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. K

    Understanding the Curl Theorem: Examples and Explanation

    Hi, this is a very simple question about the curl theorem. It says in my book: " If F is a vector field defined on all R3 whose component functions have continuous partial derivatives and curl F = 0 , then F is a conservative vector field" I might sound stupid, but what exactly does...
  2. A

    Finding Divergence of Vector Fields on a Sphere

    Homework Equations Hey guys I had a slight problem trying to find divergence of vector fields for the following equation: F(x,y,z)=(yzi-xzj-xyk)/(x^2 + y^2 + z^2) So I want to know if its possible of substitute (x^2 + y^2 + z^2) for 1 since that is the equation of a sphere? If not...
  3. F

    Description of radial vector fields?

    I just came across the term, and was hoping someone could tell me what they are and any general form of them. Thank you!
  4. S

    Non conservative vector fields

    F(x,y,z)=ax P(x,y,z)+ay Q(x,y,z)+az R(x,y,z) F is vectoral field. ax , ay and az are unit vectors. P , Q ,R are scalar functions. The question is this: If F is non-conservative vectoral field ; what are the characteristics of P Q and R? thanks in advance. have a nice day
  5. F

    Exploring Pathlines and Vector Fields

    Homework Statement Trying to get my head around the physical interpretation of pathlines and the math that describes them. The physical explanation is simple enough, they are just the path a particle of fluid will follow through the vector field. Homework Equations In the vector field...
  6. J

    Are all vector fields invariants?

    Are all vector fields invariants or is this a particular characteristic to some fields? For example, suppose the vector field E = x^2 x + xy y. If I write it in terms of covariant and contravariant basis through the polar coordinates I get the same results, or in other words \vec{E} =...
  7. K

    Differentiation of Vector Fields

    Homework Statement Let X,Y be vector fields and x(t) be a curve satisfying \dot x(t) = X(x(t)) + u(t) Y(x(t)), u(t) \in \mathbb R [/itex] and assume there exists p(t) an adjoint curve satisfying \dot p(t) = -p(t) \left( \frac{\partial X}{\partial x}(x(t)) + u(t) \frac{\partial Y}{\partial x}...
  8. M

    Curl Test for vector fields

    Homework Statement F=-ysin(x)i+cos(x)j Homework Equations Can the Curl test be applied to this vector field and state three facts you can deduce after applying the curl test. The Attempt at a Solution
  9. J

    Need Help Re-Learning Flux Integrals for Constant Vector Fields

    I have done relatively few in physics courses and I need to re-learn how to do some flux integrals for constant vector fields through rectangular and circular surfaces. If anyone has any direction to some great resources, or themselves could be a great resource for help please post what are...
  10. G

    Optimizing Work: Finding the Minimum Path in Non-Conservative Vector Fields

    I was wondering, if you have a non-conservative vector field (so that the line integral of each path from point A to point B isn't the same) that represents some sort of force, then is there a method to find the path that requires the least amount of work from a designated point A to point B...
  11. I

    Integrating Vector Fields on a Sphere

    Homework Statement Given two vector fields: i) A \frac{\vec{r}}{r^{n_{1}}} ii) Ae_{z} \times \frac{\vec{r}}{r^{n_{2}}} where A is a constant and n_{1} \neq 3 and n_{2} \neq 2 find \int \vec{F} dS through surface of a sphere of radius R Homework Equations \int \vec{F} r^{2}...
  12. A

    Vector fields in cylindrical and spherical coordinates

    I am reading the Wikipedia entry http://en.wikipedia.org/wiki/Vector_fields_in_cylindrical_and_spherical_coordinates" . There, in particular I see this: Time derivative of a vector field To find out how the vector field A changes in time we calculate the time derivatives. In cartesian...
  13. S

    Understanding the Relation between Vector Fields, Flux, and Stokes' Theorem

    What is the relation between the flux through a given surface by a vector field? And how does stokes theorem relate to the line integral around a surface in that field
  14. H

