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0
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2answers
25 views

Line integral of vector field

Compute the line integral $\int_\gamma g \cdot dx $ for an arbitrary piecewise smooth curve $\gamma$ traversing in the upper half plane from $(-a,0)$ to $(b,0)$ where $a > 0$ and $b>0$. ...
1
vote
1answer
27 views

How to show that two vector fields commute?

Could anyone help me with how to start to solve the following problem? From this problem as well as this, I have: Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such ...
0
votes
0answers
25 views

Vector fields that are smooth on the open unit ball $B$, are smooth on $\mathbb{R}^n$ if they are zero outside $B$

Could anyone help a little with the following problem? It is a continuation of this problem, but I will restate the things that are needed: Fix $\varepsilon \in (0, 1)$ and choose a smooth ...
2
votes
0answers
23 views

Existence of a differentiable function given a unit gradient field

I'm trying to prove that "Given a unit vector field $V$, it can always be uniquely determined a differentiable function $f$ that satisfies $\nabla f = V$." To provide you more information, the unit ...
0
votes
1answer
21 views

Finding the number of bases for $ \mathbb{F}_3^2 $ and isomorphisms

Find the number of bases for $ \mathbb{F}_3^2 $ and also the amount of Isomorphisms $ \mathbb{F}_3^2 \rightarrow \mathbb{F}_3^2 $ Here if $(v1,v2)$ is a basis, then $(v2, v1)$ is a different basis. ...
1
vote
0answers
16 views

Derivative matrix in polar coordinates

Considering a vector field in two dimensions, $\vec V(x,y)$, I know that the derivative matrix (Jacobian matrix) is given by: $\nabla \vec V(x,y) = \begin {bmatrix} \partial V_x/\partial x ...
1
vote
2answers
26 views

Vector spaces with complex field as scalar. [duplicate]

Sorry for stating the question informally. If we have a vector space whose scalars are the field $\mathbb{R}$, if we change the field to be $\mathbb{C}$ and "adapt" the addition and scalar ...
0
votes
0answers
27 views

One form and Vector fields on a manifold in terms of local coordinates.

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ in local coordinates where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I do not know how to ...
2
votes
2answers
44 views

Exterior differentiation of one form on a smooth manifold

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I'm fine with the right side of the equation, ...
2
votes
1answer
29 views

Does the dual basis to some basis of $T^*_pM$ looks localy like a coordinate chart?

Let $M$ be a manifold and let $\{\alpha_k\}$ be a set of $1$- forms s.t. $\{\alpha_k(p)\}$ forms a basis for $T^*_pM$. Let $(x,U)$ be a chart based in $p$ and denote $\partial_i := ...
1
vote
2answers
31 views

Lie bracket simplification

Could someone help me simplify the following: Let $$X= -x^1\frac{\partial}{\partial x^1}+x^2\frac{\partial}{\partial x^2} \qquad Y = x^2\frac{\partial}{\partial x^1}$$ Calculate $[X,Y]$ This ...
3
votes
1answer
66 views

Understanding an exercise about gradients and vector fields

In John M. Lee's Introduction to Smooth Manifolds, exercise 11.17 goes as follows: Let $f(x,y)=x^2$ on $\mathbb R^2$, and let $X$ be the vector field $$X=\operatorname{grad} ...
3
votes
0answers
39 views

Multivariable Calculus - Stokes' theorem and conservative fields - showing that a vector field does not have a potential on a domain.

I understand that the curl of a vector field $\textbf{F}$ being zero means that the field is conservative if its domain is simply connected. This was demonstrated in part a where I showed that the ...
1
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0answers
36 views

(global) Description of a vector field to a non trivial tangent bundle.

In what ways can you globally describe a vector field to a non trivial tangent bundle? I'll explain what i have trouble understanding using an example from an exercise i solved (sorry about the info ...
1
vote
0answers
14 views

In the space of polynomials of degree 2 or less, given the derivative linear transformation D and $T:=1+D+D^2$, $S:=1-D$, show that $S=T^{-1}$

Let $ P_2[X] $ be the space of polynomials of degree equal or less than 2 over the field R. Let: $$ D: P_2[X] \rightarrow P_2[X] $$ Be the derivative linear transformation, defined as follows: $$ ...
1
vote
1answer
32 views

Time derivative of an invariant probability measure

Consider a dynamical system defined through a vector field $F$ in $M \subset \mathbb{R^n}$ that generates a flow $\Phi^t$ of the form $$\bf{\Phi^t X_0 = X} \ , $$ being $X_0 \in \mathbb{R}^n$ the ...
0
votes
1answer
20 views

Work done in a vector field

Say a particle is moving along a path $\gamma$ in a vector field, then the total work done by the force $\vec{F}$ on the particle is $\displaystyle \int_{\gamma}{\vec F}.d\vec{r}$. Say if this value ...
1
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0answers
26 views

Find surface area for $f(x,y,z)=e^{−z}$, over $x^2+y^2=9, 0≤z≤3$

I am having trouble parametrizing $f$, since $z$ does not seem to be related to $x$ and $y$ in any way. Any hints?
0
votes
1answer
19 views

How would I go about differentiating this (vector function)?

