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 ...

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30 views

Vector Fields on $\mathbf R^2$ [on hold]

Let $X : \mathbf R^2 \to \mathbf R^2$ be a no-zero smooth vector field. I want to show (without background about vector bundles or manifolds, just if possible differentiable calculus in $\mathbf R^n$) ...
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15 views

Necessary condition for local existence of solution for system of two first order PDEs

Let $f,g$ be two smooth functions on $\mathbb{R}^2$ and $U,V$ be two smooth vector fields on $\mathbb{R}^2$. What is the sufficient condition for local existence of a solution $\phi$ of equations ...
3
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2answers
24 views

Can a Submanifold Become Tangent to a Nowhere Tangent Vector Field

$\newcommand{\R}{\mathbf R}$ Let $M=\R^2$ and $S=\{(0, t):-1<t < -1\}$ be a submanifold of $M$. Let $V$ be a vector field on $M$ which is nowhere tangent to $S$. Let $\theta$ be the flow of ...
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185 views

Vector Calculus Temperature Profile

Question : If $T(r) = \frac{T(0)}{r^3}$ is the temperature profile in the region R, then use the previous results to calculate the average temperature in R when $T(0) = 1000$. Verify that the average ...
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10 views

derivative of scalar vector multiplication

If we have a scalar($\mu$)-vector($X_{n \times 1}$) multiplication and take a derivative with respect to scalar $\mu$ as follows, $$\frac{\partial}{\partial \mu}\mu X$$ does it give $X$ or $X^{T}$? ...
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22 views

Normal vector field for an immersion

I know that a hypersurface $M$ of a riemannian manifold $N$ is orientable iff there exists a globally defined unit normal vector field $\eta : M \to TM^{\perp} \subset TN|_M$. Does the same hold for ...
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3 views

Vector field f and Jacobian relation

The problem is a smaller part of a bigger problem. I need help with the following: Prove $f(x)=\int_0^1 J(sx)xds$.
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1answer
37 views

Normal vector field to this hypersurface

Let $M^n$ be a hypersurface of the unit sphere $S^{n+1} \subset \mathbb{R}^{n+2}$ contained in the open upper hemisphere $S^{n+1}_+$, and let $N : M \to \mathbb{R}^{n+2}$ be a unit normal vector field ...
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1answer
29 views

Differential Equations Direction Field Problem [closed]

Can anyone do this problem? I'm struggling with it:
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1answer
31 views

Flow of a vector field?

How can I find a flow curve for a vector field given as, $$U(x,y,z)=(2z+x, y-z,z+y)$$ with a condition that $r(0)=(x_0,y_0,z_0)$.
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1answer
34 views

Flow curves in a vector field?

In $(x,y,z)$ space, we have the following vector field, $$V(x,y,z)=(V_1(x,y,z), V_2(x,y,z), V_3(x,y,z))=\left(z^2+x+1, y^2 - yz, y + \frac{z^2}{2}+\frac{1}{2}\right)$$ Let us consider the points, ...
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3answers
53 views

Is this a vector field?

One of the example questions we've been given in lectures is to plot the vector field given by the first order equation $y' = y - y^{3}$. The solution we were given looks like this: However, I ...
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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 ...
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1answer
39 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 $M$, ...
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1answer
51 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 ...
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15 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 ...
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1answer
26 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 $S^2$ ...
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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 ...
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2answers
21 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.
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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 ...
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2answers
15 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 ...
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1answer
54 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= ...
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0answers
25 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 ...
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1answer
29 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 ...
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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 ...
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31 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 ...
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29 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 ...
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0answers
50 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) ...
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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 ...
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1answer
11 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 ...
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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 $a, b \in S^n$ ...
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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} $ ...
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25 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 ...
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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 ...
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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 ...
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1answer
27 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 ...
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0answers
31 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
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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 ...
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2answers
62 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
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1answer
31 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 ...
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2answers
33 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: ...
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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 ...
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1answer
37 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 ...
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1answer
24 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) $(\phi^t_1)'|_0$ ...
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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
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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 $\overline{M}$, ...
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1answer
50 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
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65 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
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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. ...