What is Vector field: Definition and 402 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. A

    Both conservative and solenoidal vector field

    Homework Statement if a vecor A is both solenoidal and conservative; is it correct that A=-▼Φ that is A=- gradΦ Φ is a scalar function thanks
  2. D

    Understanding the Flow of a Vector Field: Basics and Calculations

    Hello, I am having a lot of trouble finding a definition of a flow generated by a vector field. I can't seem to find a good definition anywhere. I only need a basic definition, and a basic approach to calculating the flow generated by a vector field. For example, Let U = R2 , x = x(u, v, 0)...
  3. R

    Reconstruction of vector field from spherical harmonic coefficients

    The JGM3 model of Earth's gravity is expressed in the form of coefficients C and S to Legendre polynomials in r, theta and phi which give the gravitational potential U = \sum\sum CV + SW Can anyone tell me the algorithm for calculating acceleration vector g(r, theta, phi) from the...
  4. N

    Understanding the Physical Meaning of Divergence and Curl in Vector Fields

    Hello I am trying to get my head around what the divergence actually represents physically. If you have some vector field v, and the components of v, vx, vy, vz have dimensions of kg/s ("flow" - mass of material per second) the divergence will have units of kg/(s*m) (mass per time distance)...
  5. F

    Divergence and curl of vector field defined by \vec A = f(r)vec r

    Homework Statement A vector field is defined by A=f(r)r a) show that f(r) = constant/r^3 if \nabla. A = 0 b) show that \nabla. A is always equal to zeroHomework Equations divergence and curl relationsThe Attempt at a Solution I tried using spherical co-ordinates to solve this. But I am not sure...
  6. M

    Plotting Vector Field: V=(xi+yj+zk)/\sqrt{}(x^2+y^2+z^2)

    How to plot this vector field on a graph \stackrel{}{\rightarrow} V=(xi+yj+zk)/\sqrt{}(x^2+y^2+z^2)
  7. A

    Vector Field Describing Fluid Flow in a Torus

    Homework Statement Write a vector field equation which describes fluid flowing around a pipe of radius r whose axis is a circle of radius R in the (x,y)-plane. Homework Equations x2+y2=r2 Equation of a torus? The Attempt at a Solution What I've gathered from the question: the pipe...
  8. A

    Vector field equation - Find work of the field

    Homework Statement I have this vector field equation, the first part of the question is to find the potential equation for it, I found it. The second part of the question is to find the work of the field through this path. My idea is to plug t in the r equation, because I'm not sure but I...
  9. P

    Conservative vector field; classification of derivative

    Dear forum-members, Pestered by many (in my opinion, fundamental) questions and no literature at hand to answer them, I resort to posing my questions here. Let me start with the following. (Hopefully I have the correct subsection.) I am inspecting a dynamical, autonomous and conservative...
  10. A

    Find the flux of this vector field

    Homework Statement I need to find the flux of this vector field (in the pic) that goes through this plan (in the pic) and z goes from 0 to 1. How am I suppose to do that? Homework Equations The Attempt at a Solution
  11. F

    How can you use the line integral to find the work done by a conservative force?

    Homework Statement A 160-lb man carries a 25-lb can of paint up a helical staircase that encircles a silo witha radius of 20 ft. If the silo is 90 ft high and the man makes exactly three complete revolutions, how much work is done by the man against gravity in climbing to the top...
  12. A

    Finding Flux of Vector Field F Through Cube S

    Homework Statement I need to find the flux of the vector field F through S (in the pic), when S represent the edges of a cube. My question is, how do I find N (normal)? Do I need to split the curb and to find the flux through each face?Homework Equations The Attempt at a Solution
  13. K

    Potential function for conservative vector field

    [SOLVED] Potential function for conservative vector field Homework Statement Find a potential function for the conservative vector field F = <x + y, x - z, z - y> 2. The attempt at a solution OK, we know that (1) fx = x + y (2) fy = x - z og (3) fz = z - y We can then...
  14. C

    Vector calculus. Divergent vector field.

    I don't even know where to start this one. I can do all the other problems in the section, but this one makes no sense
  15. C

    Curl about an elipse. Line integral of vector field

    Homework Statement It can be shown that the line integral of F = xj around a closed curve in the xy - plane, oriented as in Green's Theorem, measures the area of the region enclosed by the curve. (You should verify this.) Use this result to calculate the area within the region of the...
  16. J

    Number of zeros of (tangent) vector field on sphere

    Is it possible to have a tangent vector field on the unit 2-sphere x^2+y^2+z^2 =1 in 3D which vanishes at exactly one point? By the Poincare-Hopf index theorem the index of such vector field at the point where it vanishes must be 2. Is that possible? If yes, can one write an explicit formula...
  17. C

    Flux of Vector Field Across Surface

    Homework Statement Compute the flux of vector field (grad x F) where F = (xz+x^2y + z, x^3yz + y, x^4z^2) across the surface S obtained by gluing the cylinder z^2 + y^2 = 1 (x is > or eq to 0 and < or eq to 1) with the hemispherical cap z^2 + y^2 (x-1)^2 = 1 (x > or eq to 1) oriented in...
  18. J

    Divergence theorem requires a conservative vector field?

