# 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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### Physical significance of $\vec{w}$ $\times$ $($curl $\vec{v})$ [migrated]

I think if curl of a vector field $\vec{v}$ corresponds to an applied rotation, it's cross product with a velocity vector field $\vec{w}$ (say) should give something analogous to the resulting torque. ...
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### Piecewise mooth closed curve in simply connected space is a boundary of surface

I'm interested in a simple proof of the following fact: Let $V \subset \mathbb{R}^3$ be a bounded, open, connected, simply connected set. Let $\gamma$ be a piecewise-smooth simple closed curve in $V$....
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### Show that $\vec\nabla\times(\vec\nabla\times\vec A )=- \nabla^ 2\vec A + \vec\nabla (\vec\nabla\cdot\vec A )$ [duplicate]

I just started learning this and I don't understand much so how can I prove this? $$\vec\nabla\times(\vec\nabla\times\vec A )=-\vec\nabla^2\vec A +\vec\nabla (\vec\nabla\cdot\vec A )$$
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### How can I simplify $\nabla (X\cdot \nabla u)$?

What is $\nabla (X\cdot \nabla u)$ where $X:U\subseteq\mathbb{R}^2\to\mathbb{R}^2$ is a vector field and $u:U\subseteq\mathbb{R}^2\to\mathbb{R}$ a scalar field?
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### Curl in spherical coordinates on example

On Wiki (https://en.wikipedia.org/wiki/Dipole_antenna) the Vector Potential of a Dipole Antenna is roughly given by: $$A = c\cdot \dfrac{e^{-i\,k\,r}}{r}\,\hat{e}_z$$ Now the curl is computed in ...
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I need to take the curl of $\frac{\partial u_i}{\partial t} = -\frac{1}{\rho}\frac{\partial p}{\partial x_i}-\frac{\partial}{\partial x_j}u_i u_j+\nu\frac{\partial^2 u_i}{\partial x_i^2}$ to get \$\...