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2
votes
2answers
35 views

Divergence of inverse square vector field

After trying to get the divergence of a vector field I got: $${\nabla \cdot \Bigg(\frac{\vec r}{|r|^3}\Bigg)=\frac{\partial}{\partial r}.\frac{1}{r^2}=-\frac{2}{r^3}}$$ But in wikipedia found that ...
3
votes
1answer
20 views

Vector fields spanning the tangent bundle at each point

Let $M$ be a smooth manifold of dimension $n.$ It is well known that it is not allways possible to find such global vector fields $X_1,\ldots, X_n$ which would be a basis for the tangent space at each ...
2
votes
1answer
27 views

Definition of “vector fields never have opposite direction”

Good day! As in my other question I am referring to the book "Differential Equations and Dynamical Systems" by Lawrence Perko, chapter 3.12. I have a question regarding Lemma 2: Lemma 2. If $v$ ...
1
vote
2answers
40 views

Quickest way to determine if a vector field is conservative?

Say I have some vector field given by $$\vec{F} (x,y,z)=(zy+\sin x)\hat \imath+(zx-2y)\hat\jmath+(yx-z)\hat k$$ and I need to verify that $\vec F$ is a conservative vector field. What would be the ...
1
vote
0answers
15 views

ONB (right handed) - vector fields - green-region

There is a right handed orthonormal basis ${a_1,a_2,a_3}$ of $R^3$ (meaning $a_3=a_1\times a_2$) and a number $\omega \geq 0$. The rotation around the axis $a_3$ with the constant angular velocity ...
0
votes
0answers
20 views

Righthanded ONB.

There is a right handed orthonormal basis ${a_1,a_2,a_3}$ of $R^3$ (meaning $a_3=a_1\times a_2$) and a number $\omega \geq 0$. The rotation around the axis $a_3$ with the constant angular velocity ...
0
votes
0answers
19 views

Vector-fields - divergence theorem.

Let $c>0$ and $A\subseteq R^3$ be a green-region (which is defined by: $A\subseteq R^2$, $A=B_1\cup ...\cup B_m$, and $B^{\circ}_i\cap B^{\circ}_j=\emptyset$) with the outward normal unit ...
0
votes
0answers
15 views

Oriented surface - sphere

$S^+$ and $S^-$ are the upper ($z\geq 0$) and lower half ($z\leq 0$) of the surface area of a sphere with Radius $R>0$ where the symmetry axis is the z-axis. $k:R^3\to R^3, ...
3
votes
1answer
29 views

Doubt about conservative fields in 2D and 3D

Regarding a conservative field $\vec{F}$ in a region $D \subseteq R^2$, I know that the requirements are: Curl of $\vec{F}$ is $0$. $\vec{F}$ is defined in D (doesn't have singularities in D). But ...
1
vote
1answer
27 views

Reverse engineering a differential equation from singular points

I've been struggling to find a way to reverse engineer a differential equation based on knowing it's singular points. In this case, I'd like to create a flow on $[-1,1] \times [-1,1]$, which has ...
2
votes
1answer
52 views

What is the level set of 1D function?

I have encountered a very strange idea today. My prof claimed that the differential of a function $df$ is the number of level curve crossed by a tangent vector $v$ at a point $p$ I tried to reason ...
1
vote
3answers
43 views

How can vector field simultaneously be a function and also an operator that acts on a function?

In elementary calculus we have definition: A vector field is a function that assigns a vector to each point in $\mathbb{R}^2$ or $\mathbb{R}^3$ i.e. F(x,y) = P(x,y) $\hat i$ + Q(x,y,) ...
0
votes
0answers
16 views

If a Point Admits an Integral Curve on an Interval then a Neighborhood Does too On the Same Interval

$\newcommand{\R}{\mathbf R}$ Let $V:U\to \R^n$ be a (continuous) vector field on an open set $U$ of $\R^n$. Suppose we have a point $\mathbf p\in U$ and an open interval $I\subseteq \R$ such that ...
0
votes
0answers
37 views

Vector fields with constraints on velocities

Let $X^{'}=f(X)$ be a system of n differential equations where $X=(X_1,..,X_n)$, $f=(f_1,..f_n)$. We assume that $f : \Bbb{R}^n \to \Bbb{R}^n$ is a smooth function. Let $F: \Bbb{R}^n \to ...
0
votes
1answer
22 views

Flux through a Hemisphere.

I need to calculate the flux of a vector field : $H = (y-z)\hat{i} + (z-x)\hat{j} + (x-y)\hat{k}$ outside the Hemisphere given by the equation : $(x-1)^2 +y^2 +z^2=1$ with $z\ge0$ Now i used ...
2
votes
1answer
29 views

Some algebraic properties of curl

Given vector fields $\mathbf E$ and $\mathbf H$ with curl $\mathbf E= - \frac 1 c \frac {\partial \mathbf H} {\partial t}$ and curl $\mathbf H= \frac 1 c \frac {\partial \mathbf E} {\partial t}$ ...
1
vote
2answers
55 views

What interpretation of the Lie braket is this?

