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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Integral of gradient of a function, times a vector fields null whatever the function implies null divergence and tangential limits conditions

At the beginning of "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris S´er. I Math. 305, no. 19, 805–808", there is: $\int_{K}...
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Divergence of a Point

I am currently reading a book titled Notes on Computational Fluid Dynamics: General Principles, which is provided by OpenFOAM at https://doc.cfd.direct/notes/cfd-general-principles/contents. In ...
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Evaluate integral of $\langle xz^2,x^2y-z^3,2xy+y^2z\rangle$ with divergence theorem

Q. Evaluate by Gauss divergence theorem $$\iint_S xz^2\,dy\,dz + \left(x^2y-z^3\right)\,dz\,dx + \left(2xy+y^2z\right)\,dx\,dy$$ where $S$ is the surface bounded by $z=0$ and $z=\sqrt{a^2-x^2-y^2}$. ...
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Divergence Theorem Exercise

The problem is as follows: Given that $\nabla \cdot \mathbf{F} = 0$ in $V$ $\mathbf{F} \cdot \mathbf{n} = 0$ on $\partial V$ Prove that: $\int_V \mathbf{F} \, dV = 0$. I understand it intuitively, ...
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Surface Integral of vector field bounded by two spheres .

enter image description here A vector field : $$D\mapsto \mathbf{a}_r\frac{\cos^2\phi}{r^3}+\mathbf{a}_\theta\sin\theta$$ Exists in the region between two spherical shells defined by r=1 cm and r=2 cm ...
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Calculating a Volume Integral with the Divergence Theorem

Hello I have been trying to apply the Divergence Theorem to the following problem but i seem to either missinterpert or not understand the problem. The following integral should be calculated with the ...
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Volume integral of gradient field

I came across the following theorem: $$ \iint_{\partial \Omega}f dS=\iiint_{\Omega}\nabla{f}dV $$ $\Omega$ is a bounded region whose boundary $\partial \Omega$ is a closed, piecewise smooth surface ...
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Clarification question on applying divergence theorem to $\nabla \cdot (u \nabla u)$ on a compact manifold without boundary

I'm following the solutions given in this post. Basically, they are trying to prove that the kernel of the Laplace operator on a compact manifold without boundary is just made up of constant functions....
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Additional terms after applying divergence theorem

following my lecture notes I stumbled upon the following integral and could not make sense of certain aspects of it. It is said that if \begin{equation} 0 = \frac{1}{2}\partial_t[(\partial_t\phi)^2 + (...
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Direct proof of Cartesian coordinates form of divergence

From the definition of divergence of a smooth vector field $F:\mathbb{R}^n\to \mathbb{R}^n$, $\mathbb{div}F:=\lim_{|V|\to 0}\frac{\int_{\partial V}F\cdot do}{|V|},$ where $\partial V$ is a piece-wise ...
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Vector field defined on the solid sphere

Let $U$ be an open set of $\mathbb{R}^3$, $f:U \to \mathbb{R}$ a $c^2$ function, $B \subset U$ an solid sphere (closed), $\mathcal{S}$ the spherical surface that bounds $B$, and $N$ the outward normal ...
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Regularity conditions needed for divergence to equal "flux density"? (I.e. for "coordinate-free definition" to be valid?)

Background: Let $\vec{V} : \mathbb{R}^n \to \mathbb{R}^n$ be treated as a vector field, $V_i$ denote the corresponding scalar coordinate functions $\mathbb{R}^n \to \mathbb{R}$, and then if these ...
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Meaning of a scalar surface integral of a vector field?