    The curl of certain vector fields

    Given the two vector fields: \vec E and \vec B Where the first is the electric vector field and the second is the magnetic vector field, we have the following identity: curl(\vec E) = -\frac{\partial \vec B } { \partial t } and further that: curl(curl(\vec E)) =...
  15. K

    Calculating Work Done by a Conservative Vector Field Along a Curve

    If C is the curve given by r(t)=<1+3sin(t), 1+5sin^2(t), 1+5sin^3(t)>, 0≤t≤π/2 and F is the radial vector field F(x, y, z)=<x, y, z>, compute the work done by F on a particle moving along C. Work= int (F dot dr) If F is the potential function(?), do I integrate F with respect to each...
  16. N

    Double Integral Syntax: WolframAlpha Solver

    Online integrator: http://www.wolframalpha.com/ \int \int (x,y,z) \times (x,y,z) \times (x,y,z) \int_{L1} \int_{L2} (dl1,0,0) \times (dl2,0,0) \times (0,-1,0) What would be correct syntax to evaluate this double integral? I tried these, but they produce wrong result: try...
  17. Y

    Can ALL Vector Fields Be Expressed as a Product?

    Hey - I'm stuck on a concept: Are ALL vector fields expressable as the product of a scalar field \varphi and a constant vector \vec{c}? i.e. Is there always a \varphi such that \vec{A} = \varphi \vec{c} ? for ANY field \vec{A}? I ask because there are some derivations from...
  18. J

    Understanding Vector Fields on the Sphere S^2: A Student's Guide

    i need a vector field on S^2 that vanishes at 1 point. there was a thread like this here, but the answer was ((v1/1+x^2),(v2/1+y^2)) and i really don't see how this vanishes at a point although i do get it intuitively. My professor hinted that i should take a non zero vector fiels in S^2 x R^2...
  19. L

    Vector Laplacian: Exploring Vector Fields

    Can all vector fields be described as the vector Laplacian of another vector field?
  20. L

    Integrating Vector Fields: Volume vs. Surface

    Homework Statement \int delxA dv = -\oint Axds where A is a vector field Left hand side is integral over volume. Right hand side is integral over closed surface. Homework Equations The Attempt at a Solution Can't understand what Axds means.
  21. L

    Can I visualize three variable functions through scalar and vector fields?

    Graphics of one variable functions are two dimensional lines. Graphics of two variable functions are three dimensional surfaces. Three variable functions cannot be plotted. But can I think of the usual 3D representations of vector and scalar fields as manners of visualizing a three variable...
  22. L

    Vector Calc Homework Help Divergence Free Vector Fields

    Homework Statement Let S be the ellipsoid where a,b, and c are all positive constants. x2/(a+1)+y2/(b2)+z2/(c2) = 1 → → → → → Let F = (r - ai) / ||r - ai|| [* r and i are vectors = I tried inserting the arrows] a)Where...
  23. R

    Line Integrals / Conservative Vector Fields

    Homework Statement F = < z^2/x, z^2/y, 2zlog(xy)> F = \nabla f, where f = z^2log(xy) Homework Equations Evaluate \int F \cdot ds for any path c from P = (1/2, 4, 2) to Q = (2, 3, 3) contained in the region x > 0, y > 0, z > 0 Why is it necessary to specify that the path lie in the...
  24. W

    Gluing maps from vector fields

    Take a tangent vector field with isolated singularities on a compact smooth Riemannian surface ( 2 dimensional manifold without boundary). Divide v by its norm to get a field of unit vectors with isolated discontinuities. Around each singularity chose a small open disc. The tangent circle...
  25. S

    Need assistance(Gussian curvature and differentiable vector fields)

    Need urgent assistance(Gussian curvature and differentiable vector fields) Hi I have a very difficult problem where I know some of the dots but can't connect them :( So therefore I hope that there is someone who can assist me (hopefully :)) Homework Statement Let S be a surface with...
  26. D

    Very quick question about notation of vector fields

    this is a very quick question. my teacher wants me to prove that ((kq)/(sqroot(x^2+y^2+z^2))) (x,y,z) is conservative. by this does it mean that F=((kq)/(sqroot(x^2+y^2+z^2)))i+((kq)/(sqroot(x^2+y^2+z^2)))j+((kq)/(sqroot(x^2+y^2+z^2)))k or does it she mean that...
  27. D

    Is F(x, y, z) a Conservative Vector Field?

