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. W

    Contact Vector Fields. "Flow Preserves Contact Structure?

    Hi All, I am going over a definition of a Contact Vector Field defined on a 3-manifold: this is defined as " a vector field v whose flow preserves the contact structure " . 1) Background (sorry if this is too simple) A contact structure ## \xi ##( let's stick to 3-manifolds for now ) is a...
  2. S

    General meaning of line integral in vector fields

    So, as i understand, the geometrical meaning of this type of integral should still be the area under the curve, however, I really do not see how you can obtain each infinitesimal rectangle from the dot product. I have understood the typical work example, that is, the line integral as the sum...
  3. P

    Only conservative vector fields are path independent?

    does anyone have a proof of this?
  4. J

    Another vector fields in terms of circulation and flux

    Other laws in terms of circulation and flux Why others vector fields no are studied like the magnetic and electric fields? In other words, why others vector fields, like the gravitational and the hydrodynamic, haven't "supreme laws" based in the circulation/flux or curl/divergence?
  5. TrickyDicky

    Vector fields, flows and tensor fields

    Vector fields generate flows, i.e. one-parameter groups of diffeomorphisms, which are profusely used in physics from the streamlines of velocity flows in fluid dynamics to currents as flows of charge in electromagnetism, and when the flows preserve the metric we talk about Killing vector fields...
  6. J

    Are all closed forms exact? Are all exact forms closed?

    Every conservative vector field is irrotational? Every irrotational vector field is conservative? Every solenoidal vector field is incompressible? Every incompressible vector field is solenoidal?
  7. A

    Integrals over vector fields and Ampere's Law

    Homework Statement Experiments show that a steady current I in a long wire produces a magnetic field B that is tangent to any circle in the plane perpendicular to the wire and whose center is the axis of the wire. Ampere's Law relates the electric current to its magnetic effects and states...
  8. ShayanJ

    Exploring Singular Vector Fields & Gauss's Laws

    You're probably familiar with Gauss's laws in electricity,magnetism and gravitation: \oint_{\partial V} \vec{D}\cdot \vec{d\sigma}=q_V \\ \oint_{\partial V} \vec{g}\cdot\vec{d\sigma}=-4\pi G m_V\\ \oint_{\partial V} \vec{B}\cdot\vec{d\sigma}=0 \\ It is also known that the first two integrals...
  9. M

    Help connecting vector fields in ODE and Vector Calc

    The vector field F=<y,x> looks exactly like the the direction field for the system dY/dt = {dx/dt = y} {dy/dt = x} A few questions on this: Are the direction field of a system of ODE's the same as a vector field of calculus? In vector calc we take the line integral of a vector field...
  10. W

    Finding Reeb Vector Fields Associated with Contact Forms

    Hi, All: Let w be a contact form , say in ℝ3, or in some 3-manifold M i.e., a smooth, nowhere-integrable 2-plane subbundle of TM. I'm trying to see how to find the Reeb field Rw associated with w. My ideas are: i) Using the actual definition of the Reeb field associated with a contact...
  11. C

    Connection Between Vector Fields & Potential Functions

    Hi, upon studying vector calculus and more precisely about the curl I stumbled upon a question : why is it that there is always a potential function of a vector field when the curl of this vector field is equal to 0?
  12. U

    The Lie bracket of fundamental vector fields

    Homework Statement The Lie bracket of the fundamental vector fields of two Lie algebra elements is the fundamental vector field of the Lie bracket of the two elements: [\sigma(X),\sigma(Y)]=\sigma([X,Y]) Homework Equations Let \mathcal{G} a Lie algebra, the fundamental vector field of an...
  13. C

    Question about potential functions from conservative vector fields

    Homework Statement 1) Show that ##\underline{a} = \underline{r} f(r)## is conservative and deduce a functional form for the potential if ##f(r) = r^n##. For what value of n does the potential diverge at both ##\underline{r_o} = 0## and ##\infty##? The Attempt at a Solution I have found...
  14. S

    Analysis of vector fields, fourier and harmonics

    Hi I am working on a optimization problem involving vector fields. In order to define a objective function I need a measure (scalar quantity) of some properties of the vector field. The vector field comes from a finite element analysis, that is the vector field is calculated on a discretized...
  15. S

    Conservative Vector Fields and Associated Potential

    Homework Statement Having issues determining what I am doing wrong, so perhaps one of you can pin point it. I have the solution, and I am extremely close to the same result, however, I am nonetheless wrong. Find the conservative vector fields potential. \vec{F}(x, y, z)=[(2xy-z^2)...
  16. Horv

    What is the path of a particle in a vector field?