I apologize if the title is confusing but after going through some assignments I was able to correctly derive that $\iiint_{V}(\overrightarrow{\nabla}\times A)dV=-\oint_{\partial V} (A \times \hat{n})...
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Divergence Theorem to find the flux of a field

Compute the flux of the field $(ye^z, 16xz, \arctan(\frac{z}{\sqrt{x^2+y^2}}))$ out of region $E$ that lies between spheres $x^2+y^2+z^2= 1$ and $x^2+y^2+z^2= 16$ in first octant. I first did $\frac{\...
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Divergence's Theorem Application

Let $\Omega \subset \mathbb{R}^3$ be the solid given by $$\left\{(x,y,z) \in \mathbb{R}^3:x^2+y^2+z^2 \leq 1; x + 2y − z \geq 0\right\}$$ and $N$ the unitary normal vector field to the surface $\...
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Understanding Conservation in Divergence Theorem

Ok so I'm struggling with the concept of conservation in the divergence theorem. Divergence theorem states that: $$ \iiint_O {\nabla \cdot {\bf F}}\ dV = \iint_{S=\partial O} ({\bf F}\cdot \hat{{\bf n}...
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How to prove the monotonicity of an integration?

Let $u: \Omega \subset \mathbb{R}^n \rightarrow \mathbb{R}$ be a harmonic function, i.e. $\Delta u=0$. Let $x_0 \in \Omega$ be a point. Assume $n \geq 2$. Simplify the following formula $$ \...
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How to make use of $(u · \nabla)u = \nabla P, \operatorname{div} u = 0$.

I come up a problem when solving the following question: Let $\boldsymbol{u} = (u_1(x, y), u_2(x, y))$ be a smooth planar vector field and $P = P(x, y)$ be a scalar function in $\Omega$ satisfying $(u ...
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Is the Normal vector is pointing inside or outside?

Good morning, I am posting this again because I am having trouble to understand this post (How do I check if the normal vector is pointing inside or outside?). Here is my question : I have the ...
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Verifying Gauss divergence theorem

The question says that: If $\vec{F}=(2x^2-3z)\vec{i}-2xy\vec{j}-4x\vec{k}$, we are supposed to calculate $\iiint \operatorname{div} \vec{F}\, dV$, where $V$ is the closed region bounded by the planes $...
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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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Divergence of a Vector field in Einstein soliton

An $n$-dimensional Riemannian manifold $(M,g)$ is an Einstein soliton if there exist a vector field $\zeta$ and a real constant $\lambda$ such that $\DeclareMathOperator{\Ric}{Ric} \...
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The integral of the Lie derivative over a divergence free vector field is zero

Let $M$ be a Riemannian manifold with metric $g$ and volume form $dV$. Let $L_X$ denote the Lie derivative taken over a vector field $X$. Suppose $X$ is a vector field over $M$ with compact support ...
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Divergence theorem for the set of singular positive semidefinite matrices.

Let us consider the space of symmetric positive-definite (SPD) $\mathscr{P}^d$ matrices of dimensions $d \times d$. It is well known that this space is a pointed convex cone, which represents a ...
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Can i use the mean value theorem here?

Suppose i have two functions, one of mass flow and other of momentum flow through a surface: $$ f = \int_A \rho (v \cdot n)dA $$ $$ g = \int_A \rho v (v \cdot n)dA $$ $\rho$ = density of the material $...
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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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Proving an integral inequality of the Klein-Gordon equation over a bounded volume in $\Bbb{R}^3$

I am attempting to solve part (ii) of the following problem: Let $V$ be a region in $\Bbb{R}^3$ with boundary a closed surface $S$. Consider a function $\phi$ defined in V that satisfies $$\nabla^2\...
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The propagation of the divergence condition in the incompressible Navier-Stokes equation

On page 16 of the book "Turbulence" written by Uriel Frisch, the author mentions that "...It is now sufficient to impose the divergence condition $\partial_j v_j=0$ at $t=0$, since (2....
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$\int_{\Omega}u\operatorname{div} \varphi \,\mathrm{d}x = -\int_{\Omega} \phi \cdot \nabla{u} \,\mathrm{d}x$

Let $\Omega \subset \mathbb{R}^n$ be open, $u \in C^1(\Omega)$ and $\phi \in C^{1}_{c}(\Omega, \mathbb{R}^n)$, which denotes the set of continuously differentiable functions $\colon \Omega \to \mathbb{...
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Divergence theorem with normal component of a curl to a surface

