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. J

    Calculate the circulation of vector field

    Hello there, I've got a vector field which you can see here: Sketch of the vector field . It is: \vec{v} = \cos(x)\,\sin(y)\vec{i}-\sin(x)\,\cos(y)\vec{j} Say I want to find the circulation around the square formed by -\frac{\pi}{2} \, \leq x \leq \, \frac{\pi}{2} and -\frac{\pi}{2} \...
  2. stripes

    Flow Lines of Vector Field F = sec(x) i + k

    Homework Statement Define the vector field F = sec(x) i + k (a) Express the flow lines of F in equations form. (b) Express in equations form the particular flow line through the point (0, 3, 2). My next question is a bonus question. I'm just reading up on this now but if someone could...
  3. J

    What is the correct way to calculate streamlines of a vector field

    Hello there, What is wrong with my way of finding stream lines of a vector field? Say I have this vector field: \vec{v} = x\,y\,\vec{i} + y\,\vec{j} You can see a plot here: http://kevinmehall.net/p/equationexplorer/vectorfield.html#xyi+yj%7C%5B-10,10,-10,10%5D It appears as if the stream...
  4. H

    When the Curl of a Vector Field is Orthogonal

    Simple question. It came out of lecture, so it's not homework or anything. My professor said that the curl of a vector field is always perpendicular to itself. The example he gave is that the magnetic vector potential A is always perpendicular to the direction of the magnetic field B. (I haven't...
  5. schmiggy

    Flux of vector field F = xi + yj + zk across S

    Homework Statement I've attached an image with the entire question. Homework Equations Attached an image with relevant equations. Can't use Gauss' Divergence The Attempt at a Solution In the attached image I've also included the start of my calculations, I just need to see if my...
  6. H

    What is the difference between a vector field and vector space?

    I'm unable to understand this generalization of vectors from a quality having a magnitude and direction, to the more mathematical approach. what is the difference between vector space and vector field? more of an intuitive example?
  7. S

    If F(x,y)=<M(x,y),N(x,y)> is a vector field on the plane?

    its components M(x,y) and N(x,y) are differentiable functions that satisfy (∂N(x,y)/∂x) – (∂M(x,y)/∂y) = 1-x. a. is it possible for the vector field to be conservative? Explain. b. Let C be x^2+y^2=1 centered at the origin traced counter clockwise. compute the integral ∫F.dr...
  8. D

    MHB Vector Field Curves - How to Add 4 Curves Between Manifolds

    \begin{tikzpicture}[scale = 1.5] \draw (0,0) circle (1cm); \draw[-] (1,0) -- (-1,0); \draw[->] (1.2,0) -- (.5,0); \draw[->] (-1.2,0) -- (-.5,0); \draw[-] (.907107,.907107) -- (-.907107,-.907107); \draw[<-] (.907107,.907107) -- (.307107,.307107); \draw[<-] (-.907107,-.907107) --...
  9. D

    Exploring Vector Field Line Integrals: A Sample Final Exam Problem

    This problem is about Line integral of Vector Field. I believe the equation i need to use is: \intF.dr = \intF.r'dt, with r = r(t) I try to solve it like this: C1: r1= < 1 - t , 3t , 0 > C2: r2= < 0 , 3 - 3t , t > C3: r3= < t , 0 , 1 - t > After some computation, I got stuck at the...
  10. S

    Verify that the Stokes' theorem is true for the given vector field

    This is a problem from an old final exam in my Calc 3 class. My book is very bad at having examples for these types of problems, and my instructor only went over one or two. Help would be much appreciated. Homework Statement Verify that the Stokes' theorem is true for the vector field...
  11. M

    Line integral of a vector field over a square curve

    Homework Statement Please evaluate the line integral \oint dr\cdot\vec{v}, where \vec{v} = (y, 0, 0) along the curve C that is a square in the xy-plane of side length a center at \vec{r} = 0 a) by direct integration b) by Stokes' theoremHomework Equations Stokes' theorem: \oint V \cdot dr =...
  12. S

    Finding Curl from a vector field picture

    Homework Statement I need to analyze these pictures for my homework and find out the curl of the vector field at the point (red) on the picture. Homework Equations http://i1242.photobucket.com/albums/gg525/sjrrkb/ScreenShot2012-11-26at61615PM.png The Attempt at a Solution basically...
  13. D

