# 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.

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### Computing divergence of a piecewise constant vector field

Let $n \in \mathbb R^2$ be a given vector of length 1 and let $U$ be the following vector field: U(x) : = \begin{cases} U^+ & \text{ if } x \cdot n>0 \\ U^- & \text{ if } ...
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### Application of the Divergence Theorem with change of variable

Let $S$ be the ellipse $\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2=1,$ with $\vec{n}$ oriented outwards. Compute $\int\!\!\!\int_S \vec{F}\cdot \vec{n}\,dA$ for ...
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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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### Integral on surface and boundary

Let $\Omega$ be connected bounded open set in $\mathbb{R}^{n}$. Let $U:\Omega\rightarrow \mathbb{R}^{n}$ be a $C^{1}$ vector field. The divergence theorem is given \begin{align} \int_{\Omega} \nabla\...
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### How to prove $\mathbf{M}/r$ is a continuously differentiable vector field?

I had previously asked a question here regarding the applicability of divergence theorem. $\mathbf{M'}$ is a continuous vector field in volume $V'$ (which is compact and has a piecewise smooth ...
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### Showing volume and surface integration is unaffected by the singularity at $\mathbf{r'}=\mathbf{r}$

This question is not entirely similar to the question here. Please read this question and the reader will see it is obviously not the same. $\mathbf{M'}$ is a continuous vector field in volume $V'$ ...
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### Proof of harmonic series divergence [closed]

Prove that ∑_(k=1)^∞ 1/k is divergent Proof Could anyone possibly help explain my professor’s notation in this proof? Struggling to understand where S_(2^n-1)≥n/2 came from as well as the ...