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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5
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96 views

How to solve $\mathrm{diag}(x) \; A \; x = \mathbf{1}$ for $x\in\mathbb{R}^n$ with $A\in\mathbb{R}^{n \times n}$?

I would like to solve the following equation for $x\in\mathbb{R}^{n}$ $$\mathrm{diag}(x) \; A \; x = \mathbf{1}, \quad \text{with $A\in\mathbb{R}^{n\times n}$},$$ where $\mathrm{diag}(x)$ is a ...
5
votes
0answers
51 views

Inhomogeneous Wave Equation in 3 dimensions

From section 7.5 in this source, I see that, for $\vec{x} \in \mathbb{R}^3$, if $$ \frac{\partial^2 u\left(\vec{x},t\right)}{\partial t^2} - \nabla^2u\left(\vec{x},t\right) ...
5
votes
0answers
110 views

vector field as integral

Define a vector field $ \vec{f}(\vec{R}) = \oint_C{|\vec{r} - \vec{R}|^2 d\vec{r} }$ where C is a simple closed curve. show that there are constant vectors $ \vec{P} $ and $ \vec{Q} $ such that $ ...
4
votes
0answers
105 views

$\operatorname{div}$ and $\operatorname{grad}$ in spherical coordinates. Formula from general relativity goes crazy

I try to calculate the gradient of a function and the divergence of a vector field in spherical coordinates. Nothing special so far, but a formula that I learned in a general relativity lecture ...
4
votes
0answers
129 views

Integrate by part for vector analysis

I am trying to solve problem 1.15 in Jackson's Classical Electrodynamics (3rd Edition). I found a bunch of solutions (proofs) to the problem, one of which shows: $$ \begin{align*} \delta W ...
3
votes
0answers
26 views

normal of a function

Function $$\mathbf{r}(u,v)=(2u+2v)\mathbf{i}+(-3+v^2)\mathbf{j}+(2u^2)\mathbf{k}$$ is a parametrization of a surface. What's the normal vector of this surface at the point $(u,v)=(1,-2)$. What I got: ...
3
votes
0answers
57 views

Integral curves in neighborhood of a minimum point of the potential

Let $X: \mathbb R^3 \rightarrow \mathbb R^3$ be a smooth vector field, and $f: \mathbb R^3 \rightarrow \mathbb R$ be its potential (i.e. $X= grad\ f$). When $f$ has a minimum point $p_{min}$ and all ...
3
votes
0answers
81 views

Far-field Poynting vector for time varying source [check my math please]

This is a question from classical electrodynamics, but it's the maths of it I need some help in. I have fields $\rho\left(\vec{r},t\right)$, $\vec{J}\left(\vec{r},t\right)$, ...
3
votes
0answers
64 views

How can i shape a Möbius Strip into another curve?

The normal parametrisation of the Möbius strip: $$\begin{align} x(u,v) &= (1+\frac{v}{2}\cos(u/2))\cos(u) \\ y(u,v) &= (1+\frac{v}{2}\cos(u/2))\sin(u)\\ z(u,v) &= ...
3
votes
0answers
59 views

Can I solve for a unique integral kernel?

Consider, for $\mathbf{v},\mathbf{w} \in \mathbb{R^3}$, $$ f(\mathbf{w}) := \int K(\mathbf{w,\mathbf{v}}) g(\mathbf{v}) \, d\mathbf{v} \, .$$ Is it possible to solve for the integral kernel, ...
3
votes
0answers
648 views

Three-dimensional vectors and force systems

Full disclosure: this is a homework problem. However, I find myself stuck in the middle. The problem is below As shown, a system of cables suspends a crate weighing W = 350 . (Part C 1 figure) ...
3
votes
0answers
68 views

Reference of explanation of this result about Fourier transforms in euclidean space

The paper says that since $$u = - \nabla \times (\Delta^{-1} \omega)$$ "The Fourier transforms of $\nabla u$ and $\omega$ satisfy $(\nabla u)^{\text{^}}(\xi) = S(\xi)\hat\omega(\xi)$ where $S$ is a ...
2
votes
0answers
70 views

Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 $$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$ \Delta w + \beta w < 0. $$ for some $\beta \in \mathbb R$. ...
2
votes
0answers
30 views

Stone-Weierstrass Approximation Theorem for Vector-Valued Functions?

