In vector calculus, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used to ...

learn more… | top users | synonyms

2
votes
0answers
34 views

Question about simply connected regions in $\mathbb{R}^2$

If given the vector field $\mathbf{F}(x,y)=\langle\frac{1}{x}+2xy,\frac{1}{y}+x^2-\cos{y}\rangle$, it is clear that the component functions are continuous everywhere expect at the origin. Now, when ...
3
votes
1answer
32 views

Does there exist a non-parallelisable manifold with exactly $k$ linearly independent vector fields?

For any positive integer $k$, is there a smooth, closed, non-parallelisable manifold $M$ such that the maximum number of linearly independent vector fields on $M$ is $k$? Note that any such $M$, ...
0
votes
1answer
49 views

The volume of a Torus

A torus $\mathscr{T}$ with the equation $$z^2 + \left( \sqrt{x^2 + y^2} - 2 \right)^2 = 1.$$ (a) Give an equation with a close line in the plane $Oxz$ where $\mathscr{T}$ is a surface of ...
0
votes
0answers
11 views

Line integral of vector field with singularity

Supose a vector field $\vec{F}(x,y)$ has a singularity at $(x_0,y_0)$. If I wanted to evaluate the integral $$\int_{\gamma}\vec{F}.d \vec{r}$$ along $\gamma$, knowing that $(x_0, y_0) \in \gamma$, how ...
1
vote
1answer
22 views

Non-existence of $1$-dimensional tangent distributions on the sphere $S^2$

I am reading about geometric quantization and real polarizations, and it is claimed that there exist no real polarizations on the sphere $S^2 \subset \Bbb R^3$ because every tangent field on $S^2$ ...
0
votes
0answers
13 views

Method of characteristics Pdes theory understanding

Wikipedia states (https://en.wikipedia.org/wiki/Method_of_characteristics ) At the start of the method trying to explain the geometricall meaning that the Field $F=(a(x,y,z),b(x,y,z),c(x,y,z))$ is ...
0
votes
2answers
20 views

Flow of sum of commuting vector fields

I'm trying to understand why the flow of sum of commuting vector fields is the composition of their flows. This is apparently supposed to be obvious but I don't see how.
1
vote
0answers
18 views

Flux of the amount of buffalo entering a square kilometer per minute

So Eq.(13) is just $\int F.n dS = \int \int div(F) dA$ So I did $div(F) = \nabla . F = y+1$, then I did $\int_2^3 \int_2^3 y + 1 dx dy = \int_2^3 y + 1 dy = .5(3^2) + 3 - (.5(2^2) + 2) = 3.5$ This ...
0
votes
2answers
10 views

Component test for conservative fields

I have a question concerning the component test for conservative fields. So the component test tells us that the vector field is conservative if the following three conditions are met. $$ P_y = N_z ...
0
votes
1answer
46 views

Outward flux of a vector field through a rectangular box.

Let $R$ be the rectangular box consisting of all points $(x,y,z)$ with $-1\leq x\leq 1$, $-1\leq y\leq 1$, and $-1\leq z\leq 1$. Define the vector field $$V= ...
1
vote
0answers
23 views

Construct manifold from vector fields and point in $\mathbb{R}^n$

Suppose one is given a set of vector fields $X_{\alpha} \in \mathbb{R}^n$. Is it generally possible to construct a manifold embedded in $\mathbb{R}^n$ going through a particular point and whose ...
2
votes
1answer
27 views

Lemma characterizing second fundamental form, do not understand step

Consider an excerpt of a lemma and part of its proof from a Riemannian geometry textbook. Lemma. The second fundamental form is independent of the extensions of $X$ and $Y$; bilinear ...
0
votes
0answers
39 views

Working with a vector field

Consider the vector field $\tilde{F}(x,y,z) = ax \tilde{i} + by \tilde{j} + cz\tilde{k}$ (1) Let $\tilde{c}(t) = (x(t), y(t), z(t))$ be the flow line such that $\tilde{c}(0)= (x_0, y_0, z_0)$. Find ...
0
votes
0answers
29 views

Vector field, flow line question. Need help please

Consider the vector field $\tilde{F}(x,y,z) = ax \tilde{i} + by \tilde{j} + cz\tilde{k}$ Let $\tilde{c}(t) = (x(t), y(t), z(t))$ be the flow line such that $\tilde{c}(0)= (x_0, y_0, z_0)$. Find ...
0
votes
0answers
27 views

Let $X,Y\in \mathfrak{X}(M)$ and $\Phi_{t}^{X}$ an integral curve of $X$. Show that $\frac{d}{dt}Y(m)(f\circ \Phi_{t}^{X}) =Y(m)(Xf).$

Let $M$ be a manifold, $X,Y\in \mathfrak{X}(M)$. Let $\Phi_{t}^{X}$ an integral curve or flow line of $X$. We define $$\begin{array}{rcl} Xf:M &\rightarrow & \mathbb{R} \\ m ...
1
vote
0answers
49 views

Can you comb the hair on a 4-dimensional coconut?

