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7 views

Using the method of isoclines with logistic equation to create direction field

I am a little unsure on how to use the method of isoclines to model $\frac{dp}{dt} = 3p-2p^2$. As far as I know I need to set $3p-2p^2 = c$ where $c$ is the slope of the field on that line. When I set ...
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1answer
15 views

Show that the outward unit normal is smooth vector field

Exercise 2.1.8 of Guillemin and Pollack asks us to prove that if $X^m \subset \mathbb{R}^n$ is an embedded submanifold with boundary, then the outward unit normal to $\partial X$ is a smooth function ...
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0answers
38 views

Understanding the definition of Lie bracket

This is the definition I was given of Lie bracket: Let be $M$ a differentiable manifold and $v$ and $w$ two vector fields on $M$. The Poisson bracket $[v,w]$ between $v$ and $w$ is a vector field on ...
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1answer
27 views

Analytic expression of a 1-form

Let $M$ be a differentiable manifold, $V\in\mathcal{X}(M)$ a vector field on $M$ and $\alpha\in\mathcal{X}^*(M)$ a 1-form. Let $L_{V\alpha}$ be another 1-form defined by: ...
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1answer
35 views

Vector field on $S^2 \setminus \mathsf{NP}$ looks like a magnetic dipole

The following is a question from Spivak's Differential Geometry text: Not really sure what he's going for here. Any ideas?
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0answers
34 views

Relation between integral curves

Let $M$ be a manifold, $f:M\rightarrow\mathbb{R}$ a differentiable map and $X$ a vector field on $M$. I'm trying to find a relation between the integral curves of $X$ and $e^fX$. I am not quite sure ...
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0answers
27 views

Vector fields on a manifold

Let $M$ be an arbitrary manifold, $p\in M$ and $0\neq\xi\in T_pM$. I have to proof there are vector fields $V$ and $W$ in $M$ with $V(p)=W(p)=\xi$ but $[V,W]|_p\neq 0$. If you choose a chart ...
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0answers
20 views

Representing a vector field locally

A couple of questions occured to me when reading about the Poincare-Hopf theorem. Any pointers or comments would be appreciated! Let $M$ be a closed oriented Riemannian manifold and $V$ a vector ...
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1answer
27 views

Proof of the fundamental inequality of the index form

I am looking for a proof of the fundamental inequality of the index form, which I have seen as references or statements in a lot of sources, but without a proof. This is the statement: Let $M$ be a ...
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1answer
16 views

If the curl of some vector function = 0, Is it a must that this vector function is the gradient of some other scalar function?

I know of course that If the curl of a vector function is equal to zero, then the vector function is the gradient of some other scalar function, but is this a must? if so, please give mathematical ...
3
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1answer
47 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 ...
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0answers
8 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 ...
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0answers
14 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 ...
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1answer
12 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
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2answers
69 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 ...
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1answer
49 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 ...
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3answers
75 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 ...
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1answer
51 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} ...
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2answers
25 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
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1answer
27 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
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1answer
29 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
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2answers
131 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
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1answer
59 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 ...
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1answer
37 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 ...
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1answer
19 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
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0answers
32 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 ...
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1answer
58 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
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2answers
70 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
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1answer
26 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
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1answer
32 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 $v$ ...
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2answers
45 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 ...
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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 ...
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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 ...
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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 ...
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0answers
17 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
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1answer
30 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 ...
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1answer
28 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
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1answer
53 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 ...
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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,) ...
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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 ...
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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 ...
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1answer
34 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 ...
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1answer
34 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 $\mathbf H= \frac 1 c \frac {\partial \mathbf E} {\partial t}$ ...
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2answers
57 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 ...
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2answers
103 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 ...
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0answers
50 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 ...
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0answers
7 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 ...
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1answer
44 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
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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 + ...
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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?