    Can anyone tell me whether or not the divergence theorem requires a conservative vector field? On a practice exam my professor gave a vector field that was nonconservative (I checked the curl) and proceeded to perform the divergence theorem to find the flux. On one of my homework problems I...
  19. N

    Computing work from a vector field

    Homework Statement Picture is attached. I am trying to find the work done by F (gradient vector field) in moving an object from point A to point B along the path C1. Homework Equations Work = the line integral of F along the curve C of F dot dr. The Attempt at a Solution Just...
  20. D

    Energy flux vector field problem is the isotherms are circles

    Homework Statement Suppose that the isotherms in a region are all concentric spheres centered at the origin. Prove that the energy flux vector field points either toward or away from the origin. Homework Equations J = - k (del)T The Attempt at a Solution so I know that -(del)T is...
  21. D

    Proving Existence of Vector Field X for 1-Form w on Smooth Manifold M

    Let w be a 1-form on smooth manifold M. Then is there a vector field X such that locally w(X)=f where f:M-->R continuous? How can I prove it? Thanks.
  22. R

    Mathematica Plotting Vector Field in Mathematica

    Homework Statement y'=ay-by^2-q, where a, b are positive constants, and q is an arbitrary constant. In the following, y denotes a solution of this equation that satisfies the initial condition y(0) = y_0. a. Choose a and b positive and q < a^2/4b. By plotting direction fields and...
  23. U

    Vector field vs vector function

    My related questions 1 Is there any difference between 'vector field' and 'vector function'? 'vector function' is also called 'vector-valued function' (Thomas calculus). According to their definitions, they are all the same things to me. And they are all some kind of mapping, which assigns a...
  24. J

    Conservative vector field, potential function

    Homework Statement A vector field is defined by F(x) = (y+z, x+y, x+z). Find the Jacobian and determine if the field is conservative in a finite region. If it is conservative, find the potential function. Homework Equations F = delta p AKA F = (upsidedown triangle) p The Attempt...
  25. C

    Vector field uniquely determined by rot/div

    In physics one often uses the following: If the rotation of a vector field A vanishes, one can write A as the gradient of some scalar field, i.e. rot(A)=0 \Rightarrow A=\bigtriangledown \Phi. Is this true without further restrictions? If yes: Why? Thanks in advance...Cliowa
  26. N

    Proving the Irrotational Property of Vector Fields with an Example Solution

    Homework Statement A vector field V is not irrotational.Show that it is always possible to find f such that fV is irrotational. Homework Equations The Attempt at a Solution \nablax[fV]=f\nablaxV-Vx\nablaf I have to equate the LHS to zero.But then,how can I extract f out of the...
  27. K

    Conservative vector field or not?

    To show that a vector field F=(P,Q,R) is conservative, is it enough to show that DP/DY = DQ/DX?
  28. M

    Drawing a simple vector field issue

    Hello everyone I'm not sure if this is right or not... If i have F(x,y,z) = zj; where j is the vector, j hat. Would that be all vectors are going to be pointing up if you assume z is up, and are in the y plane? If the coordinate system is, z is up, y is to the right, and x is...
  29. K

    Condition of a vector field F being conservative is curl F = 0,

    When we say condition of a vector field F being conservative is curl F=0,does it mean that F=F(r)?.I know normally it does not look so.Please,then site an example where F is not a function of r,but still curl F=0.
  30. J

    What Does the Notation in This Vector Field Equation Mean?

    Just a quick question about notation. I was given the vector field F = r + grad(1/bar(r)) where r= (x)i+(y)j+(z)k. grad is just written as the upside down delta (gradient) and the bar I wrote in the above equation looks like an absolute value around just the r (although I don't know if it...
  31. A

    If the divergence of a vector field is zero

    Homework Statement If the divergence of a vector field is zero, I know that that means that it is the curl of some vector. How do I find that vector? Homework Equations Just the equations for divergence and curl. In TeX: \nabla\cdot u=\frac{\partial u_x}{\partial x}+\frac{\partial...
  32. H

    How to interpret physically the divergence of vector field?

    Hi all. I have difficulty in visualizing the concept of divergence of a vector field. While I have some clue in undertanding, in fluid mechanics, that the divergence of velocity represent the net flux of a point, but I find no clue why the divergence of an electric field measures the charge...
  33. M

    Help: Vector field and radius vector

    Hi Guys, Given the vector field X(x,y) = ( a + \frac{b(y^2-x^2)}{(x^2+y^2)^2}, \frac{-2bxy}{(x^2+y^2)^2}}}) Show that for a point (x,y) on the circle with radius r = \sqrt(b/a) (i.e. x^2 + y^2 = b/a), the vector X(x,y) is tangent to a circle at the point. My strategy is that to first...
  34. E

