Questions tagged [vector-analysis]

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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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 ...
Julian Lee's user avatar
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How weird can the boundary be so that the fundamental theorems of vector calculus hold?

Let $\Omega$ be a connected open set in $\Bbb R^n$. Suppose that I want theorems in multivariables calculus like divergence theorem or its relative like Green's identities or even Stoke's theorem to ...
BigbearZzz's user avatar
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Potential for Monotone Operator

I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The authors claim to construct a convex ...
Pete Caradonna's user avatar
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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 ...
Will's user avatar
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Integration on manifolds vs. integration on chains

Motivation/Background: I wish to understand the more practical/computable aspects of integration of differential forms on manifolds. In this question I shall usually implicitly include "manifolds ...
Bence Racskó's user avatar
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"Line integrals" vs "integration on manifolds" in Lee's Introduction to Smooth Manifolds

I'm getting very muddled by the notion(s) of integration used in Lee's book. At first I thought the 4th chapter on the integral of 1-forms was separated from the main treatment of integration in ...
Theo Diamantakis's user avatar
8 votes
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875 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\...
Joey Dumont's user avatar
7 votes
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285 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(...
Calvin Khor's user avatar
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7 votes
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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 ...
user2052's user avatar
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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 $DF(0,0)=\begin{...
Julian Lee's user avatar
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'Tensor Calculus' by J.L. Synge and A. Schild (1979 Dover publication) . Exercise 12 page 110.

I'm solving all the exercises of 'Tensor Calculus' by J.L. Synge and A. Schild (1979 Dover publication) . Till now everything went smooth, but now I'm stuck at exercise 12. page 110 of the third ...
Bufo Viridis's user avatar
6 votes
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957 views

Understanding Complex Form of Green's Theorem

I'm reviewing complex analysis for the GRE. I've never taken a course in complex analysis before, but I do know vector calculus. I'm trying to understand the statement of the complex version of Green'...
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Does the following imply Lipschitz continuity?

Let $f: \mathbb{C}^d \rightarrow \mathbb{C}^d$ be a function such that there is a $c > 0$ with $$ |\langle f(x) - f(y),x-y \rangle| \leq c \langle x-y,x-y \rangle $$ for all $x, \; y \in \mathbb{C}...
ort96's user avatar
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Is exterior calculus efficient for simple vector calculus problems?

Exterior calculus and invariant formulations are important and lead to many breakthroughs and great insights in physics and mathematics. But for daily vector calculus tasks, I still struggle to apply ...
Steffen Plunder's user avatar
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384 views

Stokes' Theorem for discontinuous forms

A number of physics texts I've encountered recently have rather cavalierly applied variants of the Stokes' Theorem on fields with singularities in order to derive various properties of the ...
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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 ...
Marca85's user avatar
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Geometrical meaning of calculating area using Green's theorem

Green's theorem says that: $$ \int_C L \ dx + \int_C M \ dy = \iint_D \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \ dx \ dy $$ If the M and N statisfy $\frac{\partial M}{\partial x} -...
Wojak2121's user avatar
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Show that surface integral of absolute vorticity is constant over time

In a rotating frame the (unforced, incompressible) Euler equation is $$ \frac{∂\vec{u}}{∂t}+\vec{u}\cdot\nabla\vec{u}=-\nabla\left(\frac{p}{\rho_0}\right)-2\vec{\Omega}\times\vec{u}+\nabla\left(\frac{...
Azamat Bagatov's user avatar
5 votes
1 answer
81 views

Full rank condition when stacking vector valued function

Let $f: \mathbb{R}^1 \mapsto \mathbb{R}^n$ be a smooth vector-valued function. Consider the $n \times n$ matrix $A(x)$ obtained from a vector $x \in \mathbb{R}^n$ by appropriately stacking $[f(x_1),\...
saro falsaperla's user avatar
5 votes
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247 views

Helmholtz-Decomposition-weak Formulation in n-dimensional case

In the Wikipedia article about the Helmholtz-Decomposition it says in the section about Weak Formulation: For a slightly smoother vector field u ∈ H(curl, Ω), a similar decomposition holds: $\...
taylor123123's user avatar
5 votes
1 answer
861 views