It is well-known that you can't equip the surface of the unit sphere with a singularity-free coordinate system. Physicists have called this theorem (which is important for the theory of black holes) ...
0
votes
1answer
26 views

Line Integrals in Vector Fields

In Part A of the following, $\vec{F}$ goes from $\langle y-x,x \rangle$ to $\langle cos(t)-\sin(t),\sin(t) \rangle$ with very little explanation: I would have thought that $\vec{r}(t)=\langle ...
0
votes
1answer
9 views

Vector field line integral: confusion about sign of dl, order of limits

I have some confusion about simple line integrals of vector fields. If I want to calculate integral $\int_C\underline F\cdot d \underline l$ from point $(1,0)$ to $(0,0)$ in a straight line, then ...
1
vote
0answers
23 views

Simple singularities of a vector field

Let $M^n \subset \mathbb{R}^{n+1}$ a compact and orientable hypersurface of even dimension, with Gauss map $\gamma : M \to S^n$, that is, $\gamma(p)$ is normal to $M$ at $p \in M$. Let $a, b \in S^n$ ...
1
vote
3answers
31 views

Work Done when more than one field exist

Suppose we have two different electric field, $\vec{E_1}$ and $\vec{E_2}$ where $\vec{E_i}$ are elements of $\mathbb R^2$ $y>0 => \vec{E}$=$\vec{E_1} $ and $y<0 => \vec{E}$=$\vec{E_2} $ ...
0
votes
0answers
20 views

Are scalar/vector fields basically just “multi-valued” functions?

Not really familiar with terminology in higher Mathematics, so I will try to use python to express my ideas instead. From Wikipedia: a scalar field associates a scalar value to every point in a ...
-2
votes
0answers
16 views

How do I prove that a vector field's partials being equal is necessary for the field to be conservative?

Suppose there is a $C^1$ vector field on an open set in $\mathbb R^3$. Why is it necessary for the vector field's partials to be equal for the field to be conservative? I'm sure the proof is very ...
0
votes
1answer
15 views

Gradient fields in simply connected space

Consider there following statement: Let F be a vector field such that $\nabla \times \textbf{F} = 0$ over a simply connected region R. Then in R, there is a scalar potential function $\phi$ such that ...
-1
votes
1answer
40 views
1
vote
0answers
16 views

$\varphi$-related vector fields when $\varphi$ is an inclusion

I'm reading Jeffrey Lee's Manifolds and Differential Geometry. He's talking about vector fields being $\varphi$-related to each other. He says If $S$ is a submanifold of $M$ and $X \in ...
0
votes
1answer
23 views

Streamlines, streaklines, and pathlines

I still have difficulty distinguishing the streamlines, streaklines and pathlines of a vector fields. Can you help me to understand this concepts by explaining to me the difference between them? By ...
1
vote
0answers
29 views

Biot-Savart Law to construct vector potential for divergence free vector field on $\mathbb{R}^3$

I would like to confirm a method I am trying to use which uses the Biot-Savart Law to construct a vector potential $\underline{w}$ for a divergence free vector field $\underline{v}$ on $\mathbb{R}^3$. ...
1
vote
1answer
47 views

Evaluate Path Integral

Consider the vector field F=\begin{pmatrix} -3z^2sinx \\[0.3em] 8y^3z \\[0.3em] 2y^4+6zcosx \end{pmatrix} By evaluating the path integral, compute ...
1
vote
2answers
56 views

Relation between exterior derivative and Lie bracket

There is a formula connecting the exterior derivative and the Lie bracket $$d\omega (X,Y) = X \omega(Y) - Y \omega(X) - \omega([X,Y]).$$ What is a good way to remember this? By which I mean, what ...
2
votes
1answer
30 views

Quick clarification on the definition of vector field

I am having a class on differential geometry and another on ODEs In the ODE class, if we were given something of the type $$\dot x = x^2$$ The professor refers to $x^2$ as the vector field. In ...
1
vote
2answers
32 views

Can pathlines always be found?

Given a two dimensional vector field $$\vec F(x,y) = P(x,y) \hat i+Q(x,y) \hat j $$ To find the path a particle would follow in this flow, one needs to solve the following differential equations: ...
0
votes
1answer
20 views

Function times a vector field (one-forms)

I am reading an introduction to differential geometry in the book "Classical Dynamics: a contemporary approach" by Eugene and Saletan (page 135). Consider a manifold $Q$ of dimension $n$. They define ...
0
votes
1answer
33 views

Divergence of “inverse square law” vector field

Let $\vec{F}$ be an inverse square law vector field given by $$ \vec{F} = \frac{\vec{r}}{\lVert \vec{r} \rVert^3} $$ where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$. Prove $\nabla \cdot ...
1
vote
1answer
23 views

Question on the image of the parameter space under two non-commuting flows.

