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
votes
0answers
108 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
62 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
118 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
40 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
119 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
149 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 ...
4
votes
0answers
93 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
125 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
48 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
30 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
61 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
69 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
788 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
70 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
30 views

A line integral equation popped up when trying to derive Exact ODE integrating factor, can it be solved analytically?

(For convenience, for any functions, only its first instance the x,y dependence will be written out, all subsequent instance the x,y will be suppressed) I have an ODE $$M(x,y)+N(x,y)\frac{dy}{dx}=0$$ ...
2
votes
0answers
46 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
38 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
14 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
28 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
72 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
54 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
44 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
40 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
60 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
82 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
50 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
73 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
122 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
75 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
122 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
18 views

Is my interpretation of Rotation Matrices correct?

I've been asked to find the matrix which rotates vector $\vec{V}$ by angle $\alpha$ in the x-y plane. This I understand and I've constructed the matrix: $R_{\alpha}= \begin{bmatrix} cos\alpha & ...
1
vote
0answers
34 views

line integrals explanation

I am very new to this so sorry if it is obvious. Compute the line integral $\int Fdr $ where $F(x,y)=(x^2y,y^2x)$; $r(t)=(\cos t,\sin t)$; $t\in[0,2\pi]$. So what I would do is find $r'(t)=(-\sin ...
1
vote
0answers
11 views

$\int\int\int_{g(s)} (2x+y-2z)dx dy dz=\alpha\int\int\int_{s} z dx dy dz $..calculate $\alpha$

Let $g:R^{3}->R^{3}$ be defined by g(x,y,z)=(3x+4z,2x-3z,x+3y) and let $s={\{(x,y,z)\epsilon R^{3}:0\leq x\leq 1 ,0\leq y\leq 1 , 0\leq z\leq 1 }\}$. if $\int\int\int_{g(s)} (2x+y-2z)dx dy ...
1
vote
0answers
20 views

Evaluating the surface integral side of a divergence theorem problem

Edit: I realized where my mistake was...thanks for the help! In class we were discussing a problem (page 1185 in Smith and Minton's Calculus Third Edition): Let Q be the solid bounded by the ...
1
vote
0answers
40 views

Gradient of cosine

I am quite new to vector calculus and I am not sure how to calculate the following. Suppose we have three position vectors $\vec{r}_i$,$\vec{r}_j$, and $\vec{r}_k$ in $R^3$. The angle $\theta_{ijk}$ ...
1
vote
0answers
18 views

Tricky vector derivatives

If $n_i=n_i(x_1,x_2)$ are the components of a unit vector ($\sum_i n_in_i =0$), and $i=1,2$, I know that $$\sum_in_i\nabla_jn_i=\sum_i \frac{1}{2}\nabla_j(n_in_i)=0$$ If $\nabla_i := ...
1
vote
0answers
29 views

General solution to 3D linear 2nd order PDE using Wronskian and Integrals?

Using the Wronskian and Indefinite Integrals, I can write the solution to the general one dimensional second order linear non-homogeneous differential equation $$ y''+p(x)y'+q(x)y=g(x) \\ y^*(x)= ...
1
vote
0answers
31 views

Evans PDE derivation of solution for Poisson's equation - integration by parts

In Evans' PDE text, when proving the solution to Poisson's equation, when he integrates by parts he takes the inward pointing normal, whereas in the integration by parts formula in the appendix, it ...
1
vote
0answers
37 views

Textbook suggestion-Vector Analysis

I took a course in vector analysis this year. It was a two fold course. The first part covered linear algebra and basic euclidean geometry. The second took to more advanced areas such as differential ...
1
vote
0answers
26 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
42 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
43 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
26 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 ...
1
vote
0answers
24 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
26 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
23 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 ...