Integral of divergence over a closed surface

I am reading a paper, where an integral of a divergence over a closed surface is used without proof. $\oint_S [\nabla \cdot \vec{v}(\vec{r})] d\vec{s} = 0$, where $\vec{v}$ is tangential to the ...
Aleksejs Fomins's user avatar
5 votes
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482 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 ...
user avatar
5 votes
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399 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 $v_y = -(2v_x+v_x^2)^{1/2}$. ...
usumdelphini's user avatar
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806 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 ...
thyme's user avatar
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413 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\...
Avijit's user avatar
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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 $ \...
user66419's user avatar
  • 116
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1 answer
369 views

Prove the functions are unique in a volume, vector calculus problem

I am working through the following problem, but finding it hard to know where to go. Using the Divergence theorem and the following identities $\nabla \cdot (A \times B) = B\cdot(\nabla \times A) - ...
John Echo's user avatar
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Differential $1$-form and proof of an open disc and open circular annulus not being diffeomorphic

There is one example in my script about an application of a differential $1$ form in proving some subsets of $\Bbb R^2$ aren't diffeomorphic. As far as I've understood the explanation, we used a $C^2$ ...
PinkyWay's user avatar
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Solution verification of flux integral $\iint_DF\cdot\hat{n}dS$ with $F=(2x,y,z)$

Compute the flux of $F=(2x,y,z)$ through the surface $$ r = u^2v\,\hat{\imath} + uv^2\,\hat{\jmath} + v^3\,\hat{k}, \quad 0\leq u \leq 1, \quad 0 \leq v \leq 1 $$ My approach: is it correct?
Math Lover's user avatar
4 votes
0 answers
300 views

Coordinate free optimization of an integral

I was trying to solve this question using vectors and ideas of gradient, all the context is same as original except the fact that point lines on the x-axis. I want here the point to lie anywhere on ...
tryst with freedom's user avatar
4 votes
0 answers
161 views

Understanding Divergence Theorem

If $D$ is a region to which Green's theorem applies, $\mathbf{n}$ is the outward unit normal vector to $D$, and $$ \mathbf{F}=M(x, y) \mathbf{i}+N(x, y) \mathbf{j} $$ is a $C^{1}$ vector field on $D$, ...
Albedo's user avatar
  • 98
4 votes
0 answers
78 views

$dx(v) = dy(v) = dz(v) = F$ if $v = \mathrm{grad} \ F$?

I am trying to compute $i_v(dx\wedge dy\wedge dz):= (dx \wedge dy \wedge dz)(v,w_1,w_2)$, where $v$ denotes $\mathrm{grad} \ F$. I tried to rewrite this as \begin{align*} i_v(dx\wedge dy\wedge dz)(...
InsultedByMathematics's user avatar
4 votes
0 answers
94 views

Fast Simulation of Mass-Spring system - Formula wrong?

I'm reading through Fast Simulation of Mass-Spring Systems and I'm trying to understand the paper. To a first read it seems easy enough. However in section 4 formula (11) doesn't convince me $$ \frac{...
user8469759's user avatar
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4 votes
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342 views

foliation of a globally hyperbolic spacetime by Cauchy hypersurfaces

While studying semi-Riemannian geometry I thought: There is a globally hyperbolic spacetime $\zeta^{3,1}:=\zeta^{1,0}\times \zeta^{1,0} \times \zeta^{1,1}$, where $\zeta^{1,0}\simeq \Bbb R^{1,0},$ and ...
John Zimmerman's user avatar
4 votes
0 answers
168 views

Physical meaning of the dot product of a vector and its laplacian

What is the physical meaning of $$\boldsymbol{A}\cdot (\nabla^2\boldsymbol{A})$$ where $\boldsymbol{A}$ is a vector field in 3D space? What does it show?
HFL's user avatar
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Deriving boundary conditions at a surface of discontinuity: $\int \mathbf{B} \cdot \mathbf{n} \ dS = 0$

I am currently studying Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th edition, by Max Born and Emil Wolf. Chapter 1.1.3 Boundary conditions at ...
The Pointer's user avatar
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4 votes
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164 views