I want to find the gradient of this potential function \begin{align*} \phi(\mathbf{r}) = \frac{1}{|\mathbf{r} - \mathbf{r_0}|^2}. \end{align*} First, I wrote it as \begin{align*} \phi(x,y,z) = ...
0
votes
3answers
49 views

Finding potential function for a vector field

Let $\mathbf{F}(x,y,z) = y \hat{i} + x \hat{j} + z^2 \hat{k}$ be a vector field. Determine if its conservative, and find a potential if it is. Attempt at solution: We have that $\frac{\partial ...
1
vote
0answers
38 views

Vector field tangent to a proper submanifold

Let $S\subset M$ be a properly embedded submanifold (in particular, $S\subset M$ is a closed submanifold) and let $V$ be a smooth vector field on $M$ tangent to $S$. Since we have that $M\setminus ...
2
votes
1answer
28 views

Finding field lines given a plane vector field

Problem: Determine the field lines of the vector field \begin{align*} \mathbf{F}(x,y) = x\hat{i} + y \hat{j}. \end{align*} The field lines satisfy the system \begin{align*} \frac{dx}{x} = ...
0
votes
0answers
11 views

component-wise independent vector field

Suppose you have an $n$-dimensional vector field defined by $$ \mathbf{F}(\vec{x}) = \langle f_1(\vec{x}),...,f_n(\vec{x})\rangle $$ I would like to know what it is called when the $i$'th component ...
1
vote
0answers
32 views

Induced Lie Algebra Representation, Left invariant vector fields and more…

The following is an excerpt from a proof in John Lee's Introduction to Smooth Manifolds I am struggling to understand. I would appreciate if someone was able to help me with whatever it is I am ...
1
vote
2answers
70 views

Can someone clarify the definition of flux?

I am confused by the concept of flux as used in vector calculus. Suppose I have a sphere. On the inside of this sphere is a spherically symmetric electric charge distribution. Now I want to find the ...
0
votes
0answers
26 views

Left invariant vector field question

This question is an exercise from Jeff Lee's Manifolds and Differential Geometry Let $\tilde{X}$ be the left invariant vector field corresponding to $X\in L(V,V)$ (That is, $\tilde{X}_g=(dL_g)_e X$). ...
0
votes
0answers
53 views

Computing an integral over an ellipsoid

Compute $\iint\limits_S\frac{x dy\wedge dz+y dz\wedge dx+zdx\wedge dy}{(x^2+y^2+z^2)^{\frac32}}$ where $S$ is the ellipsoid $\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1$ oriented normal to the ...
1
vote
1answer
21 views

Proving that on a Lie group $G$ the space of left-invariant vector fields is isomorphic to $T_e G$

Let $G$ be a Lie Group. Given a vector $v\in T_e G$, we define the left-invariant vector field $L^v$ on $G$ by $$L^v(g)=L^v|_g=(dL_g)_e v.$$ I want to show that $v\mapsto L^v$ is a linear ...
2
votes
1answer
33 views

Pure Math Research into Operator Fields

Has any work been done on operator fields in the pure math world? They are a big piece of quantum field theory, but I can't find anything about them outside of that messy subject. Of course, I mean ...
0
votes
0answers
31 views

How can we calculate $Xf$?

Let $X$ be the vector field $x \dfrac{∂}{∂x} + y \dfrac{∂}{∂y}$ and $f (x, y, z)$ the function $x^2 + y^2 + z^2$ on $R^3$. Compute $Xf$. Could you give me some hints how we can calculate it?
3
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0answers
60 views

Generalizations of the Hairy Ball Theorem to wider classes of manifolds

In 2 dimensions the hairy ball theorem generalizes from spheres to all orientable closed manifolds with nonzero Euler characteristic. The hairy ball theorem holds for all even dimensional ...
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votes
1answer
24 views

Proof of identities of divergence of vector fields

I want to prove some identities but I don't know how to do this. First of all, $φ : R^3 → R$ and vector fields $F = (f_1, f_2, f_3), G = (g_1, g_2, g_3) : R^3 → R^3$ the two identities are: (i)$ ...
1
vote
1answer
34 views

Vector Fields on Real Numbers

I'm looking at vector fields on the manifold $\mathbb{R}$, in the sense that a vector field $v$ is a smooth function from $\mathbb{R}$ to $\mathbb{R}\times T_p\mathbb{R}$. These seem so simple that ...
1
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0answers
34 views