A classical result in analysis is the Stone-Weierstrass approximation theorem: Theorem. Let $X$ be a compact Hausdorff topological space and $(C(X,\mathbb C),\|\cdot\|_{\infty})$ denote the space ...
2
votes
0answers
48 views

Line integral - Stokes's theorem

Let $\gamma_1$ be the intersection curve between the surface $x^2+(y+z)+z^2=1$ and the plane $z=x$ and let $\gamma_2$ be $\gamma_1$ with $x\geq0$,$y\geq0$. Calculate ...
2
votes
0answers
39 views

Is there a way besides integration by parts to solve this integral?

$$\int_{0}^{2\pi} -10\cos^9(t)\sin^4(t)t^4\,dt$$ Maybe a formula for this form or something?
2
votes
0answers
45 views

Green Identities via Differential Forms

What is the actual meaning of the Green identities: Is there a picture/geometric interpretation of these, as well as intuition going beyond the usual integration-by-parts meaning associated to ...
2
votes
0answers
60 views

Green's Theorem

Hey guys I am having difficulties in problem 5. I thought I understood it, but I suppose I was mistaken. I will now explain what I planned to do to solve this problem and where I got stuck. So I ...
2
votes
0answers
32 views

A vector analysis question requiring multivariable calculus

The question is taken from a Vector-Analysis worksheet as an extension exercise, here is the first part: Let $\underline{v}$ be a vector field $\mathbb{R}^{2} \backslash (0,0)$ of the form ...
2
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0answers
44 views

Imaginary part of Log laplacian

I'm confused about how to calculate $\nabla^2 \log z$, where $z=re^{i\theta}$ is a complex number. My calculations return $$ \nabla^2 \log z = 2\pi\frac{\delta(r)}{r} [\delta(\theta) + i ...
2
votes
0answers
69 views

Stokes Theorem (Application)

Statement: If a vector field R is irrotational then a line integral is independent of path. Proof. Let $\nabla$ $\times$ $\vec A=0$ in $R$ consider the difference of two line integral from the point ...
2
votes
0answers
108 views

A nabla operator identity in ortho-normal coordinate systems

Let $u_1,~u_2,~u_3$ be components of the position vector $\vec{u}$: $$\vec{u}=u_1 \vec{e}_{u_1}+u_2 \vec{e}_{u_2}+u_3 \vec{e}_{u_3}$$ in an ortho-normal coordinate system (not necessarily Cartesian) ...
2
votes
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50 views

field lines terminating at infinity

A dipole consists of two equal and opposite point charges separated by a fixed distance. With two exceptions, all the electric field lines begin on one charge and end on the other. In the two ...
2
votes
0answers
68 views

Notation of a vector field

Usually vector field looks like $\sum_{i} a_i(x_1,x_2,...x_n) \frac{\partial}{\partial x_i}$, also it is not a problem to write it like that $\sum_{i}a_i\partial_i$ or even $a^i\partial_i$ using ...
2
votes
0answers
117 views

Proof of the Laplace-Beltrami operator

Consider the local representation of the Laplace-Beltrami operator on a 2-dimesnsional manifold (immersed in $\mathbf R^3$) with Riemannian metric $(g_{ij})$. Please, I want help in showing that: ...
2
votes
0answers
97 views

Changing parametrization with regard to arc length - is it worth it?

If you have found a parametrization $\vec \alpha(s)$ of a curve $C$ for which $\int \lVert \vec \alpha\,'(s)\rVert \mathrm ds$ cannot be expressed in terms of elementary functions, does it make sense ...
1
vote
0answers
39 views

find the angles of a given vector sum

Assume you have n vectors in 2D space, with different fixed magnitudes $l_i$. The problem is to find the angle of each vector such that vector sum is a specific vector. That is, $\sum l_i \cos ...
1
vote
0answers
13 views

Formula of flux-Single/Double integral

The flux of $\overrightarrow{F}$ at a surface $S$ is given by the formula: $$\iint_S \overrightarrow{F} \hat{n} d \sigma$$ but the flux at a curve is given by the formula: $$\int_A^B ...
1
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0answers
19 views

Vector Surface Integral problem finding ds

Question: Evaluate $∫F$.Nds where $F = 2x^2y \hat{\imath} -y^2 \hat{\imath} + 4xz^2 \hat{k}$ and $S$ is the closed surface of the region in the first ocant bounded by the cylinder $y^2+z^2 = 9$ and ...
1
vote
0answers
45 views

Questions about the divergence theorem

I am looking at the proof of the divergence theorem and I have some questions. The proof of the divergence theorem $$\iiint_D \nabla \cdot \overrightarrow{F} dV= \iint_S \overrightarrow{F} \cdot ...
1
vote
0answers
26 views

What vector theorem should be used?

I have to integrate this line integral $$\int_C \mathbf F \cdot d\mathbf r$$ Where $\mathbf F = (\frac{y}{x^2 + y^2},\frac{-x}{x^2 + y^2})$ and $C$ is the curve $x^2 + 2y^2 = 1$ oriented ...
1
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0answers
37 views

Proof of Green's Theorem

I am looking at the proof of the Green theorem. To show that $$\oint _S (Mdx+Ndy)= \iint_R \left( \frac{\partial{N}}{\partial{x}}-\frac{\partial{M}}{\partial{y}} \right) dxdy$$we do the following: ...
1
vote
0answers
14 views

Why is there a minus sign ($-$) , before the integral $\int_{S_2} M dx$?

