Vector fields Definition and 161 Threads

In this lecture, the lecturer claims that the ##C^{\infty}##-module of smooth vector fields defined on a smooth manifold can lack to admit a basis (not even infinite dimensional). Indeed the set of smooth vector fields can be given an (infinite dimensional) vector space structure over the field...

I've a doubt regarding the definition of canonical coordinates in phase space. As far as I can tell, phase space ##T^*M## is the cotangent bundle of the system configuration space ##M##. ##M## is assumed to be a differential manifold with atlas ##A=\{ U_i, \phi_i \}##. Call ##q_i## the...

Consider the following: suppose there is a smooth vector field ##X## defined on a manifold ##M##. Take a smooth curve ##\alpha(\tau)## between two different integral curves of ##X## where ##\tau## is a parameter along it. Let ##A## and ##B## the ##\alpha(\tau)## 's intersection points with the...

Hi, starting from this thread, I'd like to clarify some mathematical aspects related to the notion of hypersurface orthogonality condition for a congruence. Let's start from a congruence filling the entire manifold (e.g. spacetime). The condition to be hypersurface orthogonal basically means...

To my understanding, an orientation can be expressed by choosing a no-where vanishing top form, say ##\eta := f(x^1,...,x^n) dx^1 \wedge ... \wedge dx^n## with ##f \neq 0## everywhere on some manifold ##M##, which is ##\mathbb{H}^n := \{ x \in \mathbb{R}^n : x^n \geq 0 \}## here specifically. To...

Hi, suppose you have a non-zero smooth vector field ##X## defined on a manifold (i.e. it does not vanish at any point on it). Can its integral curves cross at any point ? Thanks. Edit: I was thinking about the sphere where any smooth vector field must have at least one pole (i.e. at least a...

-Verify that the space ##Vect(M)## of vector fields on a manifold ##M## is a Lie algebra with respect to the bracket. -More generally, verify that the set of derivations of any algebra ##A## is a Lie algebra with respect to the bracket defined as ##[δ_1,δ_2] = δ _1◦δ_2− δ_2◦δ_1##. In the first...

Apostol defines limit for vector fields as > ##\quad \lim _{x \rightarrow a} f(x)=b \quad(\rm or\; f(x) \rightarrow b## as ##x \rightarrow a)## means that : ##\lim _{\|x-a\| \rightarrow 0}\|f(x)-b\|=0## Can't we say it's equivalent to ##\lim _{x \rightarrow a}(f(x)-b)=0##

I identified $$(\Phi_{SN})_{*})$$ as $$J_{(\Phi_{SN})}$$ where J is the Jacobian matrix in order to $$(\Phi_{SN})$$, also noticing that $$\frac{\partial}{\partial u} = \frac{\partial s}{\partial u}\frac{\partial}{\partial s} + \frac{\partial t}{\partial u} \frac{\partial}{\partial t} $$, I wrote...

I am looking at antenna theory and just came upon scalar fields. I found an site giving an example of a scalar field as measuring the temperature in a pan on a stove with a small layer of water. The temperature away from the heat source will be cooler than near it but it doesn't have a...

This week, I've been assigned a problem about a 3-sphere. I am confused how to approach this problem and any comments would be greatly appreciated. (a) - would I be correct to assume the metric G is simply the dot product of two vector fields with dx^2 dy^2 du^2 and dv^2 next to their...

I have a vector field which is originallly written as $$ \mathbf A = \frac{\mu_0~n~I~r}{2} ~\hat \phi$$ and I translated it like this $$\mathbf A = 0 ~\hat{r},~~ \frac{\mu_0 ~n~I~r}{2} ~\hat{\phi} , ~~0 ~\hat{\theta}$$(##r## is the distance from origin, ##\phi## is azimuthal angle and ##\theta##...

Hi, I just have a quick question about a problem involving Gauss' Theorem. Question: Vector field F = \begin{pmatrix} x^2 \\ 2y^2 \\ 3z \end{pmatrix} has net out flux of 4 \pi for a unit sphere centred at the origin (calculated in earlier part of question). If we are now given a vector...

I have been at this exercise for the past two days now, and I finally decided to get some help. I am learning General Relativity using Carrolls Spacetime and Geometry on my own, so I can't really ask a tutor or something. I think I have a solution, but I am really unsure about it and I found 6...

