Vector field Definition and 402 Threads

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. D

    Conservative Vector Field: Understanding Circulation and Potential Function

    Let f(x,y) = ( \frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2}) with f : D \subset \mathbb{R}^2 \to \mathbb{R}^2 I know if I take D = D_1 = \mathbb{R}^2 - \{ (0,0) \} the vector field is not conservative, for the circulation over a circunference centered at the origin does not equal zero. But...
  2. T

    Flux through a sphere given a vector field

    Homework Statement Vector field F = (2siny-cosz, 2cosx+3sinz, cosy-2sinx) Compute the flux of F through a sphere of radius one centered at the origin with respect to the outer unit normal. Homework Equations Divergence theorem/Gauss's Law The Attempt at a Solution The only...
  3. J

    Work done and limit of a vector field.

    Homework Statement Consider the vector field F = r/|r|^p = <x,y,z>/|r|^p where p>1. Let C be the line from point (1,1,1) to the point (a,a,a) for a>1. (a) Compute work done in moving an object along C. (b) Find the limit a->infinity (that is, the work done in moving an object to...
  4. B

    From Conservative Vector Field to Potential Energy function

    What if when I'm finding the repeated terms from integrating the vector field with respect to x and then y, i come across two terms that are same except one is negative and the other positive. What does that mean, and how do i represent that in the overall potential function I'm finding?
  5. D

    Quantization of vector field in the Coulomb gauge

    I have a technical question and at the time being I can't ask it to a professor. So, I'm here: If I try to quantize the vector field in the Coulomb gauge (radiation gauge) A_0(x)=0,\quad \vec\nabla\cdot\vec A=0. by imposing the equal-time commutation relation...
  6. G

    Show that this field is orthogonal to each vector field.

    Homework Statement If a, b, and c are any three vector fields in locally Minkowskain 4-manifold, show that the field ε_{ijkl}a^{i}b^{k}c^{l} is orthogonal to \vec{a}, \vec{b}, and \vec{c}. Homework Equations The Attempt at a Solution I know I have to show that multiplying the...
  7. Matterwave

    Shutz's derivation of the Lie Derivative of a vector field

    I have a question about Bernard Shutz's derivation of the Lie derivative of a vector field in his book Geometrical Methods for Mathematical Physics. I will try to reproduce part of his argument here. Essentially, we have 2 vector fields V and U which are represented by \frac{d}{d\lambda} and...
  8. S

    Confusion over the definition of Lie Derivative of a Vector Field

    Hello all, I was hoping someone would be able to clarify this issue I am having with the Lie Derivative of a vector field. We define the lie derivative of a vector field Y with respect to a vector field X to be L_X Y :=\operatorname{\frac{d}{dt}} |_{t=0} (\phi_t^*Y), where \phi_t is the...
  9. B

    When the gradient of a vector field is symmetric?

    Homework Statement "A gradient of a vector field is symmetric if and only if this vector field is a gradient of a function" Pure Strain Deformations of Surfaces Marek L. Szwabowicz J Elasticity (2008) 92:255–275 DOI 10.1007/s10659-008-9161-5 f=5x^3+3xy-15y^3 So the gradient of this function...
  10. P

    Divergence of left invariant vector field

    Let's assume that a compact Lie group and left invariant vector filed X are given. I wonder why the divergence (with respect to Haar measure) of this field has to be equall 0. I found such result in one paper but I don't know how to prove it. Any suggestions?
  11. A

    Vector Field Homework: Learn What's Involved

    Homework Statement Homework Equations The Attempt at a Solution What's surprise? And is my work correct?
  12. P

    Two problems while reading Feynman lectures (vector field))

    Question 1: solved! Question 2: Why it's zero? I think we cannot get zero unless it's an exact differential form? Many thanks.
  13. B

    Vector field calculations- dumb question

    Given 2 vectors, say x and y with a relationship of barXx-barYy (barX =1,0,0 and barY=0,1,0) calculations for the vector field in x and y plane are: Magnitude = sqrt(x2+y2) tails of vectors begin at input points (IE: if I choose 1,0 or 0,-1 etc) and the magnitudes are calcluated from these...
  14. B

    Mathematica Mathematica: Graph for a single vector instead of vector field?

