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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.

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Show that $\int\int\limits_A\left(\frac{\partial f}{\partial x}+ \frac{\partial g}{\partial y}\right) dx dy=0$

Let $\dot x= f(x,y)$ and $\dot y=g(x,y)$ be continuously differentiable vectorfield on $\mathbb R^2$ with flow $\varphi_t$. Show that $\int\int\limits_A\left(\frac{\partial f}{\partial x}+ \frac{\...
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Integration by parts for multivariable functions using the Divergence Theorem

I came across the following formula used in a calculation of the distributional derivative (though the formula itself is not really to do with distributions), which apparently follows from the ...
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Need to take limit of divergence in spherical coordinates on the $\theta=0$ axis

The title pretty much says it all. I need to take the standard expression (one of the standard expressions, I guess) for the divergence of a vector and of a rank-2 tensor in spherical coordinates and ...
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Inner product of a vector field and gradient - Adjoint of the gradient

On page 9 in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.639.5952&rep=rep1&type=pdf it is being shown why the negative divergence is the adjoint of the gradient. $V: \mathbb R^n \...
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Showing convergence of an infinite series?

I'm trying to show whether the below series converges/diverges, and I have very little clue on how to do it. I know about the Comparison Test, but I can't think of a sequence $b_n > a_n$ to perform ...
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$ \lim_{a\to 0^{+}} \int_0^{\infty} \cosh^n{\theta}e^{-a\cosh{(\theta)}}d\theta$ with $n$ a positive integer

I am sure that $$\lim_{a\to 0^{+}} \int_0^{\infty} \cosh^n{(\theta)}e^{-a\cosh{\theta}}d\theta,$$ with $n$ a positive integer, diverges; but I don't know how to demostrate this affirmation. I tried to ...
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Relation between vector surface integral and volume integral

I tried to solve the surface integral of $I+j+k$ over a hemisphere $x^2+y^2+z^2=1$ , but when I solved it using Gauss divergence theorem it comes to be 0 ? If one could explain.
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Convergence of $\sum_{n=1}^\infty \ a_n$ when $a_n=f(\frac 1 n)$ [duplicate]

I need help with this problem:  Let $f$ be a continous function over an interval that contains $0$. Let $a_n=f(\frac 1 n)$ (for $n$ large enough). I've already showed that: If $\sum_{n=1}^\infty \ ...
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An example of divergence in spherical coordinates

I've found the following example in a vector calculus book: the divergence of the vector field $\vec F(x,y,z) = x\vec i + y\vec j - z \vec k$ in spherical coordinates is $$ \nabla \cdot \vec F(\rho,\...
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Convergence of $\sum_{n=1}^\infty \ a_n$, $a_n = f(\frac 1 n)$

I need help with this problem: Let $f$ be a continous function over an interval that contains $0$. Let $a_n = f(\frac1 n)$ (for n large enough). Show that if $\sum_{n=1}^\infty \ a_n$ ...
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Looking for explanation to solution - divergence theorem, differential equations

What is the volume of the domain to which the cube $|x| < 1, |y| < 1, |z| < 1$ move to after 20 time units, when its motion is governed by the system $\begin{cases}\dot x = \cos(x+y+z) \\ \...
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I need help using the Divergence Theorem to calculate a cubic volume

Problem: Calculate the left and right hand sides of Gauss' Divergence Theorem explicitly for a cubic volume: u(r) = (u_0 + u'x_1)\delta_1 for a cube defined by 0 \leq x_1, x_2, x_3 \leq L Also I can'...
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divergence theorem for a rank 2 tensor

I wanted to double-check on something. We all know that for a vector field $T$: $\int dV \nabla \cdot T = \int d\hat{n}\cdot T$ My understanding is that this is still true if $T$ is a rank-2 tensor,...
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Convergence and divergence of $\sum_{n=2}^\infty {1\over{\sqrt[3]{n^2-1}}}$ and $\sum_{n=1}^\infty {1\over{\sqrt[3]{n^2+1}}}$

I need to decide if the next series are convergent or divergent, using the comparison test, the quotient test, the integral test or the Leibniz Theorem: $$\sum_{n=2}^\infty {1\over{\sqrt[3]{n^2-1}}}$$...
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I need to prove a few vector identities using Cartesian Tensor Notation, and I can't figure out how!

I have been all over the internet, but I just can't make sense of this stuff. I have done my best to learn from my textbook and different websites, but this is confusing for me. I haven't taken any ...
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Show $a_n$ is unbounded if $a_n= a_{n-1} \left(1+ \frac{1}{\sqrt n}\right)$ to determine that $a_n$ diverges.

I wish to determine the limit of $a_n$, where we recursively define: $$a_n= a_{n-1} \left(1+ \frac{1}{\sqrt n}\right)$$ where $a_0=1$ I already noticed it is increasing, because $$a_n - a_{n-1}= \...
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Diverging Integral with Bessel Function

I am looking for the solution to the integral: $$\int_{a}^{\infty} x J_n(\alpha x)\;dx$$ where $a<< \alpha$ and $n$. I get something out of Mathematica for $a=1,2,0.1...$ in terms of the ...
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Is there a preference if one of the functions in convolution of Mellin transform is divergent?