Let $\phi^t_1$ and $\phi^s_2$ be two smooth flows on $\mathbb{R}^2$ defined for $t\in[-1,1]$ and $s\in [-1,1]$. Assume (1) $\phi^0_1\big((0,0)\big)=\phi_2^0\big((0,0)\big)=0$ (2) $(\phi^t_1)'|_0$ ...
1
vote
0answers
15 views

Time-dependent sections of a vector bundle?

Are there any books dealing with time-dependent vector fields and more generally with time-dependent sections of a vector bundle? Thanks
1
vote
0answers
6 views

Extension to a normal field

Let $M$ be a submanifold of the riemannian manifold $\overline{M}$, with the induced metric. Denote by $\nabla$ and $\overline{\nabla}$ the riemannian connections of $M$ and $\overline{M}$, ...
1
vote
1answer
39 views

Continuous vector field tangent to even dimensional sphere which only vanish at one point

I am working on path motion planning on different topological spaces. In order to prove the existence of some motion planning algorithms I would like to use that given an even dimensional sphere, we ...
3
votes
0answers
63 views

Calculate integral curve through a discretely sampled vector field

given a finite sample of a smooth vector field, i.e. a set $$S=\{(p_i,v_i)\mid i\in \{1,\dots, n\}, p_i,v_i\in\mathbb{R}^2\}$$ where the $p_i$ are the points at which we have a vector $v_i$, how do I ...
2
votes
1answer
55 views

How does mathematicians reconcil the fact that the gradient not drawn with respect to the origin

For example, consider the gradient vector of a function Obviously, $\nabla f(x_o,y_o)$ on the above figure is not attached to the origin. But attached to the point for which it is evaluated at. ...
3
votes
1answer
59 views

On the existence of a point in the plane where repulsive central forces exerted by $ n $ fixed points cancel

This is a physics-inspired question. In what follows, $ \alpha \in (1,\infty) $ is a fixed constant, $ n \in \mathbb{N} $ a fixed integer $ \geq 2 $, and $ [n] \stackrel{\text{df}}{=} ...
4
votes
1answer
42 views

Symplectic manifold, flow $\phi_t$ generated by unique vector fiel $X_f$ preserves symplectic form $\omega$?

Let $M$ be a symplectic manifold with symplectic form $\omega$, let $f$ be a smooth function on $M$, and let $X_f$ be the unique vector field on $M$ so that $df(Y) = \omega(X_f, Y)$ for all vector ...
2
votes
2answers
51 views

Symplectic manifold $M$, unique vector field $X_f$ on $M$ so that $df(Y)= \omega(X_f, Y)$ for all vector fields $Y$?

Let $M$ be a manifold. A symplectic form is a closed $2$-form $\omega$ suh that $X(p) \to \omega(X,\cdot)(p)$ is an isomorphism from $T_pM$ to $T_p^*M$ for each $p \in M$. A manifold together with a ...
0
votes
0answers
7 views

Variation of diffeomorphism mapping

I don't understand the lines of the following paper: Beg, M. Faisal, Michael I. Miller, Alain Trouvé, and Laurent Younes. “Computing Large Deformation Metric Mappings via Geodesic Flows of ...
1
vote
1answer
22 views

Steepest part of a surface.

Related to Steepest part of $\cos x + \cos y$ Is there a general way of finding the steepest part of a surface? I know that to find the steepest part of a normal function $f$, you'd look for ...
0
votes
0answers
23 views

Is this way of solving a line integral on a vector field correct?

Given $\nabla\varphi = \mathbf{F}$ and $\mathbf{r}(t)$ from $a$ to $b$, $$\int_c\mathbf{F}\cdot\mathrm{d}\mathbf{r} = \int_c\nabla\varphi\cdot\mathrm{d}\mathbf{r} = ...
1
vote
1answer
44 views

Determine every vector field such that its field lines are contour lines to $g(x, y) = x^2 + 4y^2$

Is it possible to determine every vector field such that it's field lines are contour lines to $g(x, y) = x^2 + 4y^2$? If so, how?
2
votes
1answer
63 views

Determine the field lines to the vector field $F(x, y) = (x^3 y, x y^3)$

I can't find a proper way to determine the field lines to a vector field! For example, if I have the field $F(x, y) = (x^3 y, x y^3)$, how can I find its field lines? Any help would be much ...
2
votes
1answer
32 views

Proving continuity/smoothness for a special function on a Lie group.

So I asked this question, yesterday, forgetting the compactness requirement. Jack Lee commented shortly afterwards, noting that, if we take an inner product on the Lie algebra $T_eG$ of a Lie group ...
0
votes
0answers
10 views

How can I find the stability of the equilibria of this vector field?

Consider the vector field given by $y' = y - y^{3}$. This clearly has equilibria at the points $y = 0, \; y = 1, \; y = -1$. How would I find the stability of these points though? I understand that I ...
0
votes
0answers
11 views

Vector integration in n dimensions

In n dimensions I want to do an integral of the flux through an n-1 D surface. The usual vector calculus integration theorems say I can integrate around the perimeter of the surface. OK, but that ...