    Hi guys, I would like to construct 2 vectors on a coordinate grid.(or a vector field for only one t) of the forces between 2 point particles on a certain moment t. Can I do that? When I try the VectorPlot function and insert all values instead of also inserting a variable it gives the error that...
  15. A

    Given divergence and curl determine vector field

    the divergence and the curl of a vector field "A" are specified everywhere in a volume V. The normal component of curl A is also specified on the surface S bounding V. Show that these data enable one to determine the vector field in the region
  16. R

    Mathematica How to plot 3D vector field in spherical coordinates with Mathematica

    I want to graph this vector field -^r/r^2 but I don't know how to do. Any help would be appreciated.
  17. L

    Vector Field Derivatives: Why & How?

    Hi I thought about putting this topic in physics subforum, but I think it's overall more fitting here. So, I'm having problems understanding some basic stuff, and I'm kinda embarassed. I'm an engineer and I'm trying to sort things I already know but on a more rigorous mathematic foundation...
  18. T

    Creating a Vector Field from a 3D Parametric Equation

    I am trying to turn a 3D parametric equation into a vector field for an experiment, but I am not having much luck. [x,y,z]=[r*cos(u),r*sin(u),a*u] is the equation, I'm using grapher on the Mac. I want it all going in a helix, which is what the equation is for. Thanks!
  19. F

    Just what does it mean when a vector field has 0 divergence?

    Homework Statement Yeah I've been pondering over that, my book doesn't really do the justice of nailing it down for me. Does having 0 divergence means having "absolute convergence", like maybe at every point (or at a certain point) all the vectors are pointing towards a point? Like...
  20. D

    How Is Work Calculated in a Vector Field?

    Homework Statement Find the work done by the force field F in moving an object from P to Q. F(x,y,z)=10y^(3/2)i+15x\sqrt{y}j P(1,1), Q(2,9)Homework Equations W = \intF dot drThe Attempt at a Solution I have no clue how to do it
  21. WannabeNewton

    Integral Curves of Vector Field B in $\mathbb{R}^3$

    Homework Statement For \mathbb{R}^{3} find the integral curves of the vector field B = xy\frac{\partial }{\partial x} - y^{2}\frac{\partial }{\partial y}. Homework Equations The Attempt at a Solution I am having a hard time understand just how to set up the differential equations in order to...
  22. C

    Computing Line Integrals Related to Vector Field F in R2

    We are given a vector field: F=\frac{-y}{x^2+y^2} , \frac{x}{x^2+y^2} Then asked if F is conservative on R2 \ (0,0). I just solved the partial derivatives of each part of the vector field and they did indeed equal each other, but I don't under stand what the "\(0,0)" part means. We are then...
  23. T

    Compute Flux of Vector Field Through Surface S

    Homework Statement Compute the flux of the vector field, , through the surface, S. \vec{F}= 3xi + yj + zk and S is the part of the surface z + 4x + 2y = 12 in the first octant oriented upward. Homework Equations by definition from my book the integral is \intF(x,y,f(x,y)\circ<-fx,fy,1>dxdy...
  24. Z

    Line Integral and Vector Field Problem

    Homework Statement Find the work done by the force field F(x,y) = x sin(y)i + yj on a particle that moves along on the parabola y = x^2 from (-1,1) to (2,4). Homework Equations Work = line integral of the dot product of Field vector and change in the path The path is parabola equation...
  25. J

    Confused- Integrating a vector field along a curve in 3D.

    Homework Statement Let f be a vector function, f = (xz, 0, 0), and C a contour formed by the boundary of the surface S S : x^2 + y^2 + z^2 = R^2 , x ≥ 0, y ≥ 0, z ≥ 0 , and oriented counterclockwise (as seen from the origin). Evaluate the integral (Closed integral sign) f · dr , directly as...
  26. S

    Representing Gravity as a Vector Field

    In my book, it says that gravity can be thought of as a force in the form of this vector: F= (-GMm)/(x2+y2+z2)*u where u is a unit vector in the direction from the point to the origin. How would this be represented as a vector field (this is not a homework problem, just me wondering...)? Is...
  27. I

    Calculating Vector Field for f(z)=-iz

    Homework Statement f(z) = -iz is an analytic function [i being the complex number] Calculate the vector field v=u(x,y)i -v(x,y)j Homework Equations z = x+iy [Im assuming z here is the complex z not a varaible z] The Attempt at a Solution I've not learned polar vector calculus yet...
  28. Y

    What Defines a Uniform Vector Field?