I was reading exercise 1.96 of Gadea's Analysis and Algebra on Differentiable Manifolds. I don't know what definition of Lie braket the author used, but I'm confused, as far as I know ...
4
votes
2answers
95 views

How do ideas in differential geometry expand upon ideas from introductory calculus

I just went through first year in mathematics and used Stewart's book for calculus. I am trying to self study differential manifold and I find many concepts such as chart, atlas very similar to that ...
1
vote
0answers
30 views

Find the mass flow rate, given a surface, density and velocity field

I have a confusion, I hope you can help me (I'd like that if you will respond, please read all my post). They ask me to find the mass flow rate passing through a surface, where the velocity field is ...
1
vote
0answers
5 views

Conservative field $F$ on not simple connected set

Give an example of a field $F:D\subseteq \mathbb R^2 \to \mathbb R^2$ such that $D$ is a doubly connected set (that is $D$ has on "hole") but $F$ is conservative And if $D$ is a triply connected ...
1
vote
1answer
32 views

Compute flux of vector field F through hemisphere

I need help solving this question from my textbook. Compute the flux of the vector field: $$\vec F = 4xz\vec i + 2 y\vec k$$ through the surface $S$, which is the hemisphere: $x^2 + y^2 + z^2 = 9 , ...
3
votes
1answer
20 views

Finding the parameterization of a curve for a line integral problem

I have to calculate the work of a particle that travel along a curve, given the following vector field: $F(x, y, z) = (2z-1, 0, 2y)$ and where the curve is the intersection between: $s1: z = x^2 + ...
0
votes
0answers
19 views

Given a vector field, how would I solve for constants, if I know that the Intergral F.dr = 0 along any curve C?

Say I have a field F = (2x + az)i + (bx - 3)j +(-3x+ cy +2z)k. Or any equation in general How would I solve for a, b and c so that the closed integral F.dr = 0?
0
votes
1answer
19 views

Proving the existence of certain vector field along a piecewise differentiable curve

I am trying to understand proposition 2.5 in chapter 9 of do Carmo's "Riemannian Geometry". In the proof of the proposition he says: Let M be a Riemannian manifold and $c:[0,a]\rightarrow M$ a ...
0
votes
1answer
23 views

Finding if a group is a vector space

$\mathbb{C}^2$ is a group over field $\mathbb{C}$, with the following actions: addition is similar to the regular addition. multiplication is defined by: for every $(z,w) \in \mathbb{C}^2$ and every ...
2
votes
1answer
30 views

The Leibniz rule for the curl of the product of a scalar field and a vector field

I have some scalar field $u:D \rightarrow\mathbb R; \space \space D\subset \mathbb R^3$ and a vector field $\vec{v}: D\rightarrow \mathbb R^3$ and I want to show that: ...
1
vote
0answers
21 views

Divergence theorem for a box

I have to use the divergence theorem to calculate $$\int \int_C F.dS $$ for $$F = (x^2z^3, 2xyz^3,xz^4) $$ where S is the surface of the box with vertices at $(\pm1,\pm2,\pm3)$ with outward pointing ...
1
vote
1answer
30 views

Intuitive explanation of the potential function of a vector field

Suppose I have some vector field $$\vec{f}(x,y)=\begin{pmatrix}A(x,y)\\B(x,y)\end{pmatrix}$$ then the potential function (if the field is conservative) can be found by integrating $A$ with respect ...
1
vote
1answer
17 views

Looking for a continuous, unit-norm vector field

I want to find a 2D vector field with three characteristics: it is continuous all the vectors are unit length vectors on the unit circle point to the origin Is this possible? I haven't been able ...
1
vote
0answers
33 views

Write the vector field such that its integral curves are the meridians of $S^2$

I have problems in writing down a vector field $X$ on the sphere $S^2$ such that the integral curves of $X$ are the meridians of $S^2$. I proceed in this way. I consider this parametrisation of ...
2
votes
0answers
19 views

What is an intuitive/geometric definition of line integrals? Do they work in 2-dimensions?

I understand that we are finding the area of a curve given by some function f(x) over the area of another curve C. (I've also successfully plugged and chugged my way through my homework, without ...
2
votes
3answers
61 views

Checking if a vector field is conservative

I have three different vector fields and I want to check if they are conservative: $$1)\space \space\vec{f}(\vec{x}):=\frac{1}{||\vec{x}||}\vec{a}, \space \space D=\mathbb R^2 / (\vec{0}), \space ...
4
votes
2answers
68 views

Line Integrals FT usage on this strange vector field: so what are the exact conditions?

