# 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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### 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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### 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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### 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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### 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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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 (...