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3
votes
1answer
35 views

Symmetry of Killing Vectors in Covariant Derivative

Several times, I've seen statements along the lines of "$\nabla_X Y=\nabla_Y X$ because $X$ is a Killing vector field." One example I found on Stack Exchange is here. I have yet to understand why ...
-1
votes
1answer
22 views

Work done on a particle moving in a force field

Given a path C and a force field F, the work done on the particle can be found by $$ \int_C \vec{F}\cdot \vec{r} \,dr $$ This seems to suggest that you can find the work done for any path. Doesnt ...
0
votes
1answer
26 views

Two ways of finding a Potential of a Vector Field

If $\vec{F}(x,y)$ is a conservative vector field and we want to find a function $V$ such that $\nabla(V)=\vec{F}$, then one way to do it is to take an arbitrary point $(x_0,y_0)$ and then define ...
2
votes
2answers
28 views

Divergence Theorem to calculate flux

Take the vector field given by: $F= (y^2+yz)i+(\sin(xz)+z^2)j+z^2k$ a) Calculate the divergence, $\operatorname{div}F$. b) Use the divergence theorem to calculate the flux $$\int_S F\cdot dA $$ ...
2
votes
1answer
30 views

Vector field with gradient and integral over curve

The problem is: Consider the vector field: $$\textbf{F}= 4x^3y^3 \,\textbf{i} + (1+3x^4y^2) \,\textbf{j}$$ a) Find a potential function $ϕ(x,y)$, i.e. a function $ϕ(x,y)$ such that $\nabla ϕ= ...
2
votes
0answers
21 views

When a vector field can be scaled to form a conservative vector field

Consider a vector field given by its components $g_i(x_1, \dots, x_n)$. It is well known that necessary and sufficient condition for a following system $$ \frac{\partial f}{\partial x_i} = g_i(x_1, ...
1
vote
1answer
15 views

plotting cone on matlab

Plot the portion of the cone $z=sqrt((x − 1)^2 + y^2)$ inside the cylinder $r = 2$ This is my matlab code. However, I get an error with the ezsurf line. Any ideas as to what I might be doing wrong? ...
3
votes
0answers
25 views

What is an example of a map not satisfying this rank condition?

Definition: Consider a Lie Group $G$ and a set of right invariant vector fields on $G$, denoted $\Gamma$. A point $y \in G$ is called normally accessible from a point $x \in G$ by $\Gamma$ if there ...
2
votes
0answers
20 views

multivariable calculus charge density question

The sphere given by $x^{2} + y^{2} + z^{2} = 4$ is submerged in an electric field with charge density given by $f(x, y, z) = x^{2} + y^{2}$. Find the total amount of electric charge on this surface. ...
1
vote
1answer
31 views

Vector Fields problem unsure of how to start

A fluid having density $f(x, y, z) = x^{2} + 2y^{2} + z^{2}$ flows with velocity $v(x, y, z) = x^{2}i + xy^{2}j + zk$. Determine the rate of mass flow through the sphere $ρ = 1$ in the outward ...
0
votes
0answers
19 views

How to compute $\int_C (X|dx)$?

I have the vector field $X(x,y,z) = (\frac {-y}{x^2+y^2+z^2}, \frac{x}{x^2+y^2+z^2},0)$ on $\mathbb{R} ^3$ \ $\left\{ (0,0,0) \right\}$. I need to compute $\int_C (X|dx)$ over a closed circle of ...
0
votes
0answers
12 views

When to use Stoke's over Divergence and vice versa

I'm still very confused as to when I'd use Stoke's Theorem over Divergence Theorem and vice versa. What are some very obvious things I could look for in a problem to determine which theorem to use? ...
1
vote
1answer
21 views

Evaluate a line integral using the fundamental theorem of line integrals

Consider the vector field $${\mathbf F}(x,y)=(e^x)(\sin y){\mathbf i}+(e^x)(\cos y){\mathbf j},$$ and the curve $C$ composed of the graph of $\sqrt{x}+\sqrt{y}=5$ followed by segment from $(25,0)$ to ...
1
vote
1answer
22 views

Integral curves and Vector field

For a given vector field $X(x,y,z) = (\frac {-y}{x^2+y^2+z^2}, \frac{x}{x^2+y^2+z^2},0)$ on $\mathbb{R} ^3$\ {(0,0,0)}, what is the integral curve $\int_C (X|dx)$ over a closed circle with radius of 1 ...
0
votes
0answers
14 views

Calculating a unique vector field given 2 vector fields in $\mathbb{R}$

So I have this exercise that is partially solved, but I am stuck in one of the steps. I would really appreciate it if someone could explain it to me. Thanks! :) EXERCISE: Let $u=\sum u_k ...
1
vote
0answers
17 views

How to use three complex vector components to calculate resultant complex vector

This is a practical problem related to complex vectors. Imagine you want to find the resultant electric field of multiple electromagnetic waves that have parallel and perpendicular components ...
1
vote
1answer
21 views

finding line integral with potential function [duplicate]

Evaluate the line integral $$ \int_C (\ln y) e^{-x} \,dx - \dfrac{e^{-x}}{y}\,dy + z\,dz $$ where C is the curve parametrized by $r(t)=(t-1)i+e^{t^4}j+(t^2+1)k$ for $0\leq t\leq 1$ I know that the ...
1
vote
2answers
32 views

What do gradient, curl, and div input and output?

