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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10
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0answers
85 views

A version of Ampère's law

The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law uses the fact that, if $\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3$, $\boldsymbol{J}\in C_c^2(\...
7
votes
0answers
135 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 d\...
6
votes
0answers
153 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
132 views

Help with integral from Boltzmann equation

I have a function $$ g\left(x,v,t\right) = u\left(x,t\right)\cdot v + \theta\left(x,t\right)\frac{1}{2}\left(\left\lvert v\right\rvert^2 - 5\right) $$ where $g(x,v,t),\theta(x,t)$ are scalars and $u(...
5
votes
0answers
76 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
76 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
126 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
75 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) =f\left(\vec{x},t\...
5
votes
0answers
133 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
91 views

Uniqueness of a PDE solution

Suppose $F:U \to \mathbb{R}^n$ is a $\mathcal{C}^1$ vector field on an open set $U \subseteq \mathbb{R}^n$. Let $\lambda \in \mathbb{C}$. Consider the partial differential equation $$F(x) \cdot \...
4
votes
0answers
47 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 \vec{\beta}...
4
votes
0answers
98 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
0answers
78 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
65 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 $1$...
4
votes
0answers
219 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
166 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
137 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 $\rho\left(\vec{r},t\right)$, $\vec{J}\left(\vec{r},t\right)$, $\vec{E}\left(\vec{r},t\...
3
votes
0answers
47 views

Surface Integral of a Vector Field

Consider the vector field $\vec{F}(x,y,z)=(x,y,z)$, and the surface parametrized by $\Phi(u,v)=(uv,\frac{u^{2}+v^{2}}{2},\frac{u^{2}-v^{2}}{2})$ where $0\leq u\leq 2 $ and $0\leq v \leq 4$. Evaluate ...
3
votes
0answers
25 views

Vector Calculus: solution to Poisson equation

This is problem 8.4.17. from Marsden Vector Calculus book. Let $\rho$ be a continuous function which vanishes outside a 3D region $W$. Define $\phi(\textbf{p})=\displaystyle\iiint_W\frac{\rho(\...
3
votes
0answers
23 views

solving linear gradient PDE

Let $f:\mathbb R^n \to \mathbb R$ be a function that satisfies the equation: $$\nabla^T f = - (\nabla^T \phi ) f $$ where $\phi:\mathbb R^n \to \mathbb R$ is some given function and $\nabla^T$ is the ...
3
votes
0answers
115 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
18 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
54 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 \mathbf{\hat{e}_i}}{\...
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: $$\int_{\gamma\times\gamma\...
3
votes
0answers
40 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 $$\vec{...
3
votes
0answers
41 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
191 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
190 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
143 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
61 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 \begin{...
3
votes
0answers
35 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) &= \frac{v}{2}\sin(u/2)\end{...
3
votes
0answers
65 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, $K(\...
3
votes
0answers
967 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
25 views

Proving an integral relation (isotropic function)

In Hansen-McDonald's book Theory of Simple Liquids the following relation is often used: We want to evaluate the integral $$\int_V f(\vec r, \vec r') d \vec r'$$ We observe that if the function $f$ ...
2
votes
0answers
31 views

Jacobian matrix of the parametrization of (part of) a ball

I read (in E. Sernesi, Geometria 2) that the function $\varphi:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\times\mathbb{R}^{n}\to\mathbb{R}^{n+1}$ defined by $$\varphi(\theta_1,\ldots,\theta_n,r)=\left(...
2
votes
0answers
38 views

Question about simply connected regions in $\mathbb{R}^2$

If given the vector field $\mathbf{F}(x,y)=\langle\frac{1}{x}+2xy,\frac{1}{y}+x^2-\cos{y}\rangle$, it is clear that the component functions are continuous everywhere expect at the origin. Now, when ...
2
votes
0answers
24 views

Poincaré-Hopf Index Theorem - Intuitive explanation

I heard about an interesting theorem. Poincaré-Hopf Index Theorem : If $\vec{v}$ is a smooth vector filed on the compact, oriented manifold $X$ with only finitely many zeros, then the global ...
2
votes
0answers
36 views

Green's theorem with unit normal and del operator

By appropriately choosing the functions P and Q in Green's theorem, show that $\iint_R\nabla^2 \phi\;dA =\int {\partial \phi}{\partial n} \;ds $, where $\frac{\partial}{\partial n}$ denotes ...
2
votes
0answers
23 views

Solving A System Of ODE's On MAPLE 17

I Have the velocity fields for two vortices that are located at two different points ${\bf{x_1}}(x_1,y_1)$, ${\bf{x_2}}(x_2,y_2)$ $\vec{V_1} = (u_1,v_1) = \frac{\Gamma_1}{2\pi}\frac{1}{(x-x_1)^2 + (...
2
votes
0answers
30 views

Evaluate $\int \int \vec{f} . \hat{n} \ \ ds$ using stokes theorem

Consider $\vec{f} = (2x-y)\hat{i} - yz^2 \hat{j} -yz^2 \hat{k}$ where $S$ is the upper half of the surface of the sphere $x^2 +y^2 +z^2 = 1$ and C is its boundary $Stoke's$ Theorem :Let $S$ be a ...
2
votes
0answers
35 views

What is the geometrical interpretation of this vector identity (Binet-Cauchy identity)?

Sometimes I use this identity really useful to solve the problem, $$\mathbf{\left(A\times B\right)\cdot}\left(\mathbf{C}\times\mathbf{D}\right)=\left(\mathbf{A}\cdot\mathbf{C}\right)\left(\mathbf{B}\...
2
votes
0answers
25 views

What is the significance of any single partial derivative being 0 in the gradient of a function?

Consider $J : \mathbb{R}^{n} \rightarrow \mathbb{R}$. I understand that $\nabla J(\theta)\ = 0$ implies a critical point of $J$, at the value $\theta$. But what does it mean if $\delta J/\delta\...
2
votes
0answers
53 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
36 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
25 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
47 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
41 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
44 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. $\int_{\Omega} \theta(x)dx=0$, $\nabla.u=0$ ...