I am looking at the proof of the Green theorem. $$S_1: y_1=f_1(x), a \leq x \leq b$$ $$S_2: y_2=f_2(x), a \leq x \leq b$$ $$\iint_R \frac{\partial{M}}{\partial{y}}=\int_{f_1(x)}^{f_2(x)} \int_a^b ...
1
vote
0answers
27 views

How to estimate/determine surface normals and tangent planes at points of a depth image (point cloud)?

I have depth image, that I've generated using 3D CAD data. This depth image can also be taken from a depth imaging sensor such as Kinect or a stereo camera. So basically it is depth map of points ...
1
vote
0answers
25 views

Conditions of the vector-calculus Green's theorem

While studying the Green's theorem, I think a lot about whether there is an abundance of the condition (now called $X$) of $X:\quad$ $P$ and $Q$ (in the Wikipedia article, $L$ and $M$) have ...
1
vote
0answers
16 views

Defining divergence of vector field

Curl is defined (in the plane) by imagining a wheel of radius $\epsilon$ placed in $(a,b)$. Denote the region enclosed by the boundary of the wheel with $D_\epsilon$. Let's suppose our vector field ...
1
vote
0answers
34 views

derivation of divergence from nabla operator

For a two dimensional orthogonal curvilinear coordinate system $(t_1, t_2)$, we have the position vector $r$, where $h_i = | \frac{\partial r}{\partial t_i} |$ are the scale factors and $a_i$ are the ...
1
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0answers
36 views

Field of vector fields

For every point $A$ outside a sphere with radius $a$, there's a field $$F= \frac{K}{r^4d^2} $$ where $r$ is distance between point $A$ and the center of the sphere, and $d$ is distance between point ...
1
vote
0answers
24 views

Interior Products

Over on the Wiki page for interior products: http://en.wikipedia.org/wiki/Interior_product There is a line that says $\iota_X \alpha = \alpha(X) = \langle \alpha,X \rangle$ where $\alpha$ is a ...
1
vote
0answers
22 views

Integrate a divergence-free vector field

Suppose we are given a vector field $\overrightarrow{B}$ in $\mathbb{R}^3$ whose divergence is zero : $div(\overrightarrow{B})=0$. We want to find $\overrightarrow{A}$ such that ...
1
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0answers
27 views

Elementwise normal to vector of unknowns and non-defined matrix multiplications

I wonder about statement (1). It is given that u is a column vector, A and B are constant, symmetric, square matrices of such size that the expressions on the left hand side of (1) are well defined. ...
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0answers
13 views

Differentiation of vector-function

Let $f(x) = e^{-x^Tx},$ where $x \in \mathbb{R}^n$. What will be the second derivative? The first is $~f'(x) = 2x^T e^{-x^Tx}$, and when I try to find the second, I confuse. It will be $$f''(x) = ...
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0answers
40 views

Question about vector field

If I want to draw the lines of force to a vector field F and consider whether any results are physically reasonable, then i can start with to solve the differentiable equation $$ y'(x)= {F_y(x,y) ...
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0answers
29 views

Covariant Derivative to study Holonomy

Can someone provide an explicit definition to the $\mathbb{R}^{3}$-covariant derivative? For instance, I don't understand the following calculation. Let $$E_1 = (-\cos(u) \sin(v), - \sin(u) \sin(v), ...
1
vote
0answers
25 views

Stoke's Theorem for an open cylinder

How do you use Stokes' Theorem to calculate the surface integral over a cylinder of $\nabla \times F$? Do you have to calculate the line integrals along the top and the bottom? If so, is this example ...
1
vote
0answers
37 views

approximating a sphere

Suppose that $R$ is a simple connected region in $\mathbb{R}^3$, enclosing a volume $V$. I am looking at ways to approximate $V$ using spheroidal volume elements. The traditional approach is to use ...
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0answers
52 views

packing spheroids/ellipsoids

I am thinking about a problem I'm trying to formulate, from a mathematical analysis viewpoint. Suppose that you have a finite region in 3-space, and you want to populate it with spheroids or even ...
1
vote
0answers
18 views

The difference between $(\frac{\partial F}{\partial T})_X$ and $\frac{\partial F}{\partial T}$?

Just as the headline says, what is the difference between $(\frac{\partial F}{\partial T})_X$ and $\frac{\partial F}{\partial T}$ ? The former is used at least in thermodynamics, and I find the ...
1
vote
0answers
51 views

Linearization of Implicit ODE (Equations of Motion)

let's say we have a system with vector $q_{(t)}$ representing the degrees of freedom (DoF), and state vector $ x_{(t)} = \left \{ \begin{array}{c c} q_{(t)} \\ \dot{q_{(t)}} \end{array} \right \}$ ...
1
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0answers
20 views

Evaluation of an Integral in Vector Analysis

I'm trying to calculate an individual probability $P(\hat{a})$ from a joint probability $P(\hat{a},\hat{b})$ in a physics application, where $\hat{a},\hat{b}$ are unit vectors. I need to evaluate the ...