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 model,...

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1answer
15 views

Potential of vector field is undefined on Y-axis although field is defined

I'm having the following vector field: $$\vec{F}(x,y) = (\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2})$$ The field is conservative in $\mathbb{R}^2 \backslash (0,0)$ as long as your curve doesn't encircle $(...
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1answer
25 views

What is $\vec{v}(\vec{\gamma}(t))$?

If we got the curve $\vec{\gamma}:[0,1]\rightarrow\mathbb{R^3}$ $$\vec{\gamma}(t) = \left(\! \begin{array}{c} t \\ t^2+1 \\ t \end{array} \!\right) $$ And the vector field $\vec{v}:\mathbb{R^3}\...
1
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0answers
34 views

A vector field in a star shaped set

I'm having problems trying to proof Poincaré's lemma for Star-Shaped sets Let $F:U\to \mathbb R^{2}$ be $C^{1}$ functions,where $U\subset \mathbb R^{2}$ is a Star-shaped set. If $F_x:U\to \mathbb ...
2
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1answer
28 views

divergence of a radial vector field

I wanted to calculate a simple example for the integral representation of the divergence $$\vec\nabla\cdot\vec{A}=\lim\limits_{\Delta V \rightarrow 0}{\frac{1}{\Delta V}}\iint_{\partial(\Delta V)} \...
3
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1answer
58 views

Irrotational fields and divergence

Let $F,G$ be $C^1$ vector fields from $\mathbb R^n$ in itself. The condition $$\int_{\partial A}F\cdot \nu_A\ d\sigma=\int_{\partial A} G\cdot \nu_A\ d\sigma$$ for every bounded domain $A$ whose ...
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0answers
52 views

The tension field is a vector field

Recall that the tension field of a function $f:(M,g)\rightarrow (N,G)$ is given in local coordinates by \begin{align*} & \Delta_gf^k+g^{ij}\hat{\Gamma}^k_{mn}\partial_if ^m\partial_jf^n \\ \end{...
2
votes
1answer
46 views

Lie bracket and inner product

$X, Y, N$ are vector fields on a riemannian manifold $M$. $\langle X, N \rangle(p)=0 $, $\langle Y, N \rangle(p)=0$ at some point $p$. Show $\langle [X,Y], N \rangle(p)=0$. I want to use $[X,Y]=XY-...
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0answers
16 views

Using Gauss Divergence Theorem Please help calculate

Please help me. I found div F. But I cannot handle the w.
2
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2answers
55 views

$\nabla \cdot (\nabla \times \vec A) = 0 $ Proof

Show that $\nabla \cdot (\nabla \times \vec A) = 0 $ for an arbitrary differentiable vector field $\vec A$ using Stokes' Theorem for an arbitrary closed surface S followed by Gauss' Theorem. An ...
2
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1answer
49 views

Flow of a vector field, equivalent characterizations

Given a vector field $X$ on the manifold $M$ its flow is, loosely speaking, a map $F= F(t,x)= F^t(x)$, in the variables $(t,x)\in I\times M$, such that the curve $t\mapsto F^t(x_0)$ is the unique ...
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25 views

Time-independent vector field

In a time-independent vector field I want to draw streamlines, path and streak lines. Given $v_x=-x$ and $v_y=y$, how should I draw them ?
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42 views

How would i go about solving these such problem?

The problem is for Vector Calculus. I am not sure what this question is asking. $\text{a) Assume }f\text{ is of class }C^2.\text{ Show that }\nabla\times\nabla f = \vec{0}.\\\text{b) Is }\mathbf{F}...
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1answer
38 views

(Closed) Line integral of Conservative Field.

Suppose we have a conservative Field $ \vec F: D' \subseteq R^2 \rightarrow R^2$ where D is a set of points inside a closed curve (for example all the points inside a circle). Say we have subset of D',...
1
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1answer
69 views

Prove the linearity of the following maps

Let $f,g$ and $h$ be in $C^\infty$ on $\mathbb{R}^3$. Is the following statement true? If $x^2+y^2+z^2=(f(x,y,z))^2+(g(x,y,z))^2+(h(x,y,z))^2$ then $F=(f,g,h)$ is linear Maybe this could be a ...
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0answers
20 views

Finite separable fields extensions and discriminant

I am supposed to prove that for a finite separable field extension $L/K$, the discriminant $Discr_{L/K}$ is not zero. (For a basis $\{a_1,\ldots,a_n\}$ of $L$ over $K$, the discriminant is defined by ...
0
votes
1answer
30 views

What are the equilibrium point of this coupled ODE?

Consider $$\dot x = x(a - bx - cy)$$ $$\dot y = y(-d + ex - fy)$$ $$a,b,c,d,e >0, f \geq 0$$ Find all the equilibrium points in the set $\mathbb{R}^2_{\geq 0}$ I can find by inspection the ...
2
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0answers
59 views

Can you prove that modules over a field are vector spaces in a categorical way?

Let $\mathcal{C}$ be the category of abelian groups endowed with tensor product. It is a monoidal category with unit object the integers. Given a monoid $(R, \mu_R , \iota_R)= R$ in this category (i.e....
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2answers
50 views

Compute the surface integral

I always get confused with this... Let $S$ be the surface $$z=x^2+y^2, z\leq 1,$$ oriented so that the normal vector has positive $z$-coordinate, and let $F$ be the vector field $(yz,-xz+\sin(z), ...
1
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1answer
40 views

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

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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0answers
16 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
25 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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0answers
188 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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0answers
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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0answers
23 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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0answers
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 $f(x)=\int_0^1 J(sx)xds$.
4
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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:
1
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1answer
32 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
35 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, $P=...
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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 ...
2
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0answers
38 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 ...
4
votes
2answers
76 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
54 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
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0answers
17 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
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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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0answers
14 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
26 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
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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 ...
0
votes
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 ...
0
votes
1answer
55 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= x(x^2+y^2+z^2)^{-3/2}\mathbf{i}+y(x^2+y^2+...
1
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0answers
26 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
31 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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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 $\...
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0answers
32 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
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0answers
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
51 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
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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 \cos(...
0
votes
1answer
13 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 $a, b \in S^n$ ...
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} $ ...