Prove vector-valued function is differentiable

Give $D$ is an open set in $\mathbb{R}^n$, $f: D\to \mathbb{R}^p$ is differentiable on $D$. Supposing that $f$ has second derivative at $x_0\in D$. For every $u\in \mathbb{R}^n$, give $g: D\to \...
Tung Nguyen's user avatar
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4 votes
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156 views

Laplacian operator in spherical coordinates (two particles)

Consider two vectors $\hat r_1$, $\hat r_2$ in a 3D Cartesian coordinate system $(O,x,y,z)$. For $\hat r_1$, the laplacian operator could be written in spherical coordinates as \begin{equation} \hat{...
EVM's user avatar
  • 41
4 votes
0 answers
154 views

does this function exist?if yes ,please explain

Does there exist a function $u : \mathbb R^2 \rightarrow \mathbb R$ that is not continuous at $0 \in \mathbb R^2$, but whose restriction to every polynomial curve going through $0 \in \mathbb R^2$ is ...
Matin Tahmasebpour's user avatar
4 votes
1 answer
223 views

Fundamental Theorem of Calculus for Jacobian

Let $J_x$ denote the Jacobian operator of the function $f: \mathbb{R}^n \to \mathbb{R}^n$. What is the proper statement of the fundamental theorem of calculus for this case? That is can we write \...
Lisa's user avatar
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4 votes
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294 views

Is Leibniz integral rule (basic form) allowed in this (physics) improper integral? Why?

Electric potential at a point inside the charge distribution is: $\displaystyle \psi (\mathbf{r})=\lim\limits_{\delta \to 0} \int_{V'-\delta} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'$ ...
Joe's user avatar
  • 1,121
4 votes
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101 views

What is a neat way to solve $\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{\mathbf{C}}$?

Let $\mathbf{u}:\mathbb{R}^3\to\mathbb{R}^3$ be a smooth enough vector field that satisfies the following equation $$\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{C},\tag{1}$$ where $\nabla\mathbf{u}$ ...
Hosein Rahnama's user avatar
4 votes
1 answer
274 views

Integral curves to a non-vanishing vector field on the unit square

Let $X$ be a non-vanishing vector field on the unit square $I^2$ in $\mathbb{R}^2$. I would like to show that every integral curve to $X$ exits the unit square in finite time. This fact is used in a ...
Chi Cheuk Tsang's user avatar
4 votes
0 answers
612 views

Is the following proof of the fluid dynamics relation $\frac{dp}{dR}=\rho \frac{v^2}{R}$ mathematically correct?

I was trying to derive the following equation from my knowledge of aerodynamics: $$\frac{dp}{dR}=\rho \frac{v^2}{R}$$ (Where $\rho$ is the fluid density, $p$ is the pressure and $R$ is the radius of ...
Amin Parvaresh's user avatar
4 votes
0 answers
149 views

In the context of curl, what does the difference between two partial derivatives tell me about rotation in a plane?

Let $\mathbf{A}(x,y,z)$ be a vector field. The curl of this vector field is defined as $$\nabla \times \mathbf{A} = \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) \...
Stan Shunpike's user avatar
4 votes
0 answers
221 views

The interpretation of the flux of a vector field

I would like to prove that the flux volume rate over a manifold is the flux volume rate over a manifold. I mean: Let $M$ be a Riemannian orientable manifold, let $S$ be a compact oriented submanifold,...
Carlos's user avatar
  • 307
4 votes
0 answers
606 views

Computing the gradient of a function depending on the eigenvalue and eigenvector of a matrix.

I have a real, scalar-valued function for which I would like to compute the gradient. Let $\xi = [\omega^T, \tau^T]^T$ be a 6-vector in $se(3)$, the Lie algebra associated with $SE(3)$, the Special ...
Daniel's user avatar
  • 272
4 votes
0 answers
140 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 \...
Matthew Kvalheim's user avatar
4 votes
1 answer
290 views

Divergence operator of higher order and intrinsic point of view

Let $\underline{u}$ be a $1$ - order tensor (say a column vector) I want to prove that : $\underline{\operatorname{div}} \left( (\underline{\underline{\operatorname{grad}}} \, \underline{u})^T\...
M.LTA's user avatar
  • 1,294
4 votes
0 answers
133 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 $$M(x,y)+N(x,y)\frac{dy}{dx}=0$$ ...
Secret's user avatar
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