    If X is a left invariant vector field, then L_x o x_t = x_t o L_x

    If X is a left invariant vector field, then L_x \circ x_t = x_t \circ L_x , where xt is the flow of X and Lx is the left translation map of the lie group G. In order to show this, I am trying to show that x_t = L_x \circ x_t \circ L_x^{-1} by showing that L_x \circ x_t \circ L_x^{-1}...
  14. 3

    Recovering a vector field from the divergence/flux

    Homework Statement (Long and unappealing looking question - but could please use few hints with what is going wrong with my analysis :) ) In brief: the object is to work out what vector field causes a given divergence - I've managed to get the divergence & flux but recovering the field is...
  15. J

    How Do We Express Vector Fields in Fluid Mechanics?

    Hi, I am experiencing a little bit of trouble grasping the concept of vector fields in fluid mechanics: If the vector V = u i + v j + w k, where i,j,k are unit vectors in the x,y,z directions, then how can u,v,w be functions of x,y,z,t? I.e. if say v is in the y direction, then how can it...
  16. U

    Vector field dot product integration

    Homework Statement Calculate F=∇V, where V(x,y,z)= xye^z, and computer ∫F"dot"ds, where A)C is any curve from (1,1,0) to (3,e-1) B)C is a the boundary of the square 0≤x≤1, 0≤y≤1... oriented counterclockwise. Homework Equations ∫F"dot"ds= ∫F(c(t)"dot"c'(t) The Attempt at a Solution...
  17. P

    Creating a magnetic field (vector field)

    Hi all, I have a question for all of you. I've been wanting to make a 3D vector field that would represent a magnetic field (for fun) around some segment of wire with a constant current flowing through it. I'm assuming I have a parametric equation for the wire segment. The one equation that...
  18. C

    Line integral across a vector field

    I've attached the problem as a picture. Generally, this would be a simple problem if I were to apply Green's theorem. But I can't use Green because D would not be a simply connected closed region; the vector field isn't defined at the origin. Could someone please give me an idea of where to...
  19. C

    Circulation of a 3d vector field

    Homework Statement consider the vector field v(x,y,z)=(-h(z)y,h(z)x,g(z)) wherer h:R->R and g:R—>R are differentiable .Let C be a closed curve in the horizontal plane z=z0.show that the circulation of v around C depends only on the area of the reion enclosed by C in the given plane and h(Z0)...
  20. S

    Understanding converting a vector field to cartesian coords

    Homework Statement Here is the problem and solution but I am confused as to part B http://gyazo.com/e77d05fc67cb6ac266ff021ef88052dc The Attempt at a Solution I understand the first part, but I am totally lost on how they reached their cartesian answer for part B. Firstly why did they...
  21. C

    Line integral across a vector field

    Homework Statement \int_C \mathbf F\cdot d \mathbf r where \mathbf F = x^2\vec{i}+e^{\sin^4{y}}\vec{j} and C is the segment of y=x^2 from (-1,1) to (1,1). Homework Equations \int_C \mathbf F\cdot d \mathbf r=\int_a^b \mathbf F( \mathbf r(t))\cdot r'(t) dt=\int_C Pdx+Qdy where \mathbf F =...
  22. R

    Spherical coordinates, vector field and dot product

    Homework Statement Show that the vector fields A = ar(sin2θ)/r2+2aθ(sinθ)/r2 and B = rcosθar+raθ are everywhere parallel to each other. Homework Equations \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos(0) The Attempt at a Solution So, if the dot product equals 1. They should be...
  23. R

    Stokes theorem in a cylindrical co-ordinates, vector field

    Homework Statement given a vector field v[/B=]Kθ/s θ (which is a two dimensional vector field in the direction of the angle, θ with a distance s from the origin) find the curl of the field and verify stokes theorem applies to this field, using a circle of radius R around the origin Homework...
  24. R

    Problem with vector field proof

    Suppose that F and G are vector fields and that F-G = ▽μ for some real-valued function μ(x,y). Prove that ∫ F.dx = ∫ G.dx for all piecewise smooth curves C in the xy-plane I just need some help in getting started really. Thanks
  25. T

    Taylor expansion of a vector field

    I was wondering if such an approximation is possible and plausible... The first term would have to look sth like this: \vec{f}(\vec{x_{0}}) + \textbf{J}_{\vec{f}}(\vec{x_{0}})\cdot(\vec{x}-\vec{x_{0}}) No clue about the second term though... We would have to calculate the Jacobian of the...
  26. D

    Find a Conservative Vector Field from a Non-Conservative One

    Homework Statement My problem is, I have a scalar field and I take the gradient of this field. It is known that the gradient of a scalar function is a conservative vector field; but I need to run a procedure in this field that will modify the vector field; the modified vector field could be a...
  27. O

    Question about parametization vs vector field.