Summary:: Seeding and visualization techniques Hi I am looking for resources where I can learn the following: Seeding strategies and algorithms for vector fields (texture-based, geometry, topological) Different techniques for visualizing vector fields (streamlines, glyph-based, LIC etc)

I'm struggling to get the hang of killing vectors. I ran across a statement that said energy in special relativity with respect to a time translation Killing field ##\xi^{a}## is: $$E = -P_a\xi^{a}$$ What exactly does that mean? Can someone clarify to me?

Dear everyone. I'm doing an assignment on vectorfields and for most of the assignment I have to deal with tensors and tensornotation. The first assignment asks me to express the following vector and matrixproducts in tensornotation. $$\overline c = \overline a + \overline b \\ d=(\overline a +...

In dealing with rotating objects, I have found the need to be able to transform a vector field from cylindrical coordinate systems with one set of coordinate axes to another set. For eg i'd like to transform a vector field from being measured in a set of cylindrical coordinates with origin at...

Hi, I want to plot the vector field ##\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k## The code I have tried: # The components of the vector field F_x = y*e**x F_y = x**2 + e**x F_z = z**2*e**z# The grid xf = np.linspace(-0.15, 2.25, 8) yf = np.linspace(-0.15, 2.25, 8) zf =...

Hi, I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields? Thanks!

the question: My attempt: The partial derivatives did not match so i simply tried to find f(x,y) I got the set of equations on the right but that's about it.

This is a refinement of a previous thread (here). I hope I am following correct protocol. Homework Statement I am studying Spacetime and Geometry : An Introduction to General Relativity by Sean M Carroll and have a question about commutators of vector fields. A vector field on a manifold can...

Homework Statement I am studying Spacetime and Geometry : An Introduction to General Relativity by Sean M Carroll and have a question about commutators of vector fields. A vector field on a manifold can be thought of as differential operator which transforms smooth functions to smooth functions...

Hello PF, I was reading Carroll’s definition of the commutator of two vector fields in “Spacetime and Geometry”, and I’m having (I think) a simple case of notational confusion. He says for two vector fields, ##X## and ##Y##, their commutator can be defined by its action on a scalar function...

I am trying to improve my understanding of Lie groups and the operations of left multiplication and pushforward. I have been looking at these notes: https://math.stackexchange.com/questions/2527648/left-invariant-vector-fields-example...

Hi, let ##\gamma (\lambda, s)## be a family of geodesics, where ##s## is the parameter and ##\lambda## distinguishes between geodesics. Let furthermore ##Z^\nu = \partial_\lambda \gamma^\nu ## be a vector field and ##\nabla_\alpha Z^\mu := \partial_\alpha Z^\mu + \Gamma^\mu_{\:\: \nu \gamma}...

Hello everyone, my question is: what are the criteria that must be satisfied for the pushforward of a smooth vector field to be a smooth vector field on its own right? Consider a smooth map \phi : M \longrightarrow N between the smooth manifolds M and N. The pushforward associated with this map...

Homework Statement Example 2:[/B] Homework Equations Flux=integrate -Pgx-Qgy+R of the proj. area on xy plane for z=g(x,y) The Attempt at a Solution Why do my attempt is wrong? The example is using the foundational formula while I use the stock formula from the book, why is there a negative...

Homework Statement The stream function Ψ(x,y) = Asin(πnx)*sin(πmy) where m and n are consitive integers and A is a constant, describes circular flow in the region R = {(x,y): 0≤x≤1, 0≤y≤1 }. Graph several streamlines with A=10 and m=n=1 and describe the flow. Explain why the flow is confined to...

I'm reading a textbook on electromagnetism. It says that for two vector fields ##\textbf{F}(\textbf{r})## and ##\textbf{G}(\textbf{r})## their inner product is defined as ##(\textbf{F},\textbf{G}) = \int \textbf{F}^{*}\cdot \textbf{G} \thinspace d^3\textbf{r}## And that if ##\textbf{F}## is...

I need some guidance regarding the directional derivative ... Two books I am reading introduce the directional derivative somewhat differently ... these books are as follows: Theodore Shifrin: Multivariable Mathematics and Susan Jane Colley: Vector Calculus (Second Edition)Colley...

Homework Statement (a) Consider the line integral I = The integral of Fdr along the curve C i) Suppose that the length of the path C is L. What is the value of I if the vector field F is normal to C at every point of C? ii) What is the value of I if the vector field F is is a unit vector...

I'm learning Differential Geometry on my own for my research in ML/AI. I'm reading the book "Gauge fields, knots and gravity" by Baez and Muniain. An exercise asks to show that "if \phi:M\to N we can push forward a vector field v on M to obtain a vector field (\phi_*v)_q = \phi_*(v_p) whenever...

