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2
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0answers
15 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
4 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
18 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
37 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
17 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
15 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
25 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
53 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
48 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
vote
0answers
56 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 ...
0
votes
1answer
39 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 $x = \chi^{-1}$ ...
30
votes
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
25 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
26 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
35 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
45 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 ...
-1
votes
1answer
29 views

Work done on a particle moving in a force field

Given a path C and a force field F, the work done on the particle can be found by $$ \int_C \vec{F}\cdot \vec{r} \,dr $$ This seems to suggest that you can find the work done for any path. Doesnt ...
0
votes
1answer
29 views

Two ways of finding a Potential of a Vector Field

If $\vec{F}(x,y)$ is a conservative vector field and we want to find a function $V$ such that $\nabla(V)=\vec{F}$, then one way to do it is to take an arbitrary point $(x_0,y_0)$ and then define ...
2
votes
2answers
33 views

Divergence Theorem to calculate flux

Take the vector field given by: $F= (y^2+yz)i+(\sin(xz)+z^2)j+z^2k$ a) Calculate the divergence, $\operatorname{div}F$. b) Use the divergence theorem to calculate the flux $$\int_S F\cdot dA $$ ...
2
votes
1answer
41 views

Vector field with gradient and integral over curve

The problem is: Consider the vector field: $$\textbf{F}= 4x^3y^3 \,\textbf{i} + (1+3x^4y^2) \,\textbf{j}$$ a) Find a potential function $ϕ(x,y)$, i.e. a function $ϕ(x,y)$ such that $\nabla ϕ= ...
2
votes
0answers
22 views

When a vector field can be scaled to form a conservative vector field

Consider a vector field given by its components $g_i(x_1, \dots, x_n)$. It is well known that necessary and sufficient condition for a following system $$ \frac{\partial f}{\partial x_i} = g_i(x_1, ...
1
vote
1answer
31 views

plotting cone on matlab

Plot the portion of the cone $z=sqrt((x − 1)^2 + y^2)$ inside the cylinder $r = 2$ This is my matlab code. However, I get an error with the ezsurf line. Any ideas as to what I might be doing wrong? ...
3
votes
1answer
42 views

What is an example of a map not satisfying this rank condition?

Definition: Consider a Lie Group $G$ and a set of right invariant vector fields on $G$, denoted $\Gamma$. A point $y \in G$ is called normally accessible from a point $x \in G$ by $\Gamma$ if there ...
2
votes
1answer
33 views

multivariable calculus charge density question

The sphere given by $x^{2} + y^{2} + z^{2} = 4$ is submerged in an electric field with charge density given by $f(x, y, z) = x^{2} + y^{2}$. Find the total amount of electric charge on this surface. ...
1
vote
1answer
33 views

Vector Fields problem unsure of how to start

A fluid having density $f(x, y, z) = x^{2} + 2y^{2} + z^{2}$ flows with velocity $v(x, y, z) = x^{2}i + xy^{2}j + zk$. Determine the rate of mass flow through the sphere $ρ = 1$ in the outward ...
0
votes
0answers
21 views

How to compute $\int_C (X|dx)$?

I have the vector field $X(x,y,z) = (\frac {-y}{x^2+y^2+z^2}, \frac{x}{x^2+y^2+z^2},0)$ on $\mathbb{R} ^3$ \ $\left\{ (0,0,0) \right\}$. I need to compute $\int_C (X|dx)$ over a closed circle of ...
0
votes
0answers
14 views

When to use Stoke's over Divergence and vice versa

I'm still very confused as to when I'd use Stoke's Theorem over Divergence Theorem and vice versa. What are some very obvious things I could look for in a problem to determine which theorem to use? ...
1
vote
1answer
24 views

Evaluate a line integral using the fundamental theorem of line integrals

Consider the vector field $${\mathbf F}(x,y)=(e^x)(\sin y){\mathbf i}+(e^x)(\cos y){\mathbf j},$$ and the curve $C$ composed of the graph of $\sqrt{x}+\sqrt{y}=5$ followed by segment from $(25,0)$ to ...
1
vote
1answer
31 views

Integral curves and Vector field

For a given vector field $X(x,y,z) = (\frac {-y}{x^2+y^2+z^2}, \frac{x}{x^2+y^2+z^2},0)$ on $\mathbb{R} ^3$\ {(0,0,0)}, what is the integral curve $\int_C (X|dx)$ over a closed circle with radius of 1 ...
0
votes
0answers
16 views

Calculating a unique vector field given 2 vector fields in $\mathbb{R}$

So I have this exercise that is partially solved, but I am stuck in one of the steps. I would really appreciate it if someone could explain it to me. Thanks! :) EXERCISE: Let $u=\sum u_k ...
1
vote
0answers
25 views

How to use three complex vector components to calculate resultant complex vector

This is a practical problem related to complex vectors. Imagine you want to find the resultant electric field of multiple electromagnetic waves that have parallel and perpendicular components ...
1
vote
1answer
23 views

finding line integral with potential function [duplicate]

Evaluate the line integral $$ \int_C (\ln y) e^{-x} \,dx - \dfrac{e^{-x}}{y}\,dy + z\,dz $$ where C is the curve parametrized by $r(t)=(t-1)i+e^{t^4}j+(t^2+1)k$ for $0\leq t\leq 1$ I know that the ...
1
vote
2answers
38 views

What do gradient, curl, and div input and output?

