# What is Vector fields: Definition and 168 Discussions

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised 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, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.
The elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow).
In coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector).
More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.

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1. ### I Induced orientation on boundary of ##\mathbb{H}^n## in ##\mathbb{R}^n##

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...
2. ### I Integral curves of (timelike) smooth vector field

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...
3. ### About lie algebras, vector fields and derivations

-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...
4. ### I Definition of Limit for vector fields

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##
5. ### Vector Field associated with Stereographic Projection

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...
6. ### Vector Fields vs Scalar Fields

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...
7. ### Tangent vector fields and covariant derivatives of the 3-sphere

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...
8. ### How to find the curl of a vector field which points in the theta direction?

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##...
9. ### Gauss' Theorem - Net Flux Out - Comparing two vector Fields

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...
10. ### Finding killing vector fields of specific spacetime

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...
11. ### Visualization of vector fields

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)
12. ### I Getting Used to Killing Vector Fields: Explained

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?
13. ### I Larger assignment on Vector Fields

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 +...
14. ### I Transforming Vector Fields between Cylindrical Coordinates

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...
15. ### Python How to plot vector fields in Matplotlib

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 =...
16. ### I The Commutator of Vector Fields: Explained & Examples

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!
17. ### Vector Calculus (non conservative vector fields

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.
18. ### Spacetime and Geometry: Vanishing commutators#2

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...
19. ### Spacetime and Geometry: Vanishing commutators

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...
20. ### I Commutator of two vector fields

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...
21. ### I Computation of the left invariant vector field for SO(3)

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...
22. ### A Compute Commutator of Covariant Derivative & D/ds on Vector Fields

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}...
23. ### A Pushforward of Smooth Vector Fields

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...
24. ### Understanding Flux of Vector Fields: Equations, Solutions, and Common Mistakes

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...
25. ### Vector fields question; not sure how to approach?

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...
26. ### I Define inner product of vector fields EM

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...
27. ### MHB The Directional Derivative .... in Scalar Fields and Vector Fields ....

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...
28. M

### Question about Vector Fields and Line Integrals

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...
29. ### A Pushing forward a vector field

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...
30. ### Lie derivative vector fields, show the Leibniz rule holds

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)...
31. ### I Constructing left invariant vector fields on SO(3)

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...
32. ### I Lie groups left invariant vector fields

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...
33. ### E and B fields for charged particle's parabolic motion

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...
34. ### I Do time-dependent and conservative vector fields exist?

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!
35. ### I Different types of vector fields?

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...
36. ### Surface integral of vector fields (sphere)

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...
37. ### A Question about Killing vector fields

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...
38. ### Topology Sources about Killing vector fields?

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.
39. ### Covariant derivative of vector fields on the sphere

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 +...
40. ### P,Q,R notation in Vector Fields

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...
41. ### I Velocity Vector Fields: Differentiating a Vector Function to Scalar

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
42. ### Question about vector fields, div, curl grad

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
43. ### Lorentz transformations and vector fields

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...
44. ### Differences between solenoidal and rotational vector fields?

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...
45. ### Time, spacelike foliations and timelike vector fields in GR

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...
46. ### Order or Grassmann, vector fields and tensors

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 ...
47. ### Vector fields transverse to the boundary of a manifold

I'm trying to work up some examples to help me understand this concept. Would the periodic flow on a solid torus be transverse to it's boundary?
48. ### Complex analysis and vector fields

I'm going to ask a very general question where I just would want to hear different possible methods that can be thought of in this kind of problem. I am trying to solve a very specific problem with this but I won't talk about that because I don't want someone to give me the answer but ideas for...
49. ### Using Green's Theorem for Vector Fields

Homework Statement Homework Equations n/a The Attempt at a Solution I don't understand how the book went from calculating Green's theorem on ##\int _c Pdx + Qdy + \int _{-c'} Pdx + Qdy = ## (1 in the attached picture) to getting (labeled 2) ##\int _c Pdx + Qdy = \int _{c'} Pdx + Qdy ##...
50. ### Vector fields and line equations Problem

Homework Statement We are giving to lines: r1(t)=<1-t,4,5+2t> r2(s)=<2,1+s,-s> 1. Find an equation perpendicular to the two lines and passing point P(1,1,1) 2. Find Coordinates of points of intersection of the line found in #1 with planes x=-1, xz-plane 3. Parametrize the line segment joining...
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