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  28. M

    Calc III Vector Fields: Finding Conservativity and Potential Function

    Homework Statement Show that the vector field given is conservative and find its potential function. F(x, y, z)= (6x^2+4z^3)i +(4x^3y+4e^3z)j + (12xz^2+12ye^3z)k. Homework Equations The Attempt at a Solution When I take partial derivative with respect of y for...
  29. D

    Surface Integral of Vector Fields

    Evaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = y i + x j + z^2 k S is the helicoid (with upward orientation) with vector...
  30. S

    Solving Differential Equations Involving Vector Fields

    Given the curl and divergence of a vector field, how would one solve for that vector field? In the particular case I would like to solve, divergence is zero at all coordinates.
  31. B

    Calculating Constants for Conservative Vector Field F

    We are told that the force field F=(\muz + y -x)i + (x-\lambdaz)j + (z+(\lambda-2)y - \mux)k Having already calculated in the previous parts of the question 1:the line integral (F.dx) along the straight line from (0,0,0) to (1,1,1) 2:the line integral (F.dx) along the path x(t)=(t,t,t^2)...
  32. E

    Sketching vector fields

    Homework Statement I am asked to sketch the following vector field in the xy-plane (a) F(r) = 2r (b) F(r) = -r/||r||3 (c) F(x,y) = 4xi + xj Homework Equations The Attempt at a Solution Can someone please give me some hints on how to proceed
  33. S

    Can I derive an equation for a vector field's divergence from its curl equation?

    Given an equation describing the curl of a vector field, is it possible to derive an equation for the originating vector field? The divergence of the field is known to be zero at all points
  34. W

    Vector Fields on S^2, 0 only at 1 point.

    Hi, everyone: I am trying to produce a V.Field that is 0 only at one point of S^2. I have been thinking of using the homeo. between S^2-{pt.} and R^2 to do this. Please tell me if this works: We take a V.Field on R^2 that is nowhere zero, but goes to 0 as (x,y) grows (in the...
  35. V

    Calculating Div and Curl for some arbitrary vector fields

    Homework Statement Calculate the (1) divergence and (2) curl of the following vector fields. (a) \widehat{E}(\widehat{x}) = r^{n}\widehat{x} (b) \widehat{E}(\widehat{x}) = r^{n}\widehat{a} (c) \widehat{E}(\widehat{x}) = r^{n}*(\widehat{a} X \widehat{x} where r = |\widehat{x}| and...
  36. P

    Surface Integrals of Vector Fields question

    Homework Statement F=<0,3,x^2> computer the surface integral over the hemisphere x^2 + y^2 + z^2 = 9 z greater than or equal to 0, outward pointing normal. Homework Equations The Attempt at a Solution I don't know why I keep getting this problem wrong. The general formula for...
  37. J

    MATLAB Plotting vector fields in MATLAB or Maple

    Homework Statement The original problem was x'=(-2 1; 1 -2)*x and I needed to find two linearly independent solutions. Homework Equations The Attempt at a Solution I found that x1=(1;1)e^(-t) and x2=(1;-1)e^(-3t). Now I am trying to plot a vector field of this. Is there an easy way...
  38. W

    Vector Fields and particle

    Homework Statement Find the work done by the force field F(x, y, z) = 3xi +3yj + 7k on a particle that moves along the helix r(t) = 4 cos(t)i + 4 sin(t)j + 4tk, 0 ≤ t ≤ 2π (As in the previous problem, recall that the work of the force F on the helix corresponds to the circulation of this...
  39. M

