2}$ bounded by the planes $z=0$ and $z=3$. Further, let $C$ be the closed curve forming the boundary of the surface $S$. A vector field $\vec{F}$ is ...
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An intuitive explanation for Green theorem and Divergence theorem

As my vector calculus exam is getting closer, I'm looking for intuitive ways to think about the different theorems we have to memorize. I think I have found a pretty intuitive way to think about the ...
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Derivation of curl in two dimension

Let $f(x,y) = u(x,y) e_x + v(x,y) e_y$ be a vector valued function on $\mathbb{R}^2$. I'm not sure how the formula is determined. I take this definition of curl: The curl at point $P$ is the (...
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Finding a vector field such that its curl equals a given vector field

Let $$g(x,y,z) = \frac{(x,y,z+1)}{(x^2+y^2+(z+1)^2)^{3/2}}$$ where $z \gt 0$. Find a vector field $f$, such that $$\nabla \times f = g$$ My attempt. I know that $$\operatorname{curl}\vec f = \vec\...
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Question about curl of vector field

Rotating body $${\bf{v}} = {\bf{\omega }} \times {\bf{r}} = \Omega {\bf{r}}$$ $$\nabla \times {\bf{v}} = 2{\bf{\omega }}$$ How to prove that $$\Omega = \frac{{\nabla {\bf{v}} - {{\left( {\nabla {\bf{...
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Proving $\oint (\vec{K} \cdot \vec{r})d\vec{l}=-\vec{K} \times\int d \vec{a}$.

I have to show that, $$\oint (\vec{K} \cdot \vec{r})d\vec{l}=-\vec{K} \times\int d \vec{a}$$ Here, $\vec{K}$ is constant vector and $\vec{r}$ is position vector. I have approached by applying Stokes'...
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How to prove one theorem related to Stoke`s theorem

Stock`s theorem $$\oint\limits_C {{\bf{a}} \cdot {\bf{dr}}} = \iint\limits_S {\nabla \times {\bf{a}}\, \cdot {\bf{n}}dA}$$ Substituting ${\bf{a}} = {\bf{f}} \times {\bf{c}}$ we find that $$\oint\...
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Euclidean Norm of Curl / Divergence

Are there any general statements of the curl or the divergence of a 3-dimensional vectorial function, e.g. for the magnetic field: $$|\nabla\times\boldsymbol B\left(t,\vec{x}\right)| = ?$$ $$|\nabla\...
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Stokes Theorem and Circulation

Find the circulation of $F = \frac{-y}{x^2+y^2}i + \frac{x}{x^2+y^2}j$ along the unit circle. I decided to express $F$ in cylindrical coordinates: $$F = \frac{1}{r}{\hat{\theta}}.$$ However, I found ...
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51 views

Surface integral of curl

Let $\vec F(x,y,z)=(x^2+y^2+\sin(xy)$, $e^x$+$2xy, -yz$). Calculate the flow of $\nabla\times\vec F$ through the following surface: $$ S=\{(x,y,z) x^2 + 2y^2 + 3z^2=10, y\ge 0\} $$ I figured that if ...
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58 views

Divergence and Curl

I was given a function $F(x,y,z)=(z^c,x^c,y^c)$ and asked to find divergence and curl. My initial answer was $0$, but i don’t think that’s right. I noticed the brackets weren’t the typical $\langle,\...
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84 views

Is curl of a position vector always zero?

Now the thing is that the given expression is zero if and only if $x,y,z$ are linearly independent ...ie when $r$ is a free position vector... Now if $r$ is the position vector on a curve or a surface ...
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If $\vec{\nabla} \cdot \vec{V} \neq 0$ at only one point, will this prevent us from saying that $\vec{V}=\vec{\nabla} \times \vec{U}$?

This question has an answer in the language of high level mathematics. Can somebody explain this in the language of vector calculus. Part I: Let us consider Cartesian coordinate system with origin $O$...
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129 views

Finding curl in spherical coordinates

I've been asked to find the curl of a vector field in spherical coordinates. The question states that I need to show that this is an irrotational field. I'll start by saying I'm extremely dyslexic ...
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Cauchy-Riemann conditions and vector fields

Can Cauchy-Riemann conditions of Complex Functions valid for vector fields, I observe when vector fields are irrotational and incompressible they possess a result similar to Cauchy-Riemann conditions. ...
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Will $\vec{f}(x,y,z)=\nabla \times \vec{g}(x,y,z)$ here?

Suppose we have a vector field $\vec{f}(x,y,z)$ defined everywhere in space except a certain region say $(y=3)$. We also know divergence of $\vec{f}(x,y,z)$ is zero at all points where $\vec{f}$ is ...
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120 views

How do I use Stokes' theorem to find curl

How do I solve questions that ask me to use Stokes' Theorem to find curl F For example: Use Stoke's Theorem to find curl F: $$F(x, y, z) = \langle e^x+y^2, y^2+z^2, \sin(z)+x^2\rangle$$ $\iint_S ...
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Calculating Curl of a vector field using properties of $\nabla$.

So i need to find the curl of a vector field $v=(a\cdot r)a\times r$ where $a$ is some constant vector and $r=(x,y,z)$ is the position vector. So i know the curl is given by the cross product $$\...
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61 views

Why is curl considered the differential operator in 3-space?

