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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6
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107 views

Stokes' Theorem and Vector Fields with Jump Discontinuities

What are the continuity requirements on a vector field $\boldsymbol{A}$ such that Stokes' theorem, $$ \iint_S\nabla\times\left[(\boldsymbol{\hat{x}}\cdot\nabla\phi)\boldsymbol{A}\right]\cdot ...
6
votes
0answers
108 views

basic question from vector analysis

Let $\vec v = (a,b,c)$ be a constant vector and let $\vec r = (x,y,z)$ denote the position vector. Consider the vector field $\vec v \times \vec r$. A straightforward calculation (using determinants ...
5
votes
0answers
62 views

What is the advantage of using Feynman's trick to use rules of vector algebras on $\bf \nabla$ operator?

I was reading Field Energy & Field Momentum of Feynman's Lectures on Physics Vol.II; there he introduced a trick 'to throw out—for a while at least—the rule of the calculus notation about what the ...
5
votes
0answers
64 views

Vector Laplace equation with constraint

I want to solve Laplace equation for a vector $\boldsymbol v=(v_x,v_y)$: $$\nabla^2 \boldsymbol{v}=0$$ but under the constraint that $$(1+v_x)^2+v_y^2=1$$ which becomes $v_y = -(2v_x+v_x^2)^{1/2}$. ...
5
votes
0answers
121 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
69 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
127 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
45 views

Integral of triple vector product

I'm following a Electrodynamics course and I'm currently stack in the following calculation: $\int{ \frac{ \vec{n} \times (\vec{n}-\vec{\beta}(t)) \times \vec{\beta'}(t) }{(1-\vec{n} \cdot ...
4
votes
0answers
55 views

Let $f: A \to \mathbb{R^n}$ be of class $C^r$; $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not 1-1, the set $B=f(A)$ is open.

Let $A$ be open in $\mathbb{R^n}$; let $f: A \to \mathbb{R^n}$ be of class $C^r$; assume $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not one-to-one on $A$, the set $B=f(A)$ is open ...
4
votes
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69 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 ...
4
votes
0answers
63 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
212 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
164 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
134 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
50 views

Determinant of non-square Jacobian

Suppose I have a 3d solid in ${\bf R}^4$ which can be parametrized by the function $F:W\subset{\bf R}^3\rightarrow{\bf R}^4$. Now suppose I want to calculate the volume of this solid. Then naively I ...
3
votes
0answers
17 views

Gradient control by div and curl

For any $u\in W^{1,p}(\mathbb R^3)$ how to prove: $$\|\nabla u\|_p \leq C(\|\operatorname{div}u\|_p+\|\operatorname{curl}u\|_p)$$ Any suggestions? Thanks!
3
votes
0answers
52 views

Orthogonal curvilinear coordinates (derivatives of unit vectors)

Suppose that $\{u_i\}_{1\le i\le 3}$ is a set of orthogonal curvilinear coordinates with unit vectors $\{\mathbf{\hat{e}_i}\}_{1\le i\le 3}$. I proved that $$\frac{\partial ...
3
votes
0answers
158 views

integral of product of curve with itself in three dimensional

I have the next problem: Let $\gamma:[0,1]\to R^3$ differentiable curve piecewise, and let $\Delta_r=\{(s,t)\in [0,1]^2| |\gamma(s)-\gamma(t)|<r\}$, i want to know if: ...
3
votes
0answers
34 views

Simplified Helmholtz decomposition

Suppose we are given a vector field $$\vec{a}= (x+y+z)\vec{i}+y\vec{j}+z\vec{k} \tag{$*$}$$ And we want to write it as a linear combination of a potential field and a solenoidal field, meaning ...
3
votes
0answers
35 views

Any unit speed reparametrization $\beta=\alpha(h)$ of $\alpha$ is reparametrized by an arc length function.

These are definitions given from Barret O'neill's Elementary Differential Geometry. Definition $1$. A curve in $\mathbb{R}^3$ is a differentiable function $\alpha: I\to \mathbb{R}^3$ from an open ...
3
votes
0answers
128 views

Chain rule for the curl of a vector-valued function

I am looking for a vector expression for the curl of a composite vector-valued function. In other words $$ \nabla\times\mathbf{A}(\mathbf{B}) = ? $$ In indicial notation, this can be ...
3
votes
0answers
161 views

A question on the definition of tangent vectors as equivalence classes and directional derivatives

I understand that a tangent vector, tangent to some point $p$ on some $n$-dimensional manifold $\mathcal{M}$ can defined in terms of an equivalence class of curves $[\gamma]$ (where the curves are ...
3
votes
0answers
141 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
60 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
34 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
73 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
64 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
960 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
47 views

Gauss´s law proof “details”

I know that this question has already been asked multiple times but I´m still not getting on the mathematical details behind the answers... So I hope that this question doesn´t get closed; also I ...
2
votes
0answers
32 views

$\nabla^2 u = 0 $ and integral $u$ around $\partial B_\rho$

I'm doing exercises from book about vector calculus. And there is problem wich I'm not sure what to do. Let's see the hypotesis. Let, $D \subseteq \mathbb{R}^2$ an open set. Let, $u:\overline{D} \to ...
2
votes
0answers
23 views

Net flux zero equivalent to vanishing solution?

