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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5 views

$f:M\to N$ smooth manifold map. $F(x)=(x,f(x))\in M\times N$. For each $X\in \mathfrak{X}(M)$ there's a F-related $Y\in \mathfrak{X}(M\times N)$.

Let $M$ and $N$ be to manifolds and $f:M\to N$ smooth map. Define $F:M\to M\times N$ by $F(x)=(x,f(x))$. Show that for each $X\in \mathfrak{X}(M)$ there's a F-related $Y\in \mathfrak{X}(M\times N)$. ...
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1answer
13 views

Let $X_i, Y_i$ be vector fields on the manifolds M and N. $X_i\oplus Y_j$ on $M\times N$. $[X_1\oplus Y_1,X_2\oplus Y_2]=[X_1,Y_1]\oplus [X_2,Y_2]$

Let $M$ and $N$ be two differentiable manifolds and $X_1,X_2$ be two vector fields on $M$ and $Y_1, Y_2$ on $N$. Using the fact that $T_p(M)\oplus T_q(N)$ is naturally isomorphic to $T_{(p,q)}(M\times ...
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17 views

Finding the vector field of the given potential function

Given the potential function $$f(V) = \cos (|X|^2), \quad X = (X_1, \dots, X_n)$$ find the vector field $V(X).$ I'm not really use to do this kind of question, usually they ask the other way ...
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23 views

Prove $F$ is a locally conservative vector field

Let $F:\Omega\rightarrow\mathbb{R}^n$ a continuous vector field. Given that for any compact set $K\subset\Omega$, $\epsilon>0$ exists $\delta>0$ such that if $\gamma(t),\eta(t), t\in[a,b]$ are ...
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0answers
15 views

Proof regarding little o for vector field

I'm struggling with one part of a proof of a theorem. Let $\gamma(t) : A \subseteq \mathbb{R} \rightarrow \mathbb{R}^m$, with $\gamma \in \mathscr{C}^1(A)$ (hence $\gamma$ differentiable). ...
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20 views

Proof of equivalence between limit of a vector field and limit of a scalar field

I have a doubt with a proof regarding the following implication. Consider $F=(f_1,..,f_m): A \subset \mathbb{R}^n \rightarrow > \mathbb{R}^m$ and $\bar{x}$ a limit point for $A$, then $$\...
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11 views

Transformation of vector from vector field coordinates

Consider an n-dimensional manifold with coordinates $x_1, x_2, \dots, x_n.$ Suppose we have a vector field defined on this manifold $V : \bar v = \bar v(\bar x).$ Let us perform a homogeneous ...
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2answers
15 views

Which linear transformations are more abundant: dimension-increasing, preserving or decreasing?

My final aim is to understand the increase of Von Neumann Entropy in quantum systems by analyzing classes of unitary matrices in finite-dimensional Hilbert spaces. I'm following a potentially very ...
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9 views

Invariance of the Divergence Operator Under Orthogonal Transformations

Consider a vector field $v : \mathbb{R}^3 \rightarrow \mathbb{R}^3$. Denote the first, second, and third components of this field by $v_1$, $v_2$, and $v_3$ and the Cartesian coordinates by $x_1$, $...
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25 views

Trajectories of a vector field on the 2-sphere

Consider the vector field given by given by $(-zx,zy,0)$, where we've identified $T_pS^2$ where we've identified the space of vectors orthogonal to $p$. How do we visualize the trajectories of the ...
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1answer
53 views

Stokes theorem concept question

I've got a conceptual question about Stokes theorem. So the way I understood it Stokes theorem is used to calculate the counterclockwise circulation through a smooth oriented surface. However one of ...
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0answers
17 views

Is the unit normal really necessary to calculate the flux of a vector field?

So I've been studying Multivariable Calculus and this definition still eludes. Let $ \gamma:\mathbb [a,b] \rightarrow \mathbb R ^2$ be a closed, simple, regular curved. It is also oriented ...
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3answers
46 views

Finding a line integral of $F(x,y) = (3x^2\cos y + 2\cos x, -x^3\sin y)$ along a given curve.

Let $F$ : $R^2 \to R^2$ be the vector field F(x,y) = ($3x^2cosy+2cosx,-x^3siny$) and $\gamma$ : [$0$,$\pi$]$\to$$R^2$ be the curve $\gamma(t)=(t,(\pi-t)^2)$. Find the line integral of $F$ along $γ$ ? ...
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1answer
58 views

Existence of closed level sets on a surface for some field

Consider an infinite 3D space with only 2 things in it: wind and a solid object. Wind evidently blows around this solid object over its rigid surface. Bascially we are trying to set up a pure field. ...
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0answers
34 views

What is the difference between $ (A\cdot\nabla )B$ and $A(\nabla\cdot B)$?

What is the difference between $ (A\cdot\nabla )B$ and $A(\nabla\cdot B)$, where $A$ and $B$ are two vectors. My guess is that $ (A\cdot\nabla )B$ means the vector $A$ is being multiplied by the rate ...
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0answers
10 views

Equality of vector & Directions

Is the direction 120° west of the South Axis same as the direction 30° north of the West Axis (both from a fixed observational point).
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7 views

Stationary solutions in dissipative fields

I am trying to solve the following exercise: Let $F:\mathbb R^d \rightarrow \mathbb R^d$ a continuous and locally Lipschitz vector field. We say that $F$ is dissipative if and only if there exists ...
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1answer
20 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
26 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}\...
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0answers
36 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
32 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)} \...
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1answer
62 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
53 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{...
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1answer
50 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-...
2
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2answers
59 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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26 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
39 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',...
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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
24 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 ...
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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
60 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
52 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), ...
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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
189 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
24 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 $f(x)=\int_0^1 J(sx)xds$.
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:
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1answer
33 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
36 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
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2answers
78 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 ...