    Calculating Line Integrals with Vector Fields on a Bounded Region in 3D Space

    Again, I'm stuck on a question: "Let C be the region in space given by 0 \leq x,y,z \leq 1 and let \partial C be the boundary of C oriented by the outward pointing unit normal. Suppose that v is the vector field given by v = (y^3 -2xy, y^2+3y+2zy, z-z^2) . Evaluate \int_{\partial...
  35. U

    Flux of the vector field

    Let S be the part of the plane 3x+y+z=4 which lies in the first octant, oriented upward. Find the flux of the vector field F=4i+2j+3k across the surface S. \int \int F\cdot dS = \int int \left( -P \frac{\partial g}{\partial x} -Q \frac{\partial g}{\partial y} +R \right) dA \int \int \left(...
  36. C

    Showing that a three-dimensional vector field is conservative

    Alright, so the field is \mathbf{F} = (z^2 + 2xy,x^2,2xz) it's a gradient only when f_x = z^2 + 2xy, f_y = x^2 and f_z = 2xz integrate the first equation with respect to x to get f(x,y,z) = \int z^2 +2xy\,dx = xz^2 + x^2y + g(y,z) now, f_z(x,y,z) = g_z(y,x) which is 2xz integrate that with...
  37. H

    How Do You Visualize a Vector Field Like v(x, y) = (2.5, -x)?

    Please could someone explain to me how to visualise a vector field? Let's say it's v(x, y) = (2.5, -x) on whatever domain. I tried it the same way as I would visualize a scalar field but the results did not correspondent at all with the results I'd expect. The same for drawing the field along...
  38. 0

    Way to express a general vector field

    Is there a simple way to express a general vector field in terms of the gradient of another (perhaps higher dimensional) function?
  39. A

    Find integral curve over vector field

    The question should be very easy, its from topics of Differential Geometry, I just want to make sure that I understands it right :shy: . My question is: in R^3 we have vector field X and for every point p(x,y,z) in R^3 space, vector field X(p) = (p; X_x(p), X_y(p), X_z(p)) has: X_x(p) =...
  40. M

    Qestion: Vector field and (n-1)-form representation of current density

    Question: Vector field and (n-1)-form representation of current density Electric current density can be represented by both a vector field and by a 2-form. Integrating them on a given surface must lead the same result. My question is, what is the relation between this vector field and the...
  41. H

    Can a 'Good Surface' Be Found for Any Continuous Vector Field?

    I am terribly sorry for not being able to write this simple equation in Latex form. (I will be really glad if someone can tell me where I can learn how to use Latex to write math symbol) Let F' be a vector field given by F' = r r' (r' = radial unit vector) and also let p be a point on the...
  42. S

    If you find the scalar potential of a conservative vector field

    Should your answer include the constant of integration? I think it should but my book's answers don't, so I dunno. Example, <2xy^3, 3y^2x^2> answer is x^2y^3, but should I include the + C? (and yes I went through and made sure h(y) was in fact a constant
  43. D

    Vector Field: Showing Divergence & Curl A = 0

    A vector field is difined by A = f(r)r. a) show that f(r) = constant/r^3 if divergence A equal to zero. b) show that curl A is alway equal to zero
  44. Reshma

    Find Divergence of Vector Field: $\vec F$

    Given a vector field: \vec F= (x^2-xy)\hat x +(y^2-yz)\hat y +(z^2-xz)\hat z Find the conditon for the divergence to be equal to zero.
  45. cepheid

    Divergence of a Radial Vector Field

    Something we did in electrostatics that's a source of confusion for me: We learned to use caution when taking the divergence of the (all important) radial vector field: \vec{v} = \frac{1}{r^2} \hat{r} Applying the formula in spherical coords gave zero...a perplexing result. The...
  46. 6

    Potential function of conservative vector field

    Hey ya'll, How do I find the potential function of this conservative vector field (It is conservative isn't it?? I did check, but i might've messed that up too!). \int (2x-3y-1)dx - (3x+y-5)dy I know to break the function: F(x,y)= (2x-3y-1)i - (3x+y-5)j apart and integrate...
  47. H

    A question about path independence and curl of a vector field

    If the curl of a vector field is zero, then we can that the vector field is path independent. But there are cases where this is not true, I was wondering how? Whats the explanation for this? Thanks in advance for any help. - harsh
  48. W

    Tagent vector and vector field difference

    Hi there Can somebody please explain shortly the difference between a tangent vector and a vector field? I'm still new to differential geometry. I read couple of sources that had mixed claims on which of them actually act on a given function f. so I'm kind of confused. Much appreciated.
  49. Mk

    Vector field, and Lorentz Symmetry

    What are they? "A fundamental property of the natural world that is of supreme importance for physics. It has two components: rotational symmetry, and boost symmetry." :confused:
  50. P

    Evaluating a Vector Field Through a Surface with the Divergence Theorem

    ok this probley seems simple but i just need to see how to do it, ok well how do u evaluate this... find the flux of the vector field... \vec{F}=<x,y,z> throught this surface above the xy-plane.. z = 4-x^2-y^2 how do u evaluate this with surface integrals method and the divergence...
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