Consider \begin{align*} \Delta u(P) = 0, \quad P\in \Omega,\\ \frac{\partial u(P)}{\partial n_{p}} = f(P),\quad P\in \Gamma, \end{align*} where $\Omega$ is the infinite domain in the plane outside ...
2
votes
0answers
42 views

How to compute the “jump”, using the divergence theorem,

Given the vector field on $R^3$∖{0}, F(x)=$(x/r^3,y/r^3,z/r^3)$, where $r=(x^2+y^2+z^2)^{\frac {1}{2}}$, let R be a simply connected bounded region, with smooth boundary, in the xy plane containing ...
2
votes
0answers
38 views

Vector/Tensor analysis, Elastic Waves

So I'm fairly confused at the moment. For reference, I'm reading this document, and the current area of interest is Section 7: Characteristic Surfaces for Planar Waves. I'm not gonna give too much ...
2
votes
0answers
43 views

Stuck while trying to simplifying this PDE identity

This is related to a fluids HW: Let $u(x)$ be incompressible vector field on n-D space, $x\in \Omega\subset R^n$ and let $\theta(x)$ be a scalar field s.t. $\int_{\Omega} \theta(x)dx=0$, $\nabla.u=0$ ...
2
votes
0answers
49 views

Area of a GREEN-region

a) Show that the area of GREEN-region B (which is defined by: $A\subseteq R^2$, $A=B_1\cup ...\cup B_m$, and $\mathring{B}_i\cap \mathring{B}_j=\emptyset$) in the plane is defined by: ...
2
votes
0answers
65 views

Directional derivative expression

We have $n=\sqrt{{\mathbf N}\cdot{\mathbf N}}$, where ${\mathbf N}$ is the normal vector to a curve, let's accept ${\mathbf N}=\ddot{{\mathbf r}}$, say the curve is unit-speed. We also have a scalar ...
2
votes
0answers
34 views

Two methods of calculating a Jacobian determinant

Suppose you have two fluid bodies, one described by a set of vectors $V$, and a perturbation of $V$ given by $V+\Delta V$. Suppose that the two regions are related by the transformation $\mathbf ...
2
votes
0answers
51 views

Simple Vector Calculus Integral

A little stuck on what I assume is probably a fairly basic vector calculus integration problem, but I haven't been able to find any resources online that deal with multivariable integration in a way ...
2
votes
0answers
84 views

Application of Implicit Function Theorem in Munkres Analysis on Manifolds

I'm studying the Implicit Function Theorem and this is a problem from Munkres' Analysis on Manifolds. Let $F:\mathbb{R^2} \to \mathbb{R}$ be of class $C^2$, with $F(0,0)=0$ and ...
2
votes
0answers
42 views

Given a nonzero vector $B$ and a vector-valued function $F$ such that $F(t)\cdot B=t$ for all $t$, show that$F''$ is orthogonal to $F'$

Given a nonzero vector $B$ and a vector-valued function $F$ such that $F(t)\cdot B=t$ for all $t$, and such that the angle between $F'(t)$ and $B$ is constant (independent of $t$). Prove that $F''(t)$ ...
2
votes
0answers
30 views

Stokes' theorem - calculating a flow round $\Delta S$

Let $F(x,y,z) = (y,0,0)$ and $$S = \{ (x,y,z);\,\,{x^2} + {y^2} + {z^2} = 1,\,\,z \ge 0\} $$ Let $\widehat N$ be the unit normal field to $S$ pointing away from the origin. Calculate ...
2
votes
0answers
35 views

A converse theorem to $\operatorname{div} (\operatorname{curl}(\mathbf{F}))=0$

I have already proved the following: Let $\mathbf{F}$ be a vector field of class $C^2$ defined on $ \mathbb{R}^3$. If $\nabla\cdot\mathbf{F}=0$, then there is some vector field $\mathbf{G}$ such that ...
2
votes
0answers
47 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 ...
2
votes
0answers
63 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
63 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
36 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 ...
2
votes
0answers
31 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
38 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
131 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 ...