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Questions tagged [vector-analysis]

Questions related to understanding line integrals, vector fields, surface integrals, the theorems of Gauss, Green and Stokes. Some related tags are (multivariable-calculus) and (differential-geometry).

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Vector Valued Integration

Can anybody suggest me books or references for the topic of VECTOR VALUED INTEGRATION? I need it asap. I looked in up the google, couldn't find it. I came across this topic as an application of ...
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Why is the tangential divergence the trace of the tangential gradient?

For a $C^1$function $\phi$ on a smooth surface $M$ one defined the tangential gradient $\nabla_M$ of $\phi$ by $$\nabla_M \phi(x):=(I-n \otimes n)\nabla \phi(x)$$ where $n$ is the unit normal and $\...
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1answer
15 views

Prove the diagonals of a parallelpiped bisect each other

I am stuck on how to Prove the diagonals of a parallelpiped bisect each other I have been given the hint to make one of the corners O. If possible I would just like a push in the right direction. ...
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Vector calculus problem involving planes

I'm working on a vector calculus problem (provided below) and the issue is I'm getting two different answers, and I'm not sure which is right. The question is as follows: Calculate the surface ...
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1answer
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How to find counter example to show that a vector field is not conservative? [on hold]

I know that to show it is not a conservation vector field I could find a closed parameterized path and then calculate the line integral such that it is not equal to zero, I've tried using a unit ...
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1answer
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How to find a scalar field, given its gradient?

Given that a vector field $\mathbf v$ satisfies $$\nabla \cdot\mathbf v =0.$$ How can I find $\phi(\mathbf r)$ such that $\mathbf v= \nabla \phi$?
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Is The Dirac Delta Communitive Over a Cross Product?

Is the statement, $\delta(\vec{x}) \vec{A}\times \vec{B} \equiv \vec{A}\times \vec{B}\ \delta(\vec{x}) $ True? If so, how would I go about proving this?
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Proof of Dirac Delta Sifting Property With Volume Integral

The Dirac delta function possess the sifting property which states, $ \int _{a}^{b} f'( x) \delta ( x-x') dx'=\begin{cases} f( x) & a< x< b\\ 0 & otherwise \end{cases} $ I suspect by ...
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Is Leibniz integral rule (basic form) allowed in this (physics) improper integral? Why?

Electric potential at a point inside the charge distribution is: $\displaystyle \psi (\mathbf{r})=\lim\limits_{\delta \to 0} \int_{V'-\delta} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'$ ...
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1answer
23 views

Finding gradient of a function

https://i.stack.imgur.com/1EIpR.jpg https://i.stack.imgur.com/O7OCP.jpg I have no idea how the answer is calculated, because there are some transpose matrix. I don't know i should treat them as ...
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Gradient field integrable

My question is, for a gradient field $G$ defined on $B_1 \setminus \{0\}$, can we show that $f \in L^1_{loc}(B_1)$ for $f$, s.t. $\nabla f =G$?
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Average of a vector field over a surface

Suppose we have a vector field $\mathbf{E}$ and a surface $S$. What is the formula for the average of $\mathbf{E}$ over the surface? My attempt: I think it is $\displaystyle\frac{1}{\text{area(S)}}\...
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Solving complex analysis problem without complex analysis

Consider the following simple problem. Let $f$ be a complex analytic function over $\mathbb{R}$ such that $f(x) in \mathbb{R}$ for each $x \in \mathbb{R}$. Prove that $f(z) = \overline{f(\overline{...
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2answers
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How to solve the derivatives of the compound functions in vector form?

for example: $f(x)=(xx^T)^{-\frac{1}{2}}x$, where $x \in \mathbb R_{+}^{1\times d}$ is a row vector. It is hoped that there will be specific theoretical basis (formula derivation and origin) (...
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1answer
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Let $G$ be a vector field defined in $A = \mathbb{R}^n -{0}$ with $\operatorname{div}G = 0$ in $A$.

Let $G$ be a vector field defined in $A = \mathbb{R}^n -{0}$ with $\operatorname{div} G = 0$ in $A$. let $M_1$ and $M_2$ be compact n-manifolds in $\mathbb{R}^n$, such that the origin is contained in ...
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2answers
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Value of $a,b,c$ that makes the vector field a gradient of a function

I am given vector field $F = (ay^2 + 2cxz)i + (ybx+ycz)j + (ay^2+cx^2)k$ and I am supposed to find values $a,b,c$ that would make $f(x,y,z)$ a function. I first took partial derivatives and I get the ...
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1answer
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in 2D dimensional plane, is it problematic to have Frenet-Serret frame with zero curvature?

