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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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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: \begin{equation} 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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Does $\sum_{i=2}^\infty \frac{1}{(\ln(n))^2}$ converge or diverge?

I've tried, the limit comparison test with several values and have tried finding some values for the direct comparison test but nothing really concrete has come out of it. $$\sum_{i=2}^\infty \frac{...
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Why curl of a vector field that is proportional to $1/r^2$ equal to $0$?

The curl of the vector field ${\bf F} = (-y {\bf i} + x {\bf j})/(x^2 + y^2)$ is $0$. I have an intuitive understanding of why the divergence of a radial field that is proportional to 1/r^2 is equal ...
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question about convergence and divergence of $\sum (n-2)^3\,e^{-n(x+2)}$

I have the problem that I could not see my fails in the my calculations and why does exists just one right way of showing the convergence or divergence of that formula. the first version is the ...
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divergence of gradient of scalar function in tensor form

I found simple expression in tensor notation for a divergence of product vector and gradient of scalar function: $$\operatorname{div}(\mathbf{j}) = 0 \text{, where } \mathbf{j} = \mathbf{m}\times \...
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Divergence of inverse square field (2D vs. 3D)

Let $\displaystyle\mathbf{v}=\frac{\mathbf{\hat{r}}}{r^2}$. Compute its divergence. My attempt: I found that $\nabla\cdot\mathbf{v}=0$ in the 3D case, in accordance with Gauss' law, but $\nabla\...
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Help on comparison test / determining Bn

This question has me stumped. I was thinking of using the comparison test, then using the integral test on bn. But I can't even figure out what to make bn in this particular situation as the form of ...
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$\Delta u = f , \operatorname{div} f=0 \Rightarrow \operatorname{div}u=0$ on non convex domain.

I am specifically referring to this paper and why equation (7.7) is the weak formulation of (7.6). My question is why $ \operatorname{div} u=0$ is implied by formulation (7.7) on a general domain. If ...
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Find a new distribution based on known distributions

Suppose there are $k$ Normal distributions $N_1,N_2,\dots,N_k$. Could you tell me which algorithm can help me to find a new Normal distribution $N_{new}$ where $N_{new} = minimize KL[N_{new}||N_1] + \...
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Why does $\frac{x^{2}}{1+x^{2}}$ diverge?

Why does $\frac{x^{2}}{1+x^{2}}$ diverge? I am trying to show that the integral $2\int\int_{\mathbb{R}^{+}}\frac{x^{2}}{(1+x^{2})(1+y^{2})}dxdy$ diverges. The solution says it diverges because $\frac{...
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About the definition of divergence

The divergence is defined as: $\nabla . \mathbf{A}=\lim \limits_{V \to 0} \dfrac{ \unicode{x222F}_{\partial V} \mathbf{A}.d\mathbf{S}}{V}$ My question is of two parts: $(1)$ If we are using ...
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Divergence of a sequence with floors and square roots

Is there any specific approach to prove the divergence of a sequence? For example, I have this problem: "Prove that the sequence $(a_n)_{n \ge 1}$, where $a_n = n\sqrt2 - [n\sqrt2] + n\sqrt3 -[n\...
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Flux integral through tricky surface.

Let $T$ be the area lying in the first octant where $x\geq0,y\geq0,z\geq0$ limited by the surfacs $z=a^2-x^2$ and $y=a^2-x^2$. Calculate $\iint_S \vec{F}\cdot\hat{N}dS$ where $\vec{F}=(x,y,z)$ for $(...
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A short way to determine the divergence of integral

The integral $$\int_0^1 \left| \frac{1}{x}\cos\left(\frac{1}{x}\right) \right| dx$$I can use the subsets in the form $$\left[ \frac{1}{(2k+7/3)\pi} , \frac{1}{(2k+5/3)\pi}\right]$$ where the cosine ...
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Volume Preserving Flow after Variable Change of Transport Equation

Suppose that we have the transport equation with non-consant coefficients in divergence from $$\partial_t f(t,x,\xi) + \nabla_{x,\xi}\cdot (F(x,\xi) f(t,x,\xi)) = 0 \\ f(0,x,\xi) = f_0(x,\xi)$$ where ...
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Prove that a quadratic form is harmonic if and only if its integral over a sphere is $0$.

Lef $F:\mathbb{R}^n\rightarrow \mathbb{R}$ be a quadratic form. I have to prove that $F$ is harmonic ($\Delta F=0$) if and only if $$\int_{\mathbb{S}^{n-1}(1)}F|_{\mathbb{S}^{n-1}(1)}dv_{g_0}=0.$$ ...
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Where does this proof of convergence fail?

Given the series, $$\sum_{n=1}^{\infty} (-1)^{n}\frac{n}{n+1}$$ I know we can immediately conclude that it is obviously divergent by the divergence test. But I want to know where exactly am I going ...
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Possible values of Radius of Convergence based on divergent and convergent x's [closed]

I am having problems with the following exercise: Given the Maclaurin series $\sum_{n=0}^{\infty} c_n x^n $ is divergent for $x=-6,8$ and convergent for $x=-4,6$. What are the possible values for ...
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Application of the Gauss's Divergence Theorem

Question: Let $S$ be the closed surface forming the boundary of the region $V$ bounded by $x^2+y^2=3$, $z=0,\ z=6$. A vector field $\vec{F}$ is defined over $V$ with $\nabla.\vec{F}=2y+z+1$. What is ...
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Applying divergence theorem to cylinder

Let $D$ be an open subset in the $(x, y)$ plane with smooth boundary and let $X = (X1, X2)$ be a continuously differentiable vector field on $\bar{D}$. Make a solid cylinder $C$ by $C = \left \{ (...
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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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Divergence of a vector field in cylindrical coordinates

Let $\bar{F}:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a vector field such that $\bar{F}(x,y,z)=(x,y,z)$. Then we know that: $$\nabla\cdot\bar{F}=\frac{\partial\bar{F}_x}{\partial x}+\frac{\partial\bar{...
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Interface conditions on electromagnetic fields

Several authors (such as Jackson in his book "Classical Electrodynamics") state the following conditions at an interface between two different media: $(\vec{D_2} - \vec{D_1})\cdot \vec{n} = \sigma$ ...
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Is this acceptable notation? (Partial Derivatives)

I have a question about the notation which I have decided to use. I'm writing lecture notes for my own Calculus course and I've introduced notation such as $[R_x-Q_y]_x$ and so on. It's supposed to ...
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Integration by parts for tensor fields on Riemannian manifold

I'm working on the following exercise in my Riemannian manifolds book: Suppose $M$ is a compact, oriented Riemannian manifold with boundary. Show that if $\omega$ is any $k$-tensor field and $\eta$...
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Proof related to divergence theorem

The exercise asks me to prove that if $u:D\cup\partial D \rightarrow \mathbb{R}^{2}$ is ${C}^{2}$ on $D$ and we define $B_{\rho}$ a circle of radius $\rho < R$ such that $B_{\rho} \subset D$. Then ...
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What's the intuition behind the shear not being possible to be obtained with exterior differentiation?

Allow me to present some context, first. If we have a $K$-vector space $E$ and a non-degenerate symmetric bilinear form (a pseudo-riemannian metric) $g$, then we can project any 2nd order tensor (the ...
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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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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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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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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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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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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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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 can I find the integral $\int_{S} 2~dydz + 1~dzdx - (3x)~dxdy$, where $S$ is the following 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 ...
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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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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 ...