Why is the curl considered the differential operator in 3-space instead of the gradient? It would seem that the gradient is the corollary to the derivative in 2-space when extending to 3-space. This ...
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Calculating the cross product of a cross product

so I really can't see what I am doing wrong. I want to use this formula: $a\times (b\times c) = b(a\cdot c) - c(a\cdot b)$ Calculate the rotation of $v(x,y,z)=(x,y,z)^T \times \omega$ with $\omega \...
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What would the curl of a longitudinal wave mean?

Is the curl of longitudinal waves even defined? Would the curls of a longitudinal wave and a transverse wave be different in any way?
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How do you compute expressions with operators like gradient, divergence or curl?

I know that it is not an objective question, but I ask for advices and tips for succesfully compute expressions like : $\nabla \cdot \nabla f$ $ \nabla\cdot \nabla \times F$ $ \nabla\cdot(fF)$ $ ...
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28 views

How to find the closed path such that $\int\limits_\Gamma F\cdot dl\ne0$

If there is some vector field $F(x,y)=(F^1(x,y),F^2(x,y))$ such that $\frac{\partial F^2}{\partial x}-\frac{\partial F^1}{\partial y}\ne0$ then by a theorem there does not exist a scalar field $f$ ...
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Wavenumbers of time-harmonic maxwell equation

I am trying to find the wavenumbers $\kappa(\omega)$ of the time-harmonic maxwell equation (eigenvalue problem) $curl \ curl E = \omega^2 E$ whereas $E(x,y,z) = u(y,z)exp(i\kappa_0x)$ and $\Omega$ ...
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How to check the exactness of a three dimensional differential equation?

How to check that a three dimensional differential equation is exact or not . My try : if you have A(x,y,z) dx + B(x,y,z) dy + C(x,y,z)dz = 0 . Shouldn't $$\partial ^2A/\partial x \partial y$$ =$$\...
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$(3x^2 +6y)dx -14yzdy +20xz^2$ is an exact differential then why the curl of $(3x^2+6y) \hat{i} -14yz\hat{j} +20xz^2\hat{k} $ is not zero?

$(3x^2 +6y)dx -14yzdy +20xz^2$ is an exact differential then why the curl of $(3x^2+6y) \hat{i} -14yz\hat{j} +20xz^2\hat{k} $ is not zero ? If you experiment if the differential equation $$\partial ^2 ...
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What are the gradient, divergence and curl of the three-dimensional delta function?

The three-dimensional delta function is defined as follows: $$\delta(\mathbf{r}-\mathbf{r'})= 0 \;\; \mathrm{for} \;\;\mathbf{r}\neq\mathbf{r'} $$ $$\delta(\mathbf{r}-\mathbf{r'})= \infty \;\; \...
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Am I finding curl correct?

I have a field $E(x,y)=2xi+yj$, and want to find the curl. I do this by the next way: $rotE=i\dfrac{d(2x+y)}{dx}+j\dfrac{d(2x+y)}{dy}$ and get $2i+j$ Is it correct?
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62 views

Show $\nabla \times \vec{u} = 2\vec{\omega}$

We need to show that velocity $\vec u$ of any point on rigid body and angular velocity of rigid body $\vec{\omega}$ are related as $\nabla \times \vec u = 2 \vec {\omega}$. Initiate using $\vec u = \...
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What is wrong with my calculating in this determinant? (Stokes theorem)

The task is: "Let $\gamma$ be the cutting curve between the surface $$z=\sqrt{2(x^2+y^2)}$$ and the plane $$z=x+1$$ orienteted opposite the clock - looking upwards. So $$F(x,y,z)=(y^2,2yz,-xy)$$ ...
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Is the divergence of the curl of a $2D$ vector field also supposed to be zero?

In three dimensions, it seems pretty straightforward to prove the identity that for any vector field $\mathbf{A}$, $$\nabla \cdot (\nabla \times \mathbf{A}) = 0$$ Does this identity still hold true ...
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Problem with Curl of $M\hat{i} + N\hat{j}$

When I take the curl of above vector I get $\partial N/\partial x$ - $\partial M/\partial y$ . But if I do take the curl in 3×3 matrix then something different comes back (As I'm in mobile I'm unable ...
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$\varphi$ verify that $\nabla \cdot\varphi=0$ but doesn't exist $G:\Bbb R^3\to \Bbb R^3$, $\mathcal C^1$ such that $\nabla \times G=\varphi$

I have this problem: Probe that $\varphi:\Bbb R^3-[0] \to \Bbb R^3, \varphi=\frac{(x,y,z)}{\vert\vert(x,y,z)\vert\vert^3}$ verify that $\nabla \cdot\varphi=0$ but does not exist $G:\Bbb R^3\to \Bbb R^...
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why is the curl of a function (which looks like a vortex) zero?

Can anyone explain why the curl of this function zero ? Clearly it's rotating then how is it irrotational ? Someone on SE said Not to mistake Curl for rotation, they aren't exactly similar. Is ...
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About the definition of divergence and curl

I was wondering about a simple detail. As you probably already know, there are two definitions curl and divergence can be defined in the following way (in $\mathbb{R^{2}}$ but the question is also ...
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Green's Theorem in the Plane: Circulation Density

The following is from Chapter 16.4: Green's Theorem in the Plane, Thomas's Calculus, 14th Edition: Circulation rate around rectangle $\approx \left( \dfrac{\partial{N}}{\partial{x}} - \dfrac{\...