I have a Frenet-Serret frame moving on a 2-D plane. As of now, I do not care about the binormal vector. So my equations are given by, \begin{align} \dot{T} = v\kappa N \\ \dot{N} = -v\kappa T \end{...
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Finding a potential function for $\vec F=\frac{-y}{x^2+y^2}\vec i+\frac{x}{x^2+y^2}\vec j$ on $\Omega=\mathbb{R}^2- \{(x,y)\,|\,y=0,\,x\geq0 \}$

In one of my Calculus III classes the professor presented the following vector field, defined on the set $S$ of all points $(x,y) \neq(0,0)$ : $$ \bbox[6px,border:1px solid black] { \vec{F}=\frac{-y}{...
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Conservative and non-conservative fields

I am currently taking multivariable calculus and mechanics classes. And both contain the concept of vector fields and their operators, so I am trying to truly understand the idea of a conservative ...
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1answer
41 views

derivate of a function of 2 variables

Can someone explain this to me : ($f$ is a differentiable function) why is $\frac{d}{dt} f(2,t) = ∇f(2,t) \cdot (0,1)$? This reminds me of the formula of directional derivate but it's not the same. ...
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Integral of gradient over all space

This came up to me while resolving an elaborate question about quantum mechanics. Assume there is a function $u : \mathbb{R}^3 \mapsto \mathbb{R}^+$ such that $\int_{\mathbb{R}^3} u(x,y,z)^2 dx dy dz =...
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Is this property of a set of vectors is independent of the basis choice?

Let $\{v_1,\ldots,v_{n}\}$ be a set of orthonormal vectors in $\mathbb{R}^n$. If these vectors have this property that for some basis $\{a_1,\ldots,a_{n}\}$, for any $j\in \{1,\ldots,n\}$, $$\sum_{i=...
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1answer
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Curl of implicit vector field?

I know I can check whether a given (continous-differentiable) vector field (with simple connected domain) is conservative by checking if its curl is zero. In the 2d case for example $$ \begin{bmatrix}...
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1answer
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How to differentiate $\frac{\partial}{\partial t} \psi(\vec r-\vec a(t))$ using the chain rule?

I am working on a homework problem that requires me to differentiate $$\frac{\partial}{\partial t} \psi(\vec r-\vec a(t)),$$ where $\vec r$ is the position vector and $\vec a$ is a vector that ...
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Explanation of Vector Area of surface of revolution

A solid shape ABCO is created by an anti-clockwise rotation of part AB of the parabola $\frac{z}{a} = 1−(\frac{x}{b})^2$, for which z > 0 and x > 0 (where a > 0 and b $\neq$ 0), about the z-axis by an ...
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How to prove that the normal line to a smooth surface an a point is independent of the function that defines said surface?

I need help with this problem: Prove that if $\mathbf{q}$ is a point of a smooth surface $S$ in $\mathbb{R}^m$, then the normal line to $S$ at $\mathbf{q}$ is independent of which smooth function $...
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2answers
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Prove that $\hat{\mathbf{F}}$ is a Conservative Field

If $\hat{\mathbf{F}}$ is a vectorial field defined by: $\hat{\mathbf{F}}(x, y) = \frac{1}{y} \hat{\mathbf{i}} - \frac{x}{y²} \hat{\mathbf{j}} $ Prove that $\hat{\mathbf{F}}$ is conservative. ...
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Vector Calculus Divergence

Prove that $$\nabla.\left(\vec{r}\,\nabla\left(\frac{1}{r^3}\right)\right)= \frac3{r^4}.$$ My doubt is that here should we consider the term $\vec{r}$ and $\nabla\left(\frac{1}{r^3}\right)$ as the ...
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1answer
192 views

Gradient of cross product of two vectors (where first is constant)

In some book about continuum mechanics I read that from principle of virtual work follows balance of rotational momentum when $\delta \boldsymbol{r} = \boldsymbol{\delta \varphi} \times \boldsymbol{r},...
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1answer
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Optimal algorithm to find maximum value of dot product of two lists

I am looking for an optimal (fastest) way to find a maximum value of dot product of two lists. A list can be rearranged freely in order to maximize the result. My idea for this was to quicksort two ...
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relationship between 2 vectors perpendicular to a particular vector $\bf1=[1,1,1,…]$

$\bf1=[1,1,1,...]$ 1) $v_1 \perp \bf1 $ 2) $v_2 \perp \bf1 $ What is the relationship between $v_1$ and $v_2$? Clearly, $\Sigma_i v_1^i=0$ and $\Sigma_i v_2^i=0$ from 1 and 2.
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1answer
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For vectors A and B, why is (A dot nabla)B treated differently from A(nabla dot B) in vector calculus identities?

I refer to such an identity as outlined here: If $\mathbf A(\nabla \cdot \mathbf B)$ was equal to $(\mathbf A \cdot \nabla)\mathbf B$ then all terms of the first line of this identity would cancel ...
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1answer
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Using the gradient to find directions of (general) increasement

The exercise says we should verify that $$f(x,y) = x^3 - xy^2 +y^4 $$ is increasing in the direction of (0,1) at the point (1,1) and decreasing in the direction (-1,0) at the point (1,1). I ...
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1answer
54 views

The differences between a vector with $n$ dimensions with a vector with $n+1$ dimensions whose $n+1$ element is zero?