The convolution of Mellin transform is $$ \sigma \left( x \right) = \int _x^1 f \left( \epsilon \right) h \left( \frac{x}{\epsilon} \right) \frac{1}{\epsilon} \mathrm{d} \epsilon , $$ if both $f \...
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How to Find the integral $\int_{S}−(xy^2)~dydz + (2x ^2 y)~dzdx − (zy^2)~dxdy$ where is the portion of the sphere $x^2 + y^2 + z^2 = 1$

Find the integral $$ \int_{S}−(xy^2)~dydz + (2x^2y)~dzdx − (zy^2)~dxdy $$ where $S$ is the portion of the sphere $x^2 + y^2 + z^2 = 1$ above the plane $z=\frac{1}{2}$. Choose the direction of ...
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Vector calculus identities using Einstein index-notation

I have a problem proving these formulas using Einstein index notation. The formulas are: 1) $$\nabla(r^n)= nr^{n-2} \vec{r}$$ 2) $$\nabla \cdot (\nabla g \times \nabla f)=0$$ 3) $$\nabla \times (\...
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How to calculate the flux through complicated Surface

How to Find the integral $\int_{S} 2~dydz + 1~dzdx - (3x)~dxdy$ Where $S$ is the surface : $x^2 + 2y^2 + 3z^2 + xyze^{(x+y+z)sin(x^2 -y+z)} = 1$ $0\leq x,y,z $ i think that i must use Gauss ...
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Problem arising while calculating surface integral by taking projection.

I was asked to verify the divergence theorem for $$\vec{A}=4x\hat{i}-2y^2\hat{j}+z^2\hat{k}$$ taken over the region bounded by $$x^2+y^2=4,z=0$$ and $$z=3$$. One part ($$\iiint\nabla.\vec{A}dV$$) is ...
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Not possible to find non-zero terms of series expansion?

I've been asked to compute the first 3 nonzero terms of a power series expansion about x=0 for two linearly independent solutions to the ODE: $$(1+x^3)y''- 6xy =0 $$ I have tried to solve this many ...
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Divergence for Series with $|a_n| \ge b_n \ge 0$

first of all i hope you get my point since English is not my native. I know the series $$ y = \sum_{k=1}^\infty \frac{1}{k} $$ is divergent. Now I want to check if the series $$x=\sum_{k=1}^\infty ...
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How to calculate flux of vector field

A vector field is given as $A = (yz, xz, xy)$ through surface $x+y+z=1$ where $x,y,z \ge 0$, normal is chosen to be $\hat{n} \cdot e_z > 0$. Calculate the flux of the vector field. I tried using ...
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Infimum of $f(\theta)= -\sum_{i=1}^n (y_i - \alpha p(y_i|x;\theta)) \ln p(y_i|x) $

Suppose $y_1 = 1, y_2 = y_3 = ...=y_n = 0\ (n\geq2)$, and $\sum_{i=1}^np(y_i|x;\theta) = 1$, $0\leq p(y_i|x;\theta)\leq1$. Meanwhile $\alpha > 0$ is a constant. Let's define a function $$f(\theta)...
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Flux. Multivariable Calculus [closed]

Find the flux of $F(x, y, z) = \langle-x, -y, z^3\rangle$ through the surface $S$ when $S$ is the part of the cone $$z =\sqrt{x^2 + y^2}$$ that lies between the planes $z = 1$ and $z = 3$, oriented ...
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An intuitive explanation for Green theorem and Divergence theorem

As my vector calculus exam is getting closer, I'm looking for intuitive ways to think about the different theorems we have to memorize. I think I have found a pretty intuitive way to think about the ...
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If the gradient of a vector is zero, does that imply that the laplacian of the vector is a null vector?

Suposse $\nabla \cdot \vec{u} = 0$ Does that imply that $\Delta \vec{u} = \vec{0}$ Thank you!
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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 ...
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Bregman projection

Given a convex body $K$ and a point $y$ outside the convex body (in the ambient space), the Bregman projection of $y$ , with respect to the regularizer $R$, is defined as $x=\rm{argmin}\{B_{R}\left(\...
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$ \int_{S_{c}} \underset{\bar{}}{F}.d \underset{\bar{}}{S}\;$ where $S_{c}$ is the curved part of the truncated cone.

