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-1
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1answer
16 views

is domain D simply connected? [on hold]

how can we know whether or not domain D is simply connected, by just looking to domain where vector field F is defined?
3
votes
1answer
44 views

How to use the $b\cdot\nabla$ operator?

While trying to prove $$[c\cdot (b\cdot\nabla) - b\cdot(c\cdot\nabla)]a = (\nabla\times a) \cdot (b\times c)$$ I had some difficulties on how to treat the term $(b\cdot\nabla)$. It seems that ...
0
votes
0answers
6 views

Find curl of vector field in 2Space implicitly in 3Space

I have a vector in 2Space, which (I'm assuming I need to implicitly treat as if it's in 3Space due to the fact curl is a cross-product which needs vectors to be in 3Space. $$ \text{ For each of the ...
2
votes
0answers
12 views

Does the concept of Lie derivative by bivector fields exist?

A cursory glance at the internet shows that perhaps the closest (if not exactly) to what I'm seeking is Albert Nijenhuis' generalization of the ordinary Lie derivative. He constructed a way to take ...
1
vote
1answer
10 views

What are intuitively the diffeomorphisms $\theta^t(p)=\Theta (t,p)$, associated to the local flux $\Theta$ of the vector field $X$?

Given a vector field $X$ on the manifold $M$ I know that I can associate to it in a unique way a local flux $\Theta: W \rightarrow M$, where $W \subset\mathbb{R} \times M$. The curve ...
2
votes
2answers
64 views

Vector fields along maps: I need another sanity check

Consider the definition of a vector field along a smooth map $f: M \to N$ where $M,N$ are smooth manifolds: A vector field along $f$ is a continuous map $W \colon M \to TN$ such that $W(m) \in ...
1
vote
1answer
48 views

Question on vector fields along maps (need a quick sanity check)

Let $f: M \to N$ be some smooth map between smooth manifolds. If $V$ is a vector field, that is, a smooth map $V: N \to TN$ then $V$ is a vector field along $f$ if the projection $\pi: TN \to N$ is ...
2
votes
3answers
64 views

Vector space or vector field?

I seem to be having a problem distinguishing between a vector space (which I know to be a set of vectors over some scalar set) and a vector field. I know that in Multivariable Calculus a vector field ...
4
votes
1answer
50 views

Completeness of the vector field $e^{-x} \frac{\partial}{\partial x} + \frac{\partial}{\partial y}$

I just want to bounce this off of the smart people on MSE to make sure I understand what's going on when we discuss complete vector fields. Consider the following field. $X = e^{-x} ...
0
votes
2answers
24 views

Estimating line paths in vector fields.

Assume I have a vector field sampled in discrete points. For simplicity let us assume it is sampled regularly on a Cartesian grid. I want to estimate flow lines through various points in this vector ...
2
votes
1answer
26 views

Simple Lotka-Volterra Slope Field in Phase Space

I'm trying to plot the slope field in phase space of a simple (all constants set equal to $1$) Lotka-Volterra system described by the following differential equations: $$\frac{dw}{dt} = w-wr$$ ...
2
votes
1answer
24 views

Is the curl of a vector field only defined on $\Bbb R^3$?

Is the curl of a vector field only defined on $\Bbb R^3$? I was wondering if the criterion $$\nabla \times \vec{F}=\vec{0} \implies \vec{F} \space\text{is conservative}$$ only applies to three ...
2
votes
2answers
126 views

Can a vector field be conservative if its domain is not a star domain?

Can a vector field be conservative if its domain is not a star domain? I was trying to figure out whether the vector field $$\vec{f}(\vec{x}):=\frac{1}{\lvert \lvert \vec{x} \rvert \rvert} ...
0
votes
1answer
52 views

Flowout Theorem

I am reading Theorem 9.20 (Flowout Thoerem) from Lee's Introduction to Smooth Manifolds, Second edition. A part of the theorem states the following: Let $M$ be a smooth manifold and $S$ be a ...
1
vote
1answer
36 views

Conservative field?

Let a vector field $F$ what it is defined by $F(x,y)=(\frac{-y}{(x-1)^2+y^2},\frac{x^2+y^2-x}{(x-1)^2+y^2})\ \forall \ (x,y)\epsilon\mathbb{R}^2$\ {$(1,0)$} then... is the vector field $F$ a ...
1
vote
1answer
16 views

Line integral of vector field/Why doesn't my solution work?