Regarding a geometrical point of view, is there any differences between a vector with n dimensions with a vector with n+1 dimensions whose first n elements are identical to elements of the first ...
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1answer
26 views

Divergence of tensor times vector equals divergence of vector times tensor

Does the following equation hold? $\vec T := \vec T(\vec x)$ is a tensor field, $\vec v := \vec v(\vec x)$ a vector field: $\text{div} \vec T \cdot \vec a = \text{div}(\vec a \cdot \vec T)$ I think ...
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1answer
25 views

Green's theorem over an annulus

I need help with this problem: Verify Green's Theorem in the plane where $S$ is the annulus $\{(x,y)\in\mathbb{R^2}|a^2\leq x^2+y^2\leq b^2\}$ and $F(x,y)=\left(\frac{-y}{\sqrt{x^2+y^2}},\...
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How do i find the acceleration of a particle in circular motion?

How do i find the acceleration of a particle in circular motion? I can find out the component $r\alpha$ using differentiation, but how can i find both rectangular components of the acceleration vector?...
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How to translate from one plane to another?

Good afternoon all, I hope I posted in the right area which i felt closest to linear algebra more than chemistry. I do apologize if its not! With that being said: I am trying to get Molecule X (from a ...
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1answer
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Test for a conservative vector field

I have a vector $\mathbf{f}$ that satisfies \begin{align} \oint_C \mathbf{f} ds = 0, \end{align} for any smooth closed contour $C$. I believe this means that \begin{gather} \oint_C \mathbf{f} \cdot \...
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Express a specific function in terms of divergence of a field

Assume $x \in \mathbb{R}^3$ and a function $y(x): \mathbb{R}^3 \rightarrow \mathbb{R}$. I want to find a closed form of the function $F=F(y,\nabla y)$ such that $\displaystyle \nabla \cdot F=\frac{\...
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1answer
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Multivariable Calculus - Finding the flux across a vector field out of a hemisphere [closed]

Past Paper Question I had an exam today and this old past paper question came up again that I had previously tried. I followed the hints and got an answer of $4π$, with the flux across v out of D ...
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Simple proof of $\operatorname{Curl} F = 0$ then F is conservative

Apparently this is just a simple proof with prerequisites for the domain to be a star-like domain. See picture below for proof: I don't however understand why $\operatorname{Curl} F = 0$ means ...
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1answer
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Vector operator of second derivatives?

Given a vector function $\mathbf{A}=A_{x}\mathbf{i}+A_{y}\mathbf{j}+A_{z}\mathbf{k}$, does the quantity given by $$\mathbf{B}=\frac{\partial^{2}}{\partial x^{2}}A_{x}\mathbf{i}+\frac{\partial^{2}}{\...
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1answer
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Is it “Valid” to prove Stokes' Theorem with Green's Theorem?

In my Vector Calculus course, the professor is rigorous enough that we do a decent number of proofs, but not rigorous enough to go all the way with manifolds/differential forms/etc. One proof in ...
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1answer
35 views

General Solution to Laplace's Equation in $\mathbb{R^3}$

I am trying to find the Green's function for $\ \nabla^2\phi=S(x)\ $ for $\ x\in\mathbb{R^3}$ and express the general solution to Laplace's equation in $\mathbb{R^3}$. To find the Green's function, I ...
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1answer
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Reference for proof of theorem on directional derivative

I require a reference for the following theorem, which is stated without proof in Matrix Groups for Undergraduates by Kristopher Tapp. In what follows, let $p \in U \subseteq \mathbb R^m$, $v \in \...
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1answer
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from cartesian coordinates to polar coordinates

The gradient is known as $\nabla u=(u_x,u_y)$ . Let $x=r\cos(\phi),\ y=r\sin(\phi)$. Does it then become $(u_r,\frac{1}{r}u_\phi)$? I have given that $u(r\cos(\phi),r\sin(\phi))=r^3\cos(\phi)$. From ...
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A vector calculus formula

Let $A, B$ be vector fields in $\mathbb R^3$. We have $$ \text{curl}\bigl((A\cdot \nabla)B\bigr)=(A\cdot \nabla)\text{curl}B -((\text{curl}A)\cdot \nabla)B+R(A,B). $$ I know that $R(A,A)=0$ and I ...
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113 views

Integrate 2-Form Along Loop to get 1-Form?

I would like generalize the following theorem from Calc III: "If $U$ is an open, connected subset of $\mathbb{R}^2$, $\omega$ is a 1-form on $U$, and $d\omega = 0$ on $U$, then TFAE 1) $\omega = df$ ...
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2answers
22 views

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 ...