Let $\mathbf{A}$ be a vector function in $\mathbf{R}^3$ and we want to find the normal and tangent components of $\nabla \times \mathbf{A}$ on a smooth and closed surface $\Gamma$. $\mathbf{n}$ is the ...
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GIven a surface area F, a normal vector, a point and a vector field, calculate the flow of the vector field through F.

the surface F, the vector field v, the point P and the normal vector n are defined as follows: $$ F = \vec{x}(s,t) = \begin{bmatrix}s \\t \\s^2+t^2 \end{bmatrix} \ s,t \in [0,1]$$ $$\vec{n} = \begin{...
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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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Why is there a singularity when I apply integration by parts to $\int_\Omega f\Delta f = \int_{\partial\Omega} f\nabla f - \int_\Omega |\nabla f|^2$?

While I was trying to write a PDE methods-centric answer to this question I ran into the problem in the title. The original problem was (with new notation) to calculate $$\int_{\Bbb{R}^3\times\Bbb{R}^...
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Using surface integral to calculate the flux of vector field through two surfaces

I really would appreciate it if someone could help me with this problem below that I am having trouble with. The problem that I am trying to ask is down below, hyperlinked. Thank you. Here is what I ...
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The Divergence Theorem - Intuition behind relating divergence with flux

I'm currently a 2nd-year math major in college. After finishing Calculus 3 from the Stewart calculus book, I'm left wanting a deeper intuition on all of the vector calculus topics (Green's Theorem, ...
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If $\vec F=\frac{\vec r}{r^3}$ then $\exists$ is no $\vec G ∶ \mathbb{R}^3 ⧵ {0} \to \mathbb{R}^3$ such that $\vec F = curl \,\vec G$.

If $\vec F=\frac{\vec r}{r^3}$ then show that $\textrm{div} \vec F = 0$ but $\exists$ is no $\vec G ∶ \mathbb{R}^3 ⧵ {(0,0,0)} \to \mathbb{R}^3$ such that $\vec F = \textrm{curl} \,\vec G$. I can show ...
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Compute the flow of a vector field through a surface

Given the vector field $X(x,y,z)=(1,y,x)$, I am asked to compute the flow of this vector field through the northern hemisphere $(z\geq 0)$ of the elipsoid with equation $$\frac{x^2}{4}+\frac{y^2}{4}+z^...
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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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Applying Divergence theorem for 1-Torus

Suppose $f=f(t,x,v)$ with $t\geq 0, x\in\mathbb{T}^1, v\in\mathbb{R}$. Here, $\mathbb{T}^1$ is the one-dimensional torus. The torus is discretized into $N$ subintervals $X_i:=\left(x_{i-\frac{1}{2}}, ...
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Integrate a vector field over a open dome using the divergence theorem

I want to evaluate $$\iint_S\mathbf F\cdot d\mathbf S$$ where S is the top half of a unit sphere $S =\{(x,y,z): x^2+y^2+z^2\leq1, z\geq0\}$ and the vector field $$\mathbf F = \langle xz^2,\; \frac 1 3 ...
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Difficulty visualising surfaces and translating them into surface integral

I am reviewing old exam questions, and I am currently doubting everything that I have worked on for the past 3 weeks. The question is as follows [Solution is below, translated from Swedish] The ...
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Divergence theorem when $\nabla \cdot \vec{F} = 0$

Calculate the flow of the vector field $$\mathbf{F}(x, y, z) = \frac{1}{(x^2 + y^2 + z^2)^{\frac{3}{2}}} (x, y, z)$$ out of a sphere with radius $10$ and center at the origin. This what I did: $$\frac{...
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Divergence theorem in 3D

I'm having a very hard time with the following problem: Consider an area $D$ given by: \begin{cases} \frac{x^2}{4} + y^2 \leq 1\\ 0 \leq z \leq -x \end{cases} Let $\vec{F} = (y+e^y)\vec{j} + z(1-e^y)\...
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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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