    I cannot find the meaning of the uniform vector field. I know \hat z k_x+\hat y k_y +\hat z k_z is a uniform vector field if k_x,k_y,k_z are constants. Does this means a uniform vector field: 1) Points to the same direction in all locations? 2) Have the same magnitude in all...
  29. U

    What is a Hamiltonian vector field in General Relativity?

    I'm researching General Relativity and have stumbled upon a bit of Hamiltonian mechanics. I roughly understand the idea behind the Hamiltonian of a system, but I'm utterly confused as to what the hell a Hamiltonian vector field is. I've taken ODE's, PDE's, Linear Algebra, and I'm just being...
  30. fluidistic

    Vector field, vortex free and sources free

    Homework Statement I must determine whether the following vector fields have sources or vortices. 1)\vec A = (\vec x )\frac{\vec a \times \vec x}{r^3} where \vec a is constant and r=||\vec x||. 2)\vec B (\vec x )= \frac{\vec a}{r+ \beta} where \vec a and r are the same as part 1) and \beta >0...
  31. J

    Vector field identity derivation using Einstein summation and kronecker delta.

    Homework Statement Let \vec{A}(\vec{r})and \vec{B}(\vec{r}) be vector fields. Show that Homework Equations \vec{\nabla}\bullet(\vec{A}\vec{B})=(\vec{A}\bullet\vec{\nabla})\vec{B}+\vec{B}(\vec{\nabla}\bullet\vec{A}) This is EXACTLY how it is written in Ch 3 Problem 2 of Schwinger...
  32. B

    Embedding vector graphics in Word docs - vector field plots

    I have a question: hope this fits under "Calculus" topic. I am trying to embed vector graphics in my Word documents. MathCAD does this quite nicely for xy-plots, but MathCAD is awkward for constructing vector fields like those you'd find in electrodynamics. I could use Maple, but those embeds...
  33. A

    Curl of Vector Field u = yi+(x+z)j+xy^(2)k: Step-by-Step Calculation Method

    Find the curl of the following vector field u = yi+(x+z)j+xy^(2)k Now using the method I've bin taught similar to finding determinant of 3x3 matrix here is my answer i(2yx-1) -j(y^2) +k(0)Just looking for confirmation if this is correct or any basic errors I have made thank you.
  34. A

    How do you find the divergence of a vector field?

    I am just curious how you find the divergence of the following vector field Heres my example u = xz^(2)i +y(x^(2)-1)j+zx^(2) y^(3)k Am I right in thinking U take the derivative with respect to x for first term derivative with respect to y for second term... giving me...
  35. E

    Bessel functions in vector field

    I need to solve this general problem. Let's consider the following vector field in cylindrical coordinates: \vec{A}=-J'_m(kr)\cos(\phi)\hat{\rho}+\frac{m^2}{k}\frac{J_m(kr)}{r}\sin(\phi)\hat{\phi}+0\hat{z} where m is an integer, and k could satisfy to: J_m(ka)=0 or J_m'(ka)=0 with a real. (the...
  36. W

    Finding a vector field perpendicular to the surface of a sphere

    I'm trying to figure out if a given vector field is perpendicular at the surface of a sphere of radius R. The vector field is given in spherical coordinates. I initially attempted to take the cross product of the vector field with the normal vector at the surface of the sphere to see if it was...
  37. C

    Vector Field Problem: Pressure Tendency at Service Station

    Homework Statement A car is driving straight southward, past a service station, at 100 km/h. The surface pressure decreaes toward the southeast at 1 Pa/km. What is the pressure tendency at the service station if the pressure measured by the car is decreasing at a rate of 50 Pa/3h? (Hint: draw...
  38. T

    Mastering Vector Field Plotting: Homework Statement and Solution Attempt

    Homework Statement [PLAIN]http://img84.imageshack.us/img84/5273/questionm.png The Attempt at a Solution How do I go about plotting this?
  39. T