The question in its entirety: Determine for which constants A & B the vector field $$\mathbb{F} = (Axln(z))\mathbb{i} + (By^2z)\mathbb{j} + ((\frac{x^2}{z})+y^3)\mathbb{j}$$ is conservative. If ...
2
votes
0answers
28 views

Describe the local flux of this vector field $X$ on $S^2$ given by $X(v)=w_0- \langle v, w_0 \rangle v.$

Let be $w_0 \in \mathbb{R}^3$ and $X: S^2 \rightarrow TS^2$ the vector field on $S^2$defined by: $$X(v)=w_0- \langle v, w_0 \rangle v.$$ ($\langle.,. \rangle$ is the standard dot product) How can I ...
5
votes
1answer
56 views

Lie bracket and flows on manifold

Suppose that $X$ and $Y$ are smooth vector fields with flows $\phi^X$ and $\phi^Y$ starting at some $p \in M$ ($M$ is a smooth manifold). Suppose we flow with $X$ for some time $\sqrt{t}$ and then ...
2
votes
2answers
62 views

Divergence of inverse square vector field

After trying to get the divergence of a vector field I got: $${\nabla \cdot \Bigg(\frac{\vec r}{|r|^3}\Bigg)=\frac{\partial}{\partial r}.\frac{1}{r^2}=-\frac{2}{r^3}}$$ But in wikipedia found that ...
3
votes
1answer
23 views

Vector fields spanning the tangent bundle at each point

Let $M$ be a smooth manifold of dimension $n.$ It is well known that it is not allways possible to find such global vector fields $X_1,\ldots, X_n$ which would be a basis for the tangent space at each ...
2
votes
1answer
30 views

Definition of “vector fields never have opposite direction”

Good day! As in my other question I am referring to the book "Differential Equations and Dynamical Systems" by Lawrence Perko, chapter 3.12. I have a question regarding Lemma 2: Lemma 2. If $v$ ...
1
vote
2answers
43 views

Quickest way to determine if a vector field is conservative?

Say I have some vector field given by $$\vec{F} (x,y,z)=(zy+\sin x)\hat \imath+(zx-2y)\hat\jmath+(yx-z)\hat k$$ and I need to verify that $\vec F$ is a conservative vector field. What would be the ...
1
vote
0answers
15 views

ONB (right handed) - vector fields - green-region

There is a right handed orthonormal basis ${a_1,a_2,a_3}$ of $R^3$ (meaning $a_3=a_1\times a_2$) and a number $\omega \geq 0$. The rotation around the axis $a_3$ with the constant angular velocity ...
0
votes
0answers
20 views

Righthanded ONB.

There is a right handed orthonormal basis ${a_1,a_2,a_3}$ of $R^3$ (meaning $a_3=a_1\times a_2$) and a number $\omega \geq 0$. The rotation around the axis $a_3$ with the constant angular velocity ...
0
votes
0answers
21 views

Vector-fields - divergence theorem.

Let $c>0$ and $A\subseteq R^3$ be a green-region (which is defined by: $A\subseteq R^2$, $A=B_1\cup ...\cup B_m$, and $B^{\circ}_i\cap B^{\circ}_j=\emptyset$) with the outward normal unit ...
0
votes
0answers
15 views

Oriented surface - sphere

$S^+$ and $S^-$ are the upper ($z\geq 0$) and lower half ($z\leq 0$) of the surface area of a sphere with Radius $R>0$ where the symmetry axis is the z-axis. $k:R^3\to R^3, ...
3
votes
1answer
29 views

Doubt about conservative fields in 2D and 3D

Regarding a conservative field $\vec{F}$ in a region $D \subseteq R^2$, I know that the requirements are: Curl of $\vec{F}$ is $0$. $\vec{F}$ is defined in D (doesn't have singularities in D). But ...
1
vote
1answer
27 views

Reverse engineering a differential equation from singular points

I've been struggling to find a way to reverse engineer a differential equation based on knowing it's singular points. In this case, I'd like to create a flow on $[-1,1] \times [-1,1]$, which has ...
2
votes
1answer
52 views

What is the level set of 1D function?

I have encountered a very strange idea today. My prof claimed that the differential of a function $df$ is the number of level curve crossed by a tangent vector $v$ at a point $p$ I tried to reason ...
1
vote
3answers
44 views

How can vector field simultaneously be a function and also an operator that acts on a function?

In elementary calculus we have definition: A vector field is a function that assigns a vector to each point in $\mathbb{R}^2$ or $\mathbb{R}^3$ i.e. F(x,y) = P(x,y) $\hat i$ + Q(x,y,) ...
0
votes
0answers
17 views

If a Point Admits an Integral Curve on an Interval then a Neighborhood Does too On the Same Interval

$\newcommand{\R}{\mathbf R}$ Let $V:U\to \R^n$ be a (continuous) vector field on an open set $U$ of $\R^n$. Suppose we have a point $\mathbf p\in U$ and an open interval $I\subseteq \R$ such that ...
0
votes
0answers
39 views

Vector fields with constraints on velocities

Let $X^{'}=f(X)$ be a system of n differential equations where $X=(X_1,..,X_n)$, $f=(f_1,..f_n)$. We assume that $f : \Bbb{R}^n \to \Bbb{R}^n$ is a smooth function. Let $F: \Bbb{R}^n \to ...
0
votes
1answer
25 views

Flux through a Hemisphere.