What do gradient, curl, and div input and output? (e.g. vector field or scalar function of several variables)
1
vote
1answer
43 views

When is a vector field on a manifold restricted to a submanifold $X$ a vector field on $X$?

Let $X$ be an embedded submanifold of $M$ and let $V$ be a vector field on $M$. One can restrict $V$ to $X$, but it may not define a vector field on $X$. Example: The vector field $x^i\partial_i$ on ...
0
votes
0answers
17 views

Find value of line integral: $\int_C(2x-3y+1)dx-(3x+y-7)dy$

Find value of line integral $$ \int_C(2x-3y+1)dx-(3x+y-7)dy $$ $$ C: from\;(0,1)\;to\;(2, e^2) $$ I know this line integral is conservative: $\frac{M}{dy}=\frac{N}{dx}=-3$ Its potential function I ...
0
votes
0answers
49 views

How to define an action of vector field on $C^{\infty}(M)$?

Let $M$ be a manifold. Let $\hat{X}$ be a vector field on $M$. How to define an action of $\hat{X}$ on $C^{\infty}(M)$? Thank you very much.
1
vote
1answer
27 views

How to define the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$?

I read a paper and on page 9, the paragraph before Proposition 5, it is said that let the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$. How to ...
0
votes
3answers
33 views

Is this a vectorial space

I can't understand why the 0 vector here is not unique?
0
votes
1answer
28 views

Exact ODE:$ \left(\frac{y}{x}+6x\right)dx + (\log(x) - 2)dy =0$

I need to solve the equation $$\left(\frac{y}{x}+6x\right)dx + (\log(x) - 2)dy=0$$ We can easly see that $$\left(\frac{y}{x}+6x\right)'_y =(\log(x) - 2)'_x$$ but the domain of $ ...
0
votes
0answers
19 views

How do I find flow of $\vec{a}$ in direction of $\vec{n}$ on surface $D$?

Problem: I have to calculate the flow (I do not know if this is the correct term in english) of vector field $\vec{a}(M)$ through triangle surface, created, when plane $(p)$ intersects planes ...
1
vote
1answer
44 views

Curl and Vector Fields

I am having real difficulty knowing how to approach this question, so any help or pointers would be appreciated. Consider the vector field: $$ \vec{G} = -3xz^2\vec{i} + z^3\vec{k} $$ FInd a vector ...
0
votes
0answers
89 views

Lie groups, Lie algebra and left invariant vector fields

Hi I'm learning about Lie Groups to understand gauge theory (in the principal bundle context) and I'm having trouble with some concepts. Now let $a$ and $g$ be elements of a Lie group $G$, the left ...
0
votes
1answer
17 views

Gradient of a function with base vectors

\begin{align} \nabla\left(x_1x_3\hat{e}_1+x_2x_3\hat{e}_2\right)&=x_3\left(\hat{e}_1\otimes\hat{e}_1\right)+x_3\left(\hat{e}_2\otimes\hat{e}_2\right)\\&+\mathbf{x_1\left(\hat ...
4
votes
1answer
76 views

Lie algebras of vector fields over $\mathbb{R}$ and over $S^1$

An exercise in the book "Moonshine beyond the Monster" has me stumped. It asks whether the real Lie algebras of smooth vector fields over the reals $V(\mathbb{R})$ and over the circle $V(S^1)$ are ...
0
votes
0answers
9 views

Helmholtz decomposition of two given vector fields

I am trying to solve the following task: Show that the following vector fields can be resolved into a scalar field $g(\vec x)$ as well as a vector field $\vec w(\vec x)$ where $\vec v(\vec x) = - ...
2
votes
0answers
50 views

Calculating the Integral of a non conservative vector field

I have no clue how to do part C because a) is non conservative What I got for b) $f(x,y)=\frac{x^3}{3}+2yx+\frac{y^3}{3}+K$ (I don't know the symbol for the thing so I used f(x,y) instead. How do I ...
0
votes
2answers
65 views

Vector space function question

Q: Let $X$ be a set and $\Bbb F$ be a field. Consider the vector space $V=Fun(X,\Bbb F)$ of functions from $X$ to $\Bbb F$ with operations of addition and mult. (i) Describe the zero element of V and ...
2
votes
1answer
39 views

Is the continuity of a vector field enough for the existence of the solution of a differential equation?

I've recently seen the existence-uniqueness theorem for ordinary differential equations from Arnold's book. I understand that the theorem as stated guarantees both existence and uniqueness if the ...
1
vote
1answer
52 views

Degree of a vector field on the boundary of a punctured ball?

Suppose $v\colon \mathbb{R}^3\setminus\{x_1,\dots,x_n\}\to S^2$ is a smooth map, where all the points $x_i$ sit inside the unit ball $B^3$, and not on the boundary $S^2$. I suppose this map can be ...
0
votes
1answer
27 views

A basic relation in spherical coordinates

Why is it that $$x\partial_x+y\partial_y+z\partial_z=r\partial_r~?$$ I know that $$r^2=x^2+y^2+z^2,$$ but how is this relation implied?