Locally Hamiltonian vector fields

Consider the following definitions (taken from [1]) Definition. Let $E$ be a Banach space and $B: E \times E \to \mathbb R$ a continuous bilinear mapping. Then $B$ induces a map $B^\natural: E \to ...
2
votes
0answers
23 views

sufficient conditions for finite time of existence of integral curves of a vector field

Let $U\subset \mathbb{R}^2$ open, $\partial U\neq \varnothing$, $V\colon U\rightarrow \mathbb{R}^2$ smooth. Let $c\colon [0,t_{max})\rightarrow U$ be an integral curve of $V$, where $t_{max}$ is the ...
0
votes
0answers
15 views

For this vector field, determine whether they are the gradient of the scalar field

I have been studying line integrals recently and I don't know, what my lecture means when he is saying about a vector field and whether it is conversative? Furthermore how do you find out if a vector ...
2
votes
0answers
21 views

Solving the PDE's naturally arising from requiring vector fields to commute

If one has two vector fields $X$ and $Y$ in $\mathbb{R}^4$ whose coefficients are indeterminate functions (some coefficients are being forced to be linearly related to others), then the condition that ...
2
votes
0answers
72 views

Differential Equation Direction field

What i want to achieve: I want to plot the direction fields of the following three differential equations: 1. Malthusian growth model: $p'(t)=\lambda*p(t)$ with $\lambda=1$ and $p(t)=t$ 2. Linear ...
0
votes
1answer
41 views

Expression for Hamiltonian vector field!

How does one prove that the Hamiltonian vector field has the following expression, what is the reasoning? \begin{equation} X_H=\sum ^n_{i=1}\frac{\partial H}{\partial q_i}\frac{\partial }{\partial ...
0
votes
1answer
26 views

Calculating flux through a square

I did not quite understand the latest lecture I've been to and would like a thorough explanation if possible. A field vector is given by $F=(\cos(xyz), \sin(xyz), xyz)$. Calculate the flux through a ...
0
votes
0answers
38 views

Existence of a fixed-point free map in a manifold.

I'm having some to proof a question. I have to show that a compact manifold that admits a nowhere vanishing smooth vector field has a smooth map fixed-point free homotopic to the identity map. I know ...
1
vote
2answers
55 views

Given a nowhere zero vector field $Z$, does there exist a one-form $\gamma$ such that $\gamma(Z) = 1$?

Take $M$ a smooth manifold, and $Z$ a vector field on $M$ such that $Z(p)\neq0$ for all $p\in M$. Is there a one form $\gamma \in \Omega^1(M)$ such that $\gamma(Z)=1$? I started to work locally, but ...
0
votes
2answers
60 views

Find the flow of a vector field

Question: Let $\mathbb{X}$ be the vector field given by $\mathbb{X}(x,y)=(x,y)$ Compute its flow $\Phi(x,y)$ Attempt: We have $\dot{x}(t)=x\therefore$$$\int_{x_0}^{x(t)}dx'=\int_{0}^{t}x(t')dt'$$ ...
1
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0answers
58 views

How do I flow in a circle on a set of vector fields?

Consider a set of complete vector fields on a manifold. They each have an associated one parameter group of diffeomorphisms related to the generated flow. What is a necessary and sufficient ...
-1
votes
1answer
46 views

last step in proof of existence of coordinate vector field

This is problem 7 on page 172 of Spivak's Differential Geometry pt. 1. Given a smooth manifold $M$ and a smooth vector field $X$ on $M$, Check that if the coordinate system $x$ is $x = \chi^{-1}$ ...
31
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4answers
2k views

Is there a vector field that is equal to its own curl?

I was wondering if there is a vector field that satisfies the following condition: $$\vec F=\nabla \times \vec F$$
0
votes
1answer
44 views

Conversion of the Gauss law $\nabla \cdot E = \frac{\rho } {\epsilon_0}$ into integral form

This may be physics related but I think it belongs here because I have some doubt about mathematical operators we have gauss law in differential form as $$\nabla \cdot E = \frac{\rho } {\epsilon_0}$$ ...
2
votes
1answer
27 views

Vector Fields and Line Integrals

I wanted to know that in order to calculate the Line Integral of a Vector Field, is it necessary that the Vector Field has to be a Conservative Vector Field?
1
vote
1answer
37 views

Finding pathline

I've been trying to find the pathline of a particle dropped in a steady flow defined by the following vector components: $$ u= \frac{-2x}{(x^2+y^2+1)^2} \hat i + \frac{-2y}{(x^2+y^2+1)^2}\hat j $$ in ...
3
votes
1answer
54 views

Symmetry of Killing Vectors in Covariant Derivative

Several times, I've seen statements along the lines of "$\nabla_X Y=\nabla_Y X$ because $X$ is a Killing vector field." One example I found on Stack Exchange is here. I have yet to understand why ...