# 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 $v_y = -(2v_x+v_x^2)^{1/2}$. ...
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### $\operatorname{div}$ 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 ...
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### 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 ...
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### 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?
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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 ...
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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 $g: D\to \... • 1,280 4 votes 0 answers 156 views ### Laplacian operator in spherical coordinates (two particles) Consider two vectors$\hat r_1$,$\hat r_2$in a 3D Cartesian coordinate system$(O,x,y,z)$. For$\hat r_1$, the laplacian operator could be written in spherical coordinates as \hat{... • 41 4 votes 0 answers 154 views ### does this function exist?if yes ,please explain Does there exist a function$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 ... 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 \... • 2,941 4 votes 0 answers 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:$\displaystyle \psi (\mathbf{r})=\lim\limits_{\delta \to 0} \int_{V'-\delta} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'$... • 1,121 4 votes 0 answers 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$\nabla\mathbf{u}$... • 14.9k 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 ... • 2,389 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 ... 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 $$\nabla \times \mathbf{A} = \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) \... • 4,939 4 votes 0 answers 221 views ### The interpretation of the flux of a vector field I would like to prove that the flux volume rate over a manifold is the flux volume rate over a manifold. I mean: Let M be a Riemannian orientable manifold, let S be a compact oriented submanifold,... • 307 4 votes 0 answers 606 views ### Computing the gradient of a function depending on the eigenvalue and eigenvector of a matrix. I have a real, scalar-valued function for which I would like to compute the gradient. Let \xi = [\omega^T, \tau^T]^T be a 6-vector in se(3), the Lie algebra associated with SE(3), the Special ... • 272 4 votes 0 answers 140 views ### Uniqueness of a PDE solution Suppose F:U \to \mathbb{R}^n is a \mathcal{C}^1 vector field on an open set U \subseteq \mathbb{R}^n. Let \lambda \in \mathbb{C}. Consider the partial differential equation$$F(x) \cdot \... • 2,010 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\...
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(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 $$M(x,y)+N(x,y)\frac{dy}{dx}=0$$ ...