# Questions tagged [divergence-theorem]

Gauss Divergence Theorem relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.

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### Confusion about divergence theorem for flux computation

I want to compute the flux of the vector field $$F = \frac{\langle x,y,z\rangle}{(x^2+y^2+z^2)^{3/2}}$$ over the unit sphere $x^2+y^2+z^2=1$. I know this is $$\iint F\cdot n \, dS$$ where $n$ is ...
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### Finding a higher dimensional analogue to 1D Sobolev space identity

I had solved a previous problem on $H^1_0(0,1)$ using the identity $$\int_0^1 b(x)u'(x) u(x) = -\frac{1}{2} \int_0^1 b'(x) u(x)^2.$$ Is there any analogue to this for nice domains $\Omega$ where we ...
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### Condition for outward pointing normal vector

I'm now trying to prove Gauss' divergence theorem with a lecture note by Professor B.K. Driver. But I'm facing a problem with outward pointing condition on page 530. Let $\Omega \subset \mathbb R^n$ ...
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### Reference to statement: A harmonic function on a compact connected riemannian manifold is constant.

I read this statement a few times, for example in the answer to this question. Does anyone have a reference to this statement?
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### Divergence theorem involving continuously differentiable function

Question: Let $\gamma \colon [0,1] \to \mathbb{R}^2$ be so that it is continuous and piecewise differentiable and defines a simple closed graph on $\mathbb{R}^2$ that encloses a bounded open connected ...
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### Need someone to clarify this vector calculus problem

The problem is: Let $S$ be a simple smooth parametric surface and let $P$ be a point such that each line that starts at $P$ intersects $S$ at most once. The solid angle $\Omega(S)$ subtended by $S$ ...
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### Divergence and Stokes theorem, Annulus and Torus, Closed path

The Divergence Theorem is generally defined as \begin{align} \int_\mathcal{V} (\nabla \cdot \vec F)\mathrm dV = \oint_{\mathcal{S}}( \vec F \cdot \vec n)\mathrm dS. \end{align} Moreover, the classical ...
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### Flux of radial vector field through sphere

Let $x^2+y^2+z^2=R^2$ be the equation of the sphere we want to calculate the flux through, and $\vec{r}=x\hat i+y\hat j+z\hat k$ be the position vector field. We can compute it via divergence theorem: ...
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### Calculate the flow (divergence) through a hemisphere

Calculate the flow of $\mathbf{F} = (0, 0, y^2 + xz)$ through the hemisphere $x^2 + y^2 + z^2 = 4$, $z \geq 0$, in the direction with a positive $z$-component. I already calculated the divergence on ...
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### Calculate the vector field given a surface using divergence theorem

Calculate the flux of the vector field $F(x, y, z) = (y + 2xz, y + z, −2x − z^2)$ through the surface given by $x^2 + y^2 + z^2 = 4$ and $x, y, z > 0$. The normal direction to the surface points ...
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### Calculate the flux of the vector field by using divergence theorem problem

Calculate the net flux of the vector field $F(x, y , z) = (x, y , 3)$ out of the region given by the inequalities$\sqrt{x^2 + y^2} ≤ z ≤ \sqrt{2 − x^2 − y^2}$ But the correct answer is \$\frac{8π}3(\...
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