    I thought i had a strong understanding of parameterizing curves and sketching vector fields. However when I was going through my practice test I came across this problem which I don't full grasp Let F^ ⃗=xi ^⃗+(x+y) j ^⃗+(x-y+z)k^⃗ . a) Find a point at which F ⃗ is parallel to the...
  28. E

    Making Flux Negative in a Constant Vector Field Help

    Homework Statement I know I posted a question yesterday also, but this homework is getting on my nerves. My prof isn't the best out there. So basically the question is as follows: http://img812.imageshack.us/img812/8130/problem6.png Homework Equations Not sure, I sort of answered the...
  29. P

    Covariant derivative of a vector field

    Homework Statement Show that \nabla_a(\sqrt{-det\;h}S^a)=\partial_a(\sqrt{-det\;h}S^a) where h is the metric and S^a a vector. Homework Equations \nabla_a V^b = \partial_a V^b+\Gamma^b_{ac}V^c \Gamma^a_{ab} = \frac{1}{2det\;h}\partial_b\sqrt{det\;h} \nabla_a\sqrt{-det\;h} (is that...
  30. R

    Finding the unit normal of a vector field

    I'm on the last chapter of a 1200 page calc book, I'm really psyched. Homework Statement The Attempt at a Solution The method I learned for finding the unit normal of a vector field, n, is take the derivative of the equation and divide that by the magnitude of the...
  31. J

    Compute the flux of vector field through a sphere

    Homework Statement Compute the flux of the vector field F(x,y,z)=(z,y,x) across the unit sphere x2+y2+z2=1 Homework Equations I believe the forumla is ∫∫D F(I(u,v))*n dudv I do not know how to do the parameterization of the sphere and then I keep getting messed up with the normal vector.*...
  32. M

    Mean of vector field using info at surface

    Hello, I have a 3d closed surface. This closed surface lies in a 3d vector field. I know the value of the vector at discrete points along the surface and the surface normal at these points. Essentially, say vector U and vector n at these points. This is the only information that I have. I...
  33. T

    5 Vector Field Proofs - apparently easy

    Homework Statement http://gyazo.com/94783c14f2d2d05e62e479ab33c73830 Homework Equations I know the dot product and cross product, but even for the first one I don't see how either helps. The Attempt at a Solution 1. the gradient of the 2 scalars multiplied together (not crossed...
  34. G

    Confirming Vector Field is Conservative: ln(x^2 + y^2)

    Homework Statement confirm that the given function is apotential for the given vector field ln(x^{2} + y^{2}) for \frac{2x}{\sqrt{x^{2}+y^{2}}} \vec{i} + \frac{2y}{\sqrt{x^{2}+y^{2}}} \vec{j} Homework Equations The Attempt at a Solution the first thing i did was let my equation...
  35. 1

    What is the divergence of vector field F(x,y,z) = (-x+y)i + (y+z)j + (-z+x)k?

    Homework Statement F(x,y,z) = (-x+y)i + (y+z)j + (-z+x)k Find divergence Homework Equations The Attempt at a Solution The gradient is -i + j + -k Dotting that with F, I get x - y + y + z + z - x = 2z My book lists the answer as -1. What the heck are they talking...
  36. M

    Integration of a vector field in polar coordinates

    Hello all, I am trying to understand how to integrate a vector field in polar coordinates. I am not looking to calculate flux here, just the sum of all vectors in a continuous region. However, there is something I am not doing properly and I am a bit lost at this point. Any help would be...
  37. M

    Vector field curvature in the complex plane

    Hey all, I have a vector field described by a complex potential function (so I have potential lines and streamlines). I am looking for a way to express its curvature at every point, but I can't find such a formula in my books. I have searched in wikipedia and I read that the way to define it...
  38. P