    Solving Vector Field with Poincare's Lemma

    Homework Statement [PLAIN]http://img130.imageshack.us/img130/8540/vecx.jpg The Attempt at a Solution I've done (i). First of all Poincare's Lemma says that if the domain U of {\bf F} is simply connected then: {\bf F} is irrotational \iff {\bf F} is conservative. So for...
  40. D

    Two covariant derivatives of a vector field

    V_{a;b} = V_{a,b} - \Gamma^d_{ad}V_d Now take the second derivative... V_{a;b;c} = (V_{a;b})_{,c} - \Gamma^f_{ac}V_{f;b} - \Gamma^f_{bc}V_{a;f} But I have no idea how to get the parts with the Christoffel symbols. V_{a;b;c} = (V_{a;b})_{,c} - \Gamma^f_{(a;b)c}V_{af} = (V_{a;b})_{,c} -...
  41. K

    Differentiation a Hamiltonian Vector Field

    Homework Statement Consider a smooth manifold M, and smooth functions H_i: T^*M \to \mathbb R, i=0,1 on the cotangent bundle. Further, let z:[t_1,t_2] \subset \mathbb R \to M define a trajectory on T^*M by \frac{dz}{dt} = \vec H_0(z(t)) + u(t) \vec H_1(z(t)) where \vec H_i is the...
  42. A

    Proof of Zero Value for Vector Field Integral on Closed Surface

    What is the value of a surface integral over a closed, continuous surface of a vector field of vectors normal to the surface? The integral of ndS over S. I believe the answer is zero. Can someone direct me to a proof for an aribitrary closed surface?
  43. H

    Vector field as smooth embedding

    We can show that any vector field V:M->TM(tangent bundle of M) is smooth embedding of M, but how do we show that these smooth embeddings are all smoothly homotopic? How to construct such a homotopy?
  44. H

    Number of Nondegenerate Zeros of Vector Field Bounded

    "if a vector field has only nondegenerate zeros then the number of zeros is bounded" With no idea how to show that without using Poincare-Hopf Theorem. Any proof possible without using any concept from algebraic topology?
  45. R

    Line integral of a vector field

    Hello, I am attempting to calculate the line integral of the vector field Line integral of a vector field \overline{A}= x^{2} \hat{i} + x y^{2} \hat{j} around a circle of radius R ( x^{2} + y^{2} = R^{2} ) using cartesian coordinates. The appropriate differential line element in cartesian...
  46. R

    Line integral of a vector field

    I am attempting to calculate the line integral of the vector field \overline{A}= x^{2} \hat{i} + x y^{2} \hat{j} around a circle of radius R (x^{2} + y^{2} = R^{2}) using cylindrical coordinates. It is simple enough to convert the x and y components to their cylindrical counterparts, but I am...
  47. F

    How Does Stokes' Theorem Apply to a Triangular Contour in Vector Calculus?

    Homework Statement Assume the vector function A = ax(3x^{2}2y^{2})-ax(x^{3}y^{2}) a) Find \ointA\cdotdl around the triangular contour shown in Fig. 2-36 [it is a triangle with base and height of one on the x and y axis. the curl travels so that the normal vector is in the -z direction] b)...
  48. X

    Finding Line Integral of Vector Field

    Homework Statement You are given a vector field A= kx2 x. a. First, calculate the line integral of A from x=-2 to x=2 along the x axis. b. Next, calculate the line integral of A between the same 2 points, but use a semicircular path with a center at the origin. Recall that in cylindrical...
  49. S

    How to inverse surface integral of a vector field

    Assume that I know the value of \iint_{S} \overrightarrow{F} \cdot \hat{n} dS over any surface in \mathbb{R}^3, where \overrightarrow{F}(x,y,z) is a vector field in \mathbb{R}^3 and \hat{n} is the normal to the surface at any point considered. Using that I would like to compute...
  50. D

    Dis-ambiguate: derivative of a vector field Y on a curve is the covariant of Y

    Homework Statement This two-part problem is from O'Neill's Elementary Differential Geometry, section 2.5. Let W be a vector field defined on a region containing a regular curve a(t). Then W(a(t)) is a vector field on a(t) called the restriction of W to a(t). 1. Prove that Cov W w.r.t...
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