$ \int_{S_{c}} \underset{\bar{}}{F}.d \underset{\bar{}}{S}\;$ where $S_{c}$ is the curved part of the truncated cone $x^2 + y^2 = (a-z)^2 , 0≤z≤(1/2)a$ Where $\underset{\bar{}}{F}$ = $(2x + y^2)\...
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A direct proof of the surjectivity of divergence operator

The main question that I want to solve is that: Let $\{X^k\}_{k=-\infty}^\infty$ be a sequence of (Banach) spaces which are subsets of distribution space $\mathcal D'(\Omega)$ where $\Omega\subset\...
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Convergent or divergent series? Asymptotic of the series?

\begin{align} A&=\sum_{n\ge 0} \frac{(-1)^n}{n!} \frac{b^n}{(n+1)^3}\\ &= \sum_{n=0}^{\infty}\frac{(-1)^{n}b^{n}}{n!}\frac{1}{(1+n)^{3}} = 1\,{}_{3}F_{3}(1,1,1;2,2,2;-b) \end{align} The series ...
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Divergence theorem for inner product of Tensor

Let $\Omega\subset \mathbb{R}^3$ be bounded domain and $f$ be smooth vector field on $\Omega$ and $f|_{\partial\Omega=0}$. I not able to compute $\int _{\Omega} grad~f:(grad~f)^{T}$, where $:$ is ...
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Divergence theorem of product

Len $\Omega$ be bounded domain in $\mathbb{R}^3$ with smooth boundary, $f$ smooth vector field on $\Omega$ with $f=0$ on $\partial \Omega$. How to use divergence theorem for $\displaystyle\int_{\Omega}...
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Intuition of definition of divergence

Intution : The divergence of a three-dimensional vector field is the extent to which the vector field flow behaves like a source at a given point. But if my vector field is $F=\langle P,Q,R\rangle$ ...
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Find vector flux of $v=(yz,y^2z, yz^2)$ through the surface of the cylinder $x^2+y^2=1, 0 \leq z \leq 1$

I'm given the following vector field : $v=(yz,y^2z, yz^2)$ and I need to find its flux through the cylinder $x^2+y^2=1, 0 \leq z \leq 1$. I don't have solutions to this exercise, so I don't know if my ...
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Finding the normal of manifold in $\mathbb{R}^3$ (Divergence theorem)

Calculate $\oint_\gamma f(X) dX$ where $f(x,y,z) = (yz,xz,xy)$, $\gamma = \{X(t)| 0 \le t \le 2\pi\}$, $X(t) = (cos(t)cos(\frac{\pi}{8} + \frac{t(2\pi-t)}{4\pi}),sin(t)cos(\frac{\pi}{8} + \frac{t(2\...
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How to find the area of a cardioid using multivariable calculus?

So I have the following three questions (and no solutions to them sadly). I wanted to know if my results for a) and b) are correct and how I should proceed for c). Given is the following ellipsoid : ...
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Divergence in non cartesian coordinates

Suppose I have some vector field $X$ in $\mathbb{R^3}$ and a submanifold $M$ such that $\dim(M)=2$. I want to calculate the flux of $X$ through $M$, $\Phi_X(M)$. I would have two ways to do it, either ...
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36 views

Divergence of a parametrized vector field

Here's the problem: Use Gauss's divergence theorem to calculate the total flux through the solid,$V$, enclosed by the set $M=\{(x,y,z)\in \mathbb{R^3}: z=x^2+y^2, 0\leq z\leq 1\}$ and the vector field ...
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Divergence theorem over non-smooth manifolds

The divergence theorem on smooth surfaces is well understood and changes the integrand from div(X) to dot(X,n) where n is the normal of the boundary of the surface. Suppose my surface is peicewise ...
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Euclidean Norm of Curl / Divergence

Are there any general statements of the curl or the divergence of a 3-dimensional vectorial function, e.g. for the magnetic field: $$|\nabla\times\boldsymbol B\left(t,\vec{x}\right)| = ?$$ $$|\nabla\...
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Find the “region of interest” of an unknown function

Given an unknown function $f:\mathbb{R} \rightarrow \mathbb{R}$, is it possible to find it's region of interest? By that I mean either the range in which $f$ does not converge or diverge, e.g. $f(x)=(...
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Divergence Theorem - Cone

Here's the question: Evaluate the surface integral $\iint _S F\cdot n \space dA$ by the divergence theorem. $ \mathit F = [xy, yz, zx]$, S the surface of the cone $x^2 + y^2 \le 4z^2, \space \space 0 ...
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38 views

Taking divergence of the gradient of a scalar field that depends only on the position vector in $\mathbb{R}^3$.

The scalar field $f$ depends only on $r=|\underline{\mathbf{r}}|$ which is the position vector in $\mathbf{R}^3$ and I need to calculate the quantity, $$\nabla \cdot \nabla f$$ i went about ...
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Intrinsic definition of Jacobian matrix on manifolds

For a vector field $X:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, the Jacobian matrix at $p\in\mathbb{R}^{3}$ is defined as $$\mathcal{J}_{p}X:=\begin{bmatrix}\left.\frac{\partial X^{i}}{\partial x^{j}}\...
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39 views

Surface integral of curl

Let $\vec F(x,y,z)=(x^2+y^2+\sin(xy)$, $e^x$+$2xy, -yz$). Calculate the flow of $\nabla\times\vec F$ through the following surface: $$ S=\{(x,y,z) x^2 + 2y^2 + 3z^2=10, y\ge 0\} $$ I figured that if ...
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51 views

Divergence and Curl

I was given a function $F(x,y,z)=(z^c,x^c,y^c)$ and asked to find divergence and curl. My initial answer was $0$, but i don’t think that’s right. I noticed the brackets weren’t the typical $\langle,\...