I need to calculate the flux of a vector field : $H = (y-z)\hat{i} + (z-x)\hat{j} + (x-y)\hat{k}$ outside the Hemisphere given by the equation : $(x-1)^2 +y^2 +z^2=1$ with $z\ge0$ Now i used ...
2
votes
1answer
31 views

Some algebraic properties of curl

Given vector fields $\mathbf E$ and $\mathbf H$ with curl $\mathbf E= - \frac 1 c \frac {\partial \mathbf H} {\partial t}$ and curl $\mathbf H= \frac 1 c \frac {\partial \mathbf E} {\partial t}$ ...
1
vote
2answers
55 views

What interpretation of the Lie braket is this?

I was reading exercise 1.96 of Gadea's Analysis and Algebra on Differentiable Manifolds. I don't know what definition of Lie braket the author used, but I'm confused, as far as I know ...
4
votes
2answers
97 views

How do ideas in differential geometry expand upon ideas from introductory calculus

I just went through first year in mathematics and used Stewart's book for calculus. I am trying to self study differential manifold and I find many concepts such as chart, atlas very similar to that ...
1
vote
0answers
37 views

Find the mass flow rate, given a surface, density and velocity field

I have a confusion, I hope you can help me (I'd like that if you will respond, please read all my post). They ask me to find the mass flow rate passing through a surface, where the velocity field is ...
1
vote
0answers
6 views

Conservative field $F$ on not simple connected set

Give an example of a field $F:D\subseteq \mathbb R^2 \to \mathbb R^2$ such that $D$ is a doubly connected set (that is $D$ has on "hole") but $F$ is conservative And if $D$ is a triply connected ...
1
vote
1answer
36 views

Compute flux of vector field F through hemisphere

I need help solving this question from my textbook. Compute the flux of the vector field: $$\vec F = 4xz\vec i + 2 y\vec k$$ through the surface $S$, which is the hemisphere: $x^2 + y^2 + z^2 = 9 , ...
3
votes
1answer
27 views

Finding the parameterization of a curve for a line integral problem

I have to calculate the work of a particle that travel along a curve, given the following vector field: $F(x, y, z) = (2z-1, 0, 2y)$ and where the curve is the intersection between: $s1: z = x^2 + ...
0
votes
0answers
19 views

Given a vector field, how would I solve for constants, if I know that the Intergral F.dr = 0 along any curve C?

Say I have a field F = (2x + az)i + (bx - 3)j +(-3x+ cy +2z)k. Or any equation in general How would I solve for a, b and c so that the closed integral F.dr = 0?
0
votes
1answer
20 views

Proving the existence of certain vector field along a piecewise differentiable curve

I am trying to understand proposition 2.5 in chapter 9 of do Carmo's "Riemannian Geometry". In the proof of the proposition he says: Let M be a Riemannian manifold and $c:[0,a]\rightarrow M$ a ...
0
votes
1answer
23 views

Finding if a group is a vector space

$\mathbb{C}^2$ is a group over field $\mathbb{C}$, with the following actions: addition is similar to the regular addition. multiplication is defined by: for every $(z,w) \in \mathbb{C}^2$ and every ...
2
votes
1answer
31 views

The Leibniz rule for the curl of the product of a scalar field and a vector field

I have some scalar field $u:D \rightarrow\mathbb R; \space \space D\subset \mathbb R^3$ and a vector field $\vec{v}: D\rightarrow \mathbb R^3$ and I want to show that: ...
1
vote
0answers
24 views

Divergence theorem for a box

I have to use the divergence theorem to calculate $$\int \int_C F.dS $$ for $$F = (x^2z^3, 2xyz^3,xz^4) $$ where S is the surface of the box with vertices at $(\pm1,\pm2,\pm3)$ with outward pointing ...
1
vote
1answer
42 views

Intuitive explanation of the potential function of a vector field

Suppose I have some vector field $$\vec{f}(x,y)=\begin{pmatrix}A(x,y)\\B(x,y)\end{pmatrix}$$ then the potential function (if the field is conservative) can be found by integrating $A$ with respect ...
1
vote
1answer
19 views

Looking for a continuous, unit-norm vector field

I want to find a 2D vector field with three characteristics: it is continuous all the vectors are unit length vectors on the unit circle point to the origin Is this possible? I haven't been able ...
1
vote
0answers
36 views

Write the vector field such that its integral curves are the meridians of $S^2$

I have problems in writing down a vector field $X$ on the sphere $S^2$ such that the integral curves of $X$ are the meridians of $S^2$. I proceed in this way. I consider this parametrisation of ...
2
votes
0answers
23 views

What is an intuitive/geometric definition of line integrals? Do they work in 2-dimensions?

I understand that we are finding the area of a curve given by some function f(x) over the area of another curve C. (I've also successfully plugged and chugged my way through my homework, without ...
2
votes
3answers
75 views

Checking if a vector field is conservative

I have three different vector fields and I want to check if they are conservative: $$1)\space \space\vec{f}(\vec{x}):=\frac{1}{||\vec{x}||}\vec{a}, \space \space D=\mathbb R^2 / (\vec{0}), \space ...