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Questions tagged [divergence]

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.

535 questions
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Ramanujan's 1/pi Series: Proving that $a_{n+1} < La_n$ (for $n \geq 1$) implies that $a_n< L^{n-1}a_1$ for $n \geq 2$.

I don't know how to go about this question regarding Ramanujan's formula: $$\frac{1}{\pi} = \sum_{n=0}^{\infty} \frac{\sqrt{8}(4n)!(1103+26390n)}{9801(n!)^4396^{4n}}$$ Let $a_n$ denote the nth term ...
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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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Does $\sum_{n=1}^{\infty}\frac{1}{2^n} + \frac{3}{n}$ converge or diverge?

Does this series converge or diverge? If it converges, determine its limit. $$\sum_{n=1}^{\infty}\frac{1}{2^n} + \frac{3}{n}$$ So far I said that $\frac{1}{2^n}$ is a geomotric series that converges,...
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Understanding this very generic divergence theorem where the open set have border $C^k$

I'm at a PDE class and my teacher gave a very generic definition of the divergence theorem. I can't find it anywhere. It's something like this: Definition: let $k\in \{1,2,\cdots,\infty\}$, $N\ge 2$ ...
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Laurent series of an integral with parameter

To find the Laurent series of function $f(a)$ at point $a=0$ $$f(a)=\int^1_0 \frac{d x}{x^2+a^2}$$ one can first do the integral $$f(a)=\frac{1}{a}\arctan(1/a)$$ then expand $\arctan(1/a)$ and ...
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Differentiate an energy containing integral in a region to derive curve evolution

I mainly aim to understand the following paper: Tomographic reconstruction of piecewise smooth images https://ieeexplore.ieee.org/document/1315083/ Here we want to minimize the energy where $p$ 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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Equivalence of notation in Divergence Theorem

Normally I read the Divergence Theorem written as (\oiint doesn't exist here): \begin{align} \oint_{\partial \Omega} \vec{F} \cdot \hat{n} \; dS = \iiint_{\Omega} \...
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Finding the flux of a cylinder using the Divergence Theorem

I have to calculate the flux of the $$F(x,y,z)=(x+\ln(yz^{2}),y+e^{-(x^{2}+z^{2})},2z)$$ through the cylinder (without its bases): $$\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$$ for $z\in[0,4]$. I know that ...
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Implying continuity from knowing divergence/curl/laplacian of vector function equals zero

my question comes from a physics question but is math related. In an electrodynamics related question I've seen the following declaration: Link to a screenshot . In the problem the author is using ...
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Does Helmholtz Decomposition make sense for $\hat{r}/r^2$?

Consider an inverse square vector field $\vec{F}$ and it's curl and divergence shown below $$\vec{F} = \hat{r}/r^2\\ \nabla \times \vec{F} = \vec{0}\\ \nabla \cdot \vec{F} = 0$$ $\vec{F}$ is ...
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Convergent integral of divergent exponential function

 The following integral is a divergent integral: $$\int_{0}^{\infty} \frac{\mathrm{e}^{- a\, x}\, \sin\!\left(x\right)}{x^5} \,d x$$ However, the following solution is provided (though ...
How to find the divergence rate of a recursive sequence defined by $s_n = s_{n-1}(1 + c_n s_{n-1})$
So my question is regarding a sequence of numbers defined recursively by $s_n = s_{n-1}(1 + c_n s_{n-1})$, where $(c_n)$ is some sequence of positive numbers when $s_0 = \epsilon > 0$ is small (how ...