Homework Statement Homework Equations ##V=V^u \partial_u ## I am a bit confused with the notation used for the Lie Derivative of a vector field written as the commutator expression: Not using the commutator expression I have: ## (L_vU)^u = V^u \partial_u U^v - U^u\partial_u V^v## (1)...

hello every one can one please construct for me left invariant vector field of so(3) rotational algebra using Euler angles ( coordinates ) by using the push-forward of left invariant vector field ? iv'e been searching for a method for over a month , but i did not find any well defined method...

hello every one . can someone please find the left invariant vector fields or the generator of SO(2) using Dr. Frederic P. Schuller method ( push-forward,composition of maps and other stuff) Dr Frederic found the left invariant vector fields of SL(2,C) and then translated them to the identity...

I am looking to find a combination of electric and magnetic fields that create something similar to the (d) crests Currently, I have the cracks flipped clockwise 90deg, so that the crests are concave up. And each crest can be defined by a parabolic function (which are uniform with each new...

Hello Forum, A conservative vector field G(x,y,z) is one that can be expressed as the gradient of a scalar field P(x,y,z). Could a time-varying vector field like D(x,y,z,t) be a conservative vector field? If not, why not? Can it be conservative (or not) at different time instants? Thanks!

Vector fields confuses me. What are the differences between (##t## could be any variable, not just time): 1. If the position vector don't have an argument, ##\mathbf{r}=x\mathbf{\hat e}_x+y\mathbf{\hat e}_y+z\mathbf{\hat e}_z=(x,y,z)## so ##\mathbf{E}(\mathbf{r},t)=E_x(\mathbf{r},t)\mathbf{\hat...

Homework Statement Hi everybody! I'm currently training at surface integrals of vector fields, and I'd like to check if my results are correct AND if there is any shortcut possible in the method I use. I'm preparing for an exam, and I found that it takes me way too much time to solve it. I...

I am trying to follow Nakahara's book. From the context, it seems that the author is trying to say if moving a point along a flow always give a isometry, the corresponding vector field X is a Killing vector field. am I right? then the book gives a proof. It only considers a linear approximation...

I'm interested in Killing vector fields and want to ask whether anybody can name me a good textbook or online-source about them, preferably with a general treatment with local coordinates as examples and not at the center of consideration.

Homework Statement Given two vector fields ##W_ρ## and ##U^ρ## on the sphere (with ρ = θ, φ), calculate ##D_v W_ρ## and ##D_v U^ρ##. As a small check, show that ##(D_v W_ρ)U^ρ + W_ρ(D_v U^ρ) = ∂_v(W_ρU^ρ)## Homework Equations ##D_vW_ρ = ∂_vW_ρ - \Gamma_{vρ}^σ W_σ## ##D_vU^ρ = ∂_vU^ρ +...

Homework Statement Vector field is given by: [PLAIN]http://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsVectorFields_files/eq0001M.gif[PLAIN]http://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsVectorFields_files/empty.gif I'm just reviewing line integrals of vector fields and...

Hi. Given a one-parameter family of maps such as Φt : ( x , y ) → ( xet + 2et -2 , ye2t ) the velocity vector field at t=0 is given by d(Φt)/dt = (x+2) ∂/∂x + 2y ∂/∂y My question is ; how does differentiating a vector function Φt with respect to t result in a scalar function ? Thanks

Homework Statement I need a pointer to a proof of the following items: if div X =0 then X = curl Y for some field Y. if curl X = 0 then X = grad Y for some field Y. Can anyone provide a pointer to a proof? Thanks. Bob Kolker Homework EquationsThe Attempt at a Solution

Hi Everyone. There is an equation which I have known for a long time but quite never used really. Now I have doubts I really understand it. Consider the unitary operator implementing a Lorentz transformation. Many books show the following equation for vector fields: U(\Lambda)^{-1}A^\mu...

In my electromagnetic theory book, there is a classification of vector fields, one of the 4 different type vector fields is "solenoidal and irrotational vector field" (both divergence-free and curl-free). If solenoidal and rotational vector fields are same thing, then it means the vector field...

Recently I've had some discussions about time in GR. I've always read in different places that people usually want a spacetime to have a hypersurface-orthogonal timelike Killing vector field so that they can assign a time dimension to that spacetime. But Why is this needed? I can understand it...

Hello. There is one thing I can not find the answer to, so I try here. For instance, writing a general superfield on component form, one of the terms appearing is: \theta \sigma^\mu \bar{\theta} V_\mu My question is if one could have written this as \theta \bar{\theta} \sigma^\mu V_\mu ...