What do gradient, curl, and div input and output? (e.g. vector field or scalar function of several variables)
1
vote
1answer
40 views

When is a vector field on a manifold restricted to a submanifold $X$ a vector field on $X$?

Let $X$ be an embedded submanifold of $M$ and let $V$ be a vector field on $M$. One can restrict $V$ to $X$, but it may not define a vector field on $X$. Example: The vector field $x^i\partial_i$ on ...
0
votes
0answers
17 views

Find value of line integral: $\int_C(2x-3y+1)dx-(3x+y-7)dy$

Find value of line integral $$ \int_C(2x-3y+1)dx-(3x+y-7)dy $$ $$ C: from\;(0,1)\;to\;(2, e^2) $$ I know this line integral is conservative: $\frac{M}{dy}=\frac{N}{dx}=-3$ Its potential function I ...
0
votes
0answers
45 views

How to define an action of vector field on $C^{\infty}(M)$?

Let $M$ be a manifold. Let $\hat{X}$ be a vector field on $M$. How to define an action of $\hat{X}$ on $C^{\infty}(M)$? Thank you very much.
1
vote
1answer
25 views

How to define the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$?

I read a paper and on page 9, the paragraph before Proposition 5, it is said that let the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$. How to ...
0
votes
3answers
33 views

Is this a vectorial space

I can't understand why the 0 vector here is not unique?
0
votes
1answer
27 views

Exact ODE:$ \left(\frac{y}{x}+6x\right)dx + (\log(x) - 2)dy =0$

I need to solve the equation $$\left(\frac{y}{x}+6x\right)dx + (\log(x) - 2)dy=0$$ We can easly see that $$\left(\frac{y}{x}+6x\right)'_y =(\log(x) - 2)'_x$$ but the domain of $ ...
0
votes
0answers
19 views

How do I find flow of $\vec{a}$ in direction of $\vec{n}$ on surface $D$?

Problem: I have to calculate the flow (I do not know if this is the correct term in english) of vector field $\vec{a}(M)$ through triangle surface, created, when plane $(p)$ intersects planes ...
1
vote
1answer
36 views

Curl and Vector Fields

I am having real difficulty knowing how to approach this question, so any help or pointers would be appreciated. Consider the vector field: $$ \vec{G} = -3xz^2\vec{i} + z^3\vec{k} $$ FInd a vector ...
0
votes
0answers
78 views

Lie groups, Lie algebra and left invariant vector fields

Hi I'm learning about Lie Groups to understand gauge theory (in the principal bundle context) and I'm having trouble with some concepts. Now let $a$ and $g$ be elements of a Lie group $G$, the left ...
0
votes
1answer
14 views

Gradient of a function with base vectors

\begin{align} \nabla\left(x_1x_3\hat{e}_1+x_2x_3\hat{e}_2\right)&=x_3\left(\hat{e}_1\otimes\hat{e}_1\right)+x_3\left(\hat{e}_2\otimes\hat{e}_2\right)\\&+\mathbf{x_1\left(\hat ...
4
votes
1answer
69 views

Lie algebras of vector fields over $\mathbb{R}$ and over $S^1$

An exercise in the book "Moonshine beyond the Monster" has me stumped. It asks whether the real Lie algebras of smooth vector fields over the reals $V(\mathbb{R})$ and over the circle $V(S^1)$ are ...
0
votes
0answers
5 views

Helmholtz decomposition of two given vector fields

I am trying to solve the following task: Show that the following vector fields can be resolved into a scalar field $g(\vec x)$ as well as a vector field $\vec w(\vec x)$ where $\vec v(\vec x) = - ...
2
votes
0answers
48 views

Calculating the Integral of a non conservative vector field

I have no clue how to do part C because a) is non conservative What I got for b) $f(x,y)=\frac{x^3}{3}+2yx+\frac{y^3}{3}+K$ (I don't know the symbol for the thing so I used f(x,y) instead. How do I ...
0
votes
2answers
61 views

Vector space function question

Q: Let $X$ be a set and $\Bbb F$ be a field. Consider the vector space $V=Fun(X,\Bbb F)$ of functions from $X$ to $\Bbb F$ with operations of addition and mult. (i) Describe the zero element of V and ...
2
votes
1answer
36 views

Is the continuity of a vector field enough for the existence of the solution of a differential equation?

I've recently seen the existence-uniqueness theorem for ordinary differential equations from Arnold's book. I understand that the theorem as stated guarantees both existence and uniqueness if the ...
1
vote
1answer
49 views

Degree of a vector field on the boundary of a punctured ball?