    Hi all! I want to ask about vector fields. So if I had any kind of field for example \bar{F}(x,y) = (0,x) which represents a river or somthing similar and I put into the river a particle, or point-like body how can I get the path, or curve (flwo line?) from the vector field? I mean that path...
  17. I

    Doubt in plotting Vector Fields

    Hi, I have a doubt in plotting the vector field. In the post https://www.physicsforums.com/showthread.php?t=155579 it is mentioned that a vector field could be plotted for F (x,y) by, marking the (x,y) as the tail and F(x,y) as the head portion. If so, then consider the function...
  18. M

    Analyzing Vector Fields: Determining Conservativeness

    Question: Which vector field is conservative? I added two pictures of the vector fields in the paint document. So far the only things I know about conservative vector fields are.. 1. Net change is 0 for a closed path 2.there is some function f such that F = ∇f (F being the conserv...
  19. U

    Evaluation of a Conservative Vector Fields

    Homework Statement Evaluate closed integral ∫sin(x)dx+zcos(y)dy+sin(y)dz where c is an ellipse 4x^2+9y^2=36, oriented clockwise. Homework Equations ∫Fds=∫F(c(t))*||c'(t)|| or ∫Fdot ds= ∫F(c(t))*c'(t) where it's clockwise...-∫F(c(t))*c'(t)The Attempt at a Solution I don't know where to go with...
  20. K

    Scalar fields/ Scalar functions / Vector fields / Vector functions

    I know that physically, they describe relationships whereby, for instance a vector field, for each point in three dimensional space (a "vector"), we have a "vector" which has a direction or magnitude. Now I once asked what the difference between a vector field and a vector function is and the...
  21. estro

    Linear Algebra - orthogonal vector fields

    I want to prove that: Ker(T*)=[Im(T)]^\bot Everything is in finite dimensions. What I'm trying: Let v be some vector in ImT, so there is v' so that Tv'=v. Let u be some vector in KerT*, so T*u=0. So now: <u,v>=<u,Tv'>=<T*u,v'>=0 so every vector in ImT is perpendicular to every vector...
  22. G

    Understanding Nonlinear Exponential Maps for Vector Fields

    I'm having trouble understanding the exponential map for nonlinear vector fields. If dσ/dt=X(σ) for vector field X, then how does one interpret the solution: σ(t)=exp[tX]σ(0) ? If X is nonlinear, then X is not a matrix, so this expression wouldn't make sense. If X is a...
  23. D

    Proof- Vector fields form vector space

    How can I prove that the set of all planar vector fields forms a vector space? Thanks for any input!
  24. K

    Non-coordinate basis for vector fields

    Hello! Im trying to read some mathematical physics and have problems with the understanding of vector fields. Th questions are regarding the explanations in the book "Geometrical methods of mathematical physics".. The author, Bernard Schutz, writes: "Given a coordinate system x^i, it is often...
  25. ShayanJ

    Non-rotating vector fields with non-zero Curl

    In some texts the author tries to interpret operations like Curl. Some say the curl of a vector field shows the amount of rotation of the vector field But some of them say,if you put a wheel in a fluid velocity field which is like the vector field at hand,if it can rotate the wheel,then it has...
  26. B

    BiPDetermining Particle Path and Velocity from Acceleration Vector Field

    Let's say we have the acceleration vector, A, which gives us the acceleration of some particle as a function of its position in a three-dimensional space. Let's say that we also know the starting point of the function, say P. Let's say we also know the starting velocity, V. Can we determine...
  27. DryRun

