Questions tagged [divergence-operator]

In vector calculus, divergence is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point. The divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. For questions about divergent sequences use [tag:convergence-divergence]

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A laplacian working on an equation containing a laplacian and a gradient

I have an equation as follows: $a \Delta \vec{u} + \vec{\nabla}(\vec{\nabla} \cdot \vec{u}) = 0$. If I take a Laplacian of the above equation, the answer is supposed to be: $\Delta \Delta \vec{u} = 0$....
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How to derive divergence operator in complex spaces [closed]

I would like to find a convinient way to understand divergence and gradient in complex space. Lets for example take a simple function f=u+vi. It might be the case that the are not harmonic conjugates. ...
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Divergence of transposed gradient of vector quantity

I would appreciate some help with this. Is: $$\nabla \cdot \left( \nabla \vec{v}\right)^T= \nabla \left( \nabla \cdot \vec{v}\right)$$ How can I show this? Is the gradient of a vector mathematically ...
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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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Surface integrals in Force Fields

So I am given the following question: On $\mathbb{R}^3\backslash \{\underline{0}\}$ consider the vector field \begin{align*} \underline{F}(x,y,z) = (0,\frac{-2yz}{r^4}, \frac{-1}{r^2}+ \frac{2y^2}{...
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Why does vector field <x, -y> have 0 divergence?

enter image description here Numerically, the divergence is 0. However, when looking at the vector field, it appears like the flow is away from the origin. Hence, why does it have 0 divergence just by ...
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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) ...
Is there any condition under which $\nabla\cdot F=0$ implies $F=0$?
On a physics course it was stated that $$\nabla\cdot\vec{D}=\rho_f=\nabla\cdot(\varepsilon_0\vec{E})$$ and then it follows that $$\vec{D}=\varepsilon_0\vec{E}$$ I know this is not generally true, ...