Suppose $v\colon \mathbb{R}^3\setminus\{x_1,\dots,x_n\}\to S^2$ is a smooth map, where all the points $x_i$ sit inside the unit ball $B^3$, and not on the boundary $S^2$. I suppose this map can be ...
0
votes
1answer
27 views

A basic relation in spherical coordinates

Why is it that $$x\partial_x+y\partial_y+z\partial_z=r\partial_r~?$$ I know that $$r^2=x^2+y^2+z^2,$$ but how is this relation implied?
1
vote
1answer
38 views

3d Laplacian vector field with constant magnitude

Are there non-trivial (i.e. non-constant) 3D Laplacian vector fields with constant magnitude? If not, how can one proof this, if yes, how many are there (how can they be classified?)
0
votes
1answer
28 views

Calculating the divergence of a central vector field.

I am trying to calculate the divergence of a central electric field, namely the electric field due to a point charge and my book begins like this: http://imgur.com/bW9tPEZ However in the last line I ...
0
votes
1answer
72 views

Converting a slope field into a vector field

I have homework on slope fields where I have to graph a bunch and find the equillibrium solution, but instead of taking such a long time to graph them, I decided to use WolframAlpha. Sadly, there is ...
0
votes
1answer
30 views

Equation for a Vector Field Spiraling to a Point

I'm building a generative animation and one of the things I'm trying to achieve is a vector field that spirals towards a point. I've discovered ...
3
votes
2answers
91 views

Symmetry group of the vector field $V=x \partial /\partial x + y \partial /\partial y$

I was trying to solve an exercise in one of Arnold's book that asks for the symmetry group of the vector field $V=x \partial /\partial x + y \partial /\partial y$, that is the diffeomorphisms $g$ of ...
2
votes
1answer
48 views

Link between the two definitions of a “hyperbolic point”

The common definition of a hyperbolic point for a flow of a vector field $f$ is a fixed point in which the eigenvalues of the Jacobian matrix of $f$ all have non-zero real parts ...
1
vote
2answers
53 views

What is the potential function of the field $\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$

The vector field is obviously conservative on every closed domain that doesn't encompass the point $(0,0)$, so there must be a potential function. I've got $\arctan(\frac{x}{y})$ for $x$ unequal to ...
1
vote
1answer
36 views

Finding a function from a vector field

The vector field $F(x, y) = \left(\displaystyle\frac{x}{r^3}, \frac{y}{r^3}\right)$ appears in electrostatics, where $r = \sqrt{x^2 + y^2}$ is the distance to the charge. Find a function $f(x, y)$ ...
1
vote
0answers
38 views

Two isomorphisms of the space of vector fields/forms

Suppose you have a diffeomorphism $$ \phi: M \to N. $$ As far as I understand, this gives rise to two distinct isomorphisms $$ a : \mathcal A^1(M) \cong \mathcal A^1(N) :b $$ between the space of ...
2
votes
1answer
32 views

What is the exterior normal to the boundary of a Riemannian manifold?

Let $(M,g)$ be a Riemannian manifold with boundary $\partial M$. Let $p \in \partial M$ and in local coordinates $(x_1,\ldots,x_n)$ near $p = (0,\ldots,0)$ the manifold $M$ is given by $\{x_n ...
0
votes
1answer
54 views

Divergence theorem

I have to use the divergence theorem to solve $\iint F \cdot ds$ where $F(x,y,z)=x^3 \hat{\imath}+y^3\hat{\jmath}+z^3\hat{k}$ and $S$ is the surface of the solid rounded by the cylinder $x^2+y^2=1$ ...
1
vote
1answer
74 views

Finding a scalar field whose gradient is a given conservative vector field

I'm studying for a course in electromagnetism, and I've been given an electric field for which I need to find the associated scalar potential. I was going to originally post this in the physics ...
4
votes
2answers
74 views

Show ker($\alpha$)=ker($\alpha$)^2 iff ker($\alpha$) and im($\alpha$) are disjoint

Let $V$ be a vector space over a field $F$ and let $\alpha$ be an element of $\operatorname{End}(V)$. Show $\ker(\alpha)=ker(\alpha^2)$ iff $\ker(\alpha)$ and $\operatorname{im}(\alpha)$ are disjoint. ...
0
votes
1answer
73 views

A question about vector fields and divergence

I am reading the paper http://www.goshen.edu/physix/mathphys/gco/TensorGuideAJP.pdf in order to gain a basic understanding about tensors. I had some difficulties about understanding some definitions. ...
1
vote
0answers
88 views

Does the index of a curve determine the asymptotic behaviour of certain vector fields?

There are a collection $C$ of charges in $\mathbb{R}^2$ which cause an electric vector field $V$ to form. Each charge's contribution to $V$ follows the inverse-square law. Let $\gamma$ be a curve ...