    Evaluating Normal Outward Flux of Vector Fields

    Homework Statement Vector field ##\vec F= 4x \hat i+4y \hat j +3 \hat k## Let S be the open surface above the xy-plane defined by ##z=4-x^2-y^2## a. Evaluate normal outward flux of F through S. b. Use Stokes' theorem to evaluate the normal outward flux of ##∇ \times \vec F## through S. c...
  28. C

    Surface integrals of vector fields

    The integral for calculating the flux of a vector field through a surface S with parametrization r(u,v) can be written as: \int\int_{D}F\bullet(r_{u}\times r_{v})dA But what's to stop one from multiplying the normal vector r_{u}\times r_{v} by a scalar, which would result in a different...
  29. T

    Mathematica Vector Fields Explaination.

    I was about to do an experiment in Wolfram Mathematica like drawing electric field lines around a charged body and other arrangements. So i saw this nifty little Function for that very purpose called VectorPlot My Problem is that i don't know what the function does exactly i went through...
  30. C

    Vector calc, gradient vector fields

    Homework Statement Is F = (2ye^x)i + x(sin2y)j + 18k a gradient vector field? The Attempt at a Solution Yeah I just don't know...I started to find some partial derivatives but I really don't know what to do here. Please help!
  31. S

    Vector Fields in Cartesian and Cylindrical Coordinates, The Curl

    All necessary information is attached except the answer in Cartesian coordinates, which is -ix-jy+2kz and my work converting back from cylindrical to Cartesian, which I used WolframAlpha for, as the trig is a mess (that is, if the way I am doing this is correct)...
  32. T

    Electromagnetic Vector Fields (Static)

    Homework Statement Use the integral form and symmetry arguments to compute the electric field produced by the following charge densities: (i) Point charge q, placed at the origin, in 3 dimensions; (ii) Point charge q, placed at the origin, in 2 dimensions; (iii) Point charge q, placed at the...
  33. S

    Vector Fields in Cartesian and Cylindrical Coordinates, The Curl

    All necessary information is attached except the answer in Cartesian coordinates, which is -ix-jy+2kz and my work converting back from cylindrical to Cartesian, which I used WolframAlpha for, as the trig is a mess (that is, if the way I am doing this is correct)...
  34. W

    Affine spaces and time-varying vector fields

    Consider the following affine space \mathbb{G} 1. a four-dimensional vector space G_{v}^{4} over field \mathbb{R} which acts (sharply transitive) on a set G_{p}^{4} 2. a surjective linear functional from G_{v}^{4} to its field, which kernel is isomorphic with three-dimensional Euclidean vector...
  35. S

    Visualizing Vector Fields

    Hello all, Just looking for tips on "visualizing" vector fields and perhaps drawing them. I have encountered a few that have given me trouble. As an example, \vec{F}(x, y)=[cos(y), -cos(x)] Applying dx/F_1=dy/F_2 I get, sin(x)+sin(y)=C I have also seen what the vector field looks like, but I...
  36. S

    Divergence free vector fields in R^n

    Prove that every divergence free vector field on R^n, n>1 is of the form: v(x)=SUM dAij/dxi *ej where Aij(x) is smooth function from R^n to R such that Aij(x)=-Aji(x) i.e. matrix $[Aij(x)]$ is skew symmetric for every vector x.
  37. D

    Surface Integral of Vector fields

    Homework Statement Use Stokes' Theorem to evaluate ∫C F · dr. C is oriented counterclockwise as viewed from above. F(x, y, z) = (x + y^2) i + (y + z^2) j + (z + x^2) k C is the triangle with vertices (9, 0, 0), (0, 9, 0), and (0, 0, 9). Homework Equations Stokes' Theorem The...
  38. T

    Polar Coordinates and Conservative Vector Fields

    Homework Statement Let F = <-y/(x2+y2, x/(x2+y2>. Recall that F was not conservative on R2 - (0,0). In this problem, we show that F is conservative on R2 minus the non-positive x-axis. Let D be all of R2 except points of the form (-x,0), where x≥0. a) If (x,y) is included on D, show that...
  39. K

    Lie derivative of two left invariant vector fields

    Hi all, I was following Nakahara's book and I really got my mind stuck with something. I would appreciate if anybody could help with this. The Lie derivative of a vector field Y along the flow \sigma_t of another vector field X is defined as L_X...
  40. S

    Line integrals and vector fields.

