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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105 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
60 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
112 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
36 views

The missing vector derivative operation

Generally, the differential operator $d$ takes differential $k$-forms to $(k+1)$-forms. In $\mathbb{R}^n$ (or more generally on an oriented Riemannian manifold), we can identify vector fields with ...
4
votes
0answers
116 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
145 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
121 views

Calculate the flux of $\underline{v}$ across the boundary of the sector.

For $a\in(0,1)$, calculate without use of the divergence theorem the flux of $\underline{v}(x,y) = g(y/x)(-1/x,1/y)$ across the boundary of the sector $ S_a := \{(x,y)\in \Omega : 1\leqslant x^2+y^2 ...
3
votes
0answers
45 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 ...
3
votes
0answers
29 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
60 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
90 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
68 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
60 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
778 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
69 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
32 views

Unable to evaluate this surface integral

Question is to find the surface integral $$\iint p(x^4 + y^4 + z^4) \, \mathbb dS.$$ The given surface is $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,$$ and $p$ is the length of the ...
2
votes
0answers
54 views

Covariant derivative for a covector field

In the lecture we had the immersion $f: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$. Now we said that a map $\lambda : U \rightarrow T^*f$ is a covector field. Okay, that's fine. Also, we ...
2
votes
0answers
32 views

Topographical survey / vector calculations

I have difficulties to find the proper way to solve the following exercise. In the swiss grid map we have so called LFPs which are (x,y) anchors. I am looking for a (x,y) position that is 12m further ...
2
votes
0answers
13 views

Setting up a Green's Theorem Problem where C is not oriented counterclockwise

Use Green's Theorem to evaluate $\mathbf{F}(x,y)=\langle y^{2}\cos x, x^{2}+2y\sin x\rangle$, where $C$ is the triangle from $(0,0)$ to $(2,6)$ to $(2,0)$ to $(0,0)$. Taking the appropriate partial ...
2
votes
0answers
27 views

Existence of gradient perpendicular to a vector field

Let $v$ be a divergence-less vector field (in $\mathbb{R}^3$). When can we find a non-constant scalar function $f$ so that $\nabla f$ is perpendicular to $v$?
2
votes
0answers
66 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
51 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
42 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
36 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 ...
2
votes
0answers
56 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
78 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
49 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
71 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
121 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
0answers
51 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
74 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
120 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
104 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
24 views

Applying Stokes' theorem

$C$ is the surface $z=y(e^{-x^2}-y-1/2)$ and conditions $z\geq 0$, $x \in [-1, 1]$ and $\varphi = dxdz+dydz+(e^{-x^2}-2y-1/2)xe^{xz}dxdy$ is a 2-form. I have to compute $\int_C\varphi$ using Stokes' ...
1
vote
0answers
37 views

Confusion with conclusion drawn from alternative to Stoke's Theorem

It's not hard to prove that $$\int_S(\vec{dS}\times\vec\nabla)\times\vec P = \oint_C\vec{dl}\times \vec P$$ Is an alternative way to write Stoke's Theorem. Now, from this alternative Theorem you ...
1
vote
0answers
40 views

Verify Stokes' Theorem, example

For this question, I have found $Curl F$, which is $xi - (y-3)j - k$. But for the equation of the plane, is it $z = 0$? If I continue, the circulation will become $2$, is that correct? I am ...
1
vote
0answers
25 views

Divergence integral evaluation

Let W be the region bounded by planes $$x=0, y=0, y=3, x+2z=6$$ Evaluate the surface integral using Gauss Divergence theorem where F= $$ 2xy \hat i + yz^2 \hat j + xz \hat k$$ I am able to set ...
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vote
0answers
19 views

Question from Stewart's Calculus regarding proof of independence of path and conservative vector fields.

Please look over this proof. In the proof, it says: "Notice that the first of these integrals does not depend on $x$, so..." How is that so? $C_1$ does depend on $x$. How/Why does it not ...
1
vote
0answers
9 views

Conservative vector field on the union of simply connected regions

I am having some problems proving the following: Let $D = \mathbb{R}^2 \setminus \{(x_0,y_0)\}$ for some point $(x_0,y_0)\in \mathbb{R}^2$. Show that $D$ cannot be written as $D = D_1 \cup D_2$ where ...
1
vote
0answers
19 views

How to rewrite this matrix form

I had this equation. \begin{equation} \begin{pmatrix} g_{1,1} & g_{1,2} & \cdots & g_{1,n} \\ g_{2,1} & g_{2,2} & \cdots & g_{2,n} \\ \vdots & \vdots & \ddots ...
1
vote
0answers
22 views

Proving a bound on the curve integral of a vector field

I want to prove $$ \left| \int_\Gamma F \cdot dl \right| \leq \max_{x \in \Gamma} \left \{ \left| F(x)\right| \right\} \int_\Gamma dl $$ where $F : R^n \rightarrow R^n$ is continuous, $\Gamma \in ...
1
vote
0answers
26 views

'Simple' Vector analysis and I need a little help prooving an identity.

I have to prove that $$\vec{S}(\vec{r}) = -\frac{c^2}{\omega} \bigg( u(\vec{r})\vec{\nabla}\phi(\vec{r}) + \frac{i}{2}\vec{\nabla}u(\vec{r}) \bigg) ~~~~(1)$$ given that $$\vec{S}(\vec{r}) = ...
1
vote
0answers
42 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
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0answers
26 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
52 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 ...
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vote
0answers
31 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
vote
0answers
87 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
30 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
23 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
45 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 ...