    Vector currents, vector fields and bosons

    Consider this quote from Mandl and Shaw, p. 237 ...this interaction coulpes the field W_{\alpha}(x) to the leptonic vector current. Hence it must be a vector field, and the W particles are vector bosons with spin 1. Could someone explain this for me? I do not understand the "hence"...
  40. J

    Some questions on vector fields on Lie groups

    Homework Statement Let G be a Lie group with unit e. For every g in G let Lg: G -> G, h-> gh be left multiplication. a) Prove that the map G x TeG -> TG, (g,v) -> De Lg (v) is a diffeomorphism, and conclude G has a basis of vector fields. b) Prove for every v in TeG the is a unique...
  41. JasonJo

    Parallel Vector Fields

    Let M be a Riemmanian manifold. Prove that a parallel vector field along a curve c(t) preserves the length of the parallel transported vector. Furthermore if M is an oriented manifold, prove that P preserves the orientation. So I want to prove that d/dt <P, P> = 0, so <P, P> = constant...
  42. A

    Calculating Homothetic Vector Fields: A Step-by-Step Guide

    can anyone please give me an example of how to calculate the homothetic vector field of say a Bianchi Type I exact solution of the Einstein field equation (refer to dynamical systems in cosmology by Wainwright and Ellis chapter 9) note : I know already how to calculate the...
  43. S

    Coordinates adapted to two vector fields

    I am considering two vector fields in the spacetime of General Relativity. One is spacelike, the other is timelike, they are normalized and orthogonal: U.U = -1 V.V = +1 U.V = 0 where dot denotes scalar product. In addition, it is known the integral curves of U and V always remain in...
  44. P

    Vector fields, metrics and two forms on a spacetime.

    Homework Statement Let (M,g) be a spacetime. (a) Let A and A' be vector fields on M such that g(A,B)=g(A',B) for any future-pointing timelike vector field Y. Show that X=X'. (b) Let w and w' be two two-forms on M. Suppose that i¬A w = i¬A w' for any future -pointing timelike vector field A...
  45. K

    Conservative vector fields

    Q: Let F= (-y/(x2+y2), x/(x2+y2), z) be a vector field and let U be the interior of the torus obtained by rotating the circle (x-2)2 + z2 = 1, y=0 about the z-axis. a) Show that curl F=0 but ∫ F . dx = 2pi where dx=(dx,dy,dz) C and C (contained in U) is the circle x2+y2=4, z=0. Therefore F...
  46. K

    How do functions and vector fields interact with normal derivatives?

    Suppose that normal derivative = \nablag . n = dg/dn, then f \nablag . n = f dg/dn [I used . for dot product] But how is this possible? For f \nablag . n, I would interpret it as (f \nablag) . n But f dg/dn = f (\nablag . n) which is DIFFERENT (note the location of brackets) I...
  47. E

    Conservative Vector Fields - Is this right?

    Conservative Vector Fields -- Is this right? Homework Statement G = <(1 + x)e^{x+y}, xe^{x+y}+2z, -2y> Evaluate \int_{C}G.dR where C is the path given by: x = (1 - t)e^{t}, y = t, z = 2t, 1=>t>=0 Homework Equations The Attempt at a Solution First, i noticed that there is a scalar potential...
  48. E

    Finding Vector Fields that Satisfy Certain Curls

    Homework Statement Is it possible to find a vector field whose curl is yi? xi? Homework Equations The Attempt at a Solution I found that if F = <0,0,y^2/2>, its curl will be yi. However, I cannot figure out a vector field whose curl will be xi. I tried using exponentials and just...
  49. N

    Singularities and conservative vector fields

    I have a question regarding conservative vectorfields and singularities. Suppose we have a vectorfield who is defined everywhere in R^2 except at the origin where it has a singularity, and suppose it's curl is zero. We then have that it is conservative in every open, simply connected subset in...
  50. Q

    Quanta of massive vector fields

    Are the W+,W- and Z0 the field quanta of the massive charged vector fields? ie Proca fields
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