    Homework Statement There is a circle of equation x^2+y^2=1 and a vector field F (x; y) =< y + .5x, x + .3y >. Imagine the field zoomed in extremely close at (0,1), to the point where it looks like a constant field of <-1,.3>. Calculate the work from say (0,1) to (-.001, 1). The constant field...
  41. V

    Finding force vector fields for 3 dimensional potential energy fields

    Homework Statement Find the force vector fields (in terms of x, y, z, and any constants) for each of the following 3-dimensional potential energy fields Question B: Assume SI units for force, energy, and lengths x, y, z: What must be the units of each of the constants? Homework Equations a)...
  42. S

    Help with plotting vector fields

    I'm having difficulty plotting a vector field of a first-order system. I understand that I am supposed to select various points, "plug" them into the equations, and then plot the vector from the resulting point. But how do I know in which way the vector is pointing? Once I get the point from...
  43. WannabeNewton

    Proof of Normal Vector Field for Hypersurface Sigma

    Hi guys, I was wondering if anyone could post or point me to a proof of the statement that given a hypersurface \Sigma , specified by setting a function f(x) = const., the vector field \xi ^{\mu } = \triangledown ^{\mu }f = g^{\mu \nu }\triangledown _{\nu }f will be normal to \Sigma in the...
  44. N

    Lie groups and non-vanishing vector fields

    I'm trying to understand why a Lie group always has a non-vanishing vector field. I know that one can somehow generate one by taking a vector from the Lie algebra and "moving it around" using the group operations as a mapping, but the nature of this map eludes me.
  45. J

    Understanding Homothetic Vector Fields: Intuition and References

    Can anyone provide a nice intuitive explanation for the main properties of homothetic vector fields? Alternatively, could anyone point me in the direct of a thorough reference?
  46. S

    About Vector fields and vector valued functions

    how do I make difference between vector valued functions and vector fields, I am confused how they differ and how are they same? Which is used with what? What about a function F(x,y,z,t) = (f1(x,y,z,t), f2(x,y,z,t), f3(x,y,z,t)) which maps R4 to R3, what type of function is this? F(x,y)...
  47. T

    Calculating the killing vector fields for axial symmetry

    hi, i need to calculate the killing vector fields for axial symmetry for a project so i can study the galaxy rotation curves. i am assuming the galaxy to be a flat disk, in addition to being axially symmetric. so i figured that the killing vector fields with respect to which the metric...
  48. A

    Exploring Vector Fields with Zero Divergence and Curl: A 2D Approach

    Homework Statement This problem is in Introduction to Eletrodynamics, of Griffiths, 3rd edition, p.20, problem 1.19. He asks a vector function v(x,y,z), other than the constant, that has: \nabla\cdot\vec{v}=0 \mbox{ and } \nabla\times\vec{v}=0 Homework Equations I hope you know them...
  49. T

    Action of Lie Brackets on vector fields multiplied by functions

    Hi, Is there a specific product rule or something one must follow when applying the lie bracket/ commutator to two vector fields such that one of them is multiplied by a function and added to another vector field? This is the expression given in my textbook but I don't see how: [fX+Z,Y] =...
  50. T

    Expansion of the commutator of two vector fields

    Hi, I don't understand a particular coordinate expansion of the commutator of 2 vector fields: [X, Y ]f = X(Y f) − Y (Xf) = X_be_b(Y _ae_af) − Y _be_b(X_ae_af) = (X_b(e_bY_ a) − Y _b(e_bX_a))e_af + X_aY _b[e_a, e_b]f X,Y = Vector fields f = function X_i = Components of X and...
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