I really tried thousands of things before deciding to ask here. Searched all over the internet for an answer, but failed to find it. Let's get started with the Fundamental Theorem of Line Integrals. ...
0
votes
0answers
15 views

Why is the rotation not zero and the divergence zero in the figures below?

Why is the rotation not zero and the divergence zero in figure 1 and figure 2 below? Figure 1 Figure 2
1
vote
0answers
27 views

Computing the value of a line integral of a vector field in the plane

We are given the vector field $ x^2dx+y^2dy $ and are interested in the line integral of it over the closed equilateral triangle with vertices (0,0) (2,0) (1,-2) Because the partial derivatives of ...
2
votes
1answer
16 views

Multivariable Calculus: Line Integrals (Directed Curve)

I have this math problem, that I got a bit confused on. I just need to know whether or not I did it correctly. Thanks! Question: Calculate $\oint_c xe^{z}dx+yzdy+xe^{y}dz$ over the directed curve ...
2
votes
5answers
54 views

Multivariable Calculus: Line Integral

I have this math problem. It states: Calculate the given line integral $\oint _c {M dx+Ndy}$ where $C$ is the triangle with vertices $P_0=(0, 1)$, $P_1=(2, 1)$, $P_2=(3, 4)$ with ...
0
votes
1answer
25 views

What is vectors straddle a plane mean?

There is a condition in a paper, saying that two vectors straddle a plane. How can we transfer this condition to a equation? Because I have another 5 equations and need this one to solve 6 unknowns. ...
1
vote
2answers
36 views

Verification of the identity $\langle \nabla _{[X,Y]}Z,Z \rangle = \frac{1}{2}[X,Y]\langle Z,Z \rangle$

In the book Riemannian Geometry, page 91, Do Carmo writes: $$\langle \nabla _{[X,Y]}Z,Z \rangle = \frac{1}{2}[X,Y]\langle Z,Z \rangle$$ I could not understand how this happens. Can someone ...
1
vote
1answer
20 views

Obvious way to $F$-relate vector fields?

An exercise in Lee's Introduction to Smooth Manifolds asks one to check that $F:\mathbb R\to\mathbb R^2$ given by $F(t)=(\cos t,\sin t)$ relates $X=d/dt\in\mathfrak X(\mathbb R)$ to $Y\in\mathfrak ...
0
votes
2answers
80 views

Line integral of conservative vector field

Compute the line integral $\int_\gamma g \cdot dx $ for an arbitrary piecewise smooth curve $\gamma$ traversing in the upper half plane from $(-a,0)$ to $(b,0)$ where $a > 0$ and $b>0$. ...
1
vote
0answers
82 views

How to show that the vector fields $X_i = f_*(\frac{\partial}{\partial x^i})$ and $X_j = f_*(\frac{\partial}{\partial x^j})$ commute?

Could anyone help me with the following problem? We have: Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for $t ...
0
votes
0answers
70 views

Vector fields that are smooth on the open unit ball $B$, are smooth on $\mathbb{R}^n$ if they are zero outside $B$

Could anyone help a little with the following problem? Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for $t \in ...
2
votes
0answers
28 views

Existence of a differentiable function given a unit gradient field

I'm trying to prove that "Given a unit vector field $V$, it can always be uniquely determined a differentiable function $f$ that satisfies $\nabla f = V$." To provide you more information, the unit ...
0
votes
1answer
35 views

Finding the number of bases for $ \mathbb{F}_3^2 $ and isomorphisms

Find the number of bases for $ \mathbb{F}_3^2 $ and also the amount of Isomorphisms $ \mathbb{F}_3^2 \rightarrow \mathbb{F}_3^2 $ Here if $(v1,v2)$ is a basis, then $(v2, v1)$ is a different basis. ...
1
vote
0answers
19 views

Derivative matrix in polar coordinates

Considering a vector field in two dimensions, $\vec V(x,y)$, I know that the derivative matrix (Jacobian matrix) is given by: $\nabla \vec V(x,y) = \begin {bmatrix} \partial V_x/\partial x ...
1
vote
2answers
33 views

Vector spaces with complex field as scalar. [duplicate]

Sorry for stating the question informally. If we have a vector space whose scalars are the field $\mathbb{R}$, if we change the field to be $\mathbb{C}$ and "adapt" the addition and scalar ...
0
votes
0answers
31 views

One form and Vector fields on a manifold in terms of local coordinates.

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ in local coordinates where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I do not know how to ...
2
votes
2answers
48 views

Exterior differentiation of one form on a smooth manifold

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I'm fine with the right side of the equation, ...
2
votes
1answer
31 views

Does the dual basis to some basis of $T^*_pM$ looks localy like a coordinate chart?

Let $M$ be a manifold and let $\{\alpha_k\}$ be a set of $1$- forms s.t. $\{\alpha_k(p)\}$ forms a basis for $T^*_pM$. Let $(x,U)$ be a chart based in $p$ and denote $\partial_i := ...