    Calculating the Vector Field from a curl function

    Homework Statement Consider the intersection,R, between two circles : x2+y2=2 and (x-2)2+y2=2 a) Find a 2-Dimensional vector field F=(M(x,y),N(x,y)) such that ∂N/∂x - ∂M/∂y=1 Homework Equations none. The Attempt at a Solution There are other parts to the main question but I don't think I will...
  39. 1

    Determining if a vector field is conservative

    I have a bad habit of deciding how I would solve the concepts presented to us in lecture before the instructor covers the method. I try my methods immediately and use the textbook method if I hit a road block. This leads to problems for me. Right now we are covering line integrals over...
  40. T

    Find Vector Field Given The Curl

    Homework Statement Find a vector field \vec{A}(\vec{r}) in ℝ3 such that: \vec{\nabla} \times \vec{A} = y2cos(y)e-y\hat{i} + xsin(x)e-x2\hat{j} The Attempt at a Solution I broke it down into a series of PDE's that would be the result of \vec{\nabla} \times \vec{A}: ∂A3/∂y - ∂A2/∂z...
  41. T

    Conservative Vector Field Potential

    Homework Statement Let \vec{E}(\vec{r}) = \vec{r}/r2, r = |\vec{r}|, \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} be a vector field in ℝ3. Show that \vec{E} is conservative and find its scalar potential. Homework Equations All of the above. The Attempt at a Solution \vec{\nabla}...
  42. S

    Linear Integration of a Vector Field over a Parametric Path

    Problem statement attached. The correct way to do this seems to plug in your given x, y, z into F then integrate the dot product of F and <x',y',z'> dp from 0 to 1, however, this results in way too messy of an integral. Answer is 3/e...
  43. T

    Showing a vector field is irrational on

    Homework Statement Let F = ( -y/(x2+y2) , x/(x2+y2) ) Show that this vector field is irrotational on ℝ2 - {0}, the real plane less the origin. Then calculate directly the line integral of F around a circle of radius 1.Homework Equations The Attempt at a Solution To show F is irrotational we...
  44. P

    Orbits of a Killing vector field

    I was wondering what the orbits of a Killing vector field are. Do you have any good sources or reading material for this?
  45. S

    Show a flowline of a vector field?

    Show a flowline of a vector field?? Homework Statement Consider the vector field F(x,y,z)=(8y,8x,2z). Show that r(t)=(e8t+e−8t, e8t−e−8t, e2t) is a flowline for the vector field F. r'(t)=F(r(t)) = (_,_,_) Now consider the curve r(t)=(cos(8t), sin(8t), e2t) . It is not a flowline of...
  46. Z

    Curl is a measure of the tendency of a vector field

    \nabla\timesgrad(f) is always the zero vector. Can anyone in terms of physical concepts make it intuitive for me, why that is so. I get that the curl is a measure of the tendency of a vector field to rotate or something like that, but couldn't really assemble an understanding just from that.
  47. P

    Flux of a vector field through warped sphere

    Homework Statement Consider the surface S with the graph z = 1-x^{2}-y^{2} with z≥0, and also the unit disc in the xy plane. Give this surface an outer normal. Compute: \int\int_{S}\vec{F}\bulletd\vec{S} where \vec{F}(x,y,z) = (2x,2y,z)Homework Equations \int\int_{S}\vec{F}\bulletd\vec{S} =...
  48. I

    Prove a Statement about the Line Integral of a Vector Field

    Homework Statement Given the vector field \vec{v} = (-y\hat{x} + x\hat{y})/(x^2+y^2) Show that \oint \vec{dl}\cdot\vec{v} = 2\pi\oint dl for any closed path, where dl is the line integral around the path.Homework Equations Stokes' Theorem: \oint_{\delta R} \vec{dl}\cdot\vec{v} = \int_R...
  49. G

    Solving Non-Conservative Vector Field Line Integrals

    Hi, I'm studying calculus 3 and am currently learning about conservative vector fields. ============================= Fundamental Theorem for Line Integrals ============================= Let F be a a continuous vector field on an open connected region R in ℝ^{2} (or D in ℝ^{3}). There exists...
  50. W

    Surface integral of a vector field w/o div. theorem

    Homework Statement First a thanks for the existence of this site, i find it quite useful but had no need to actually post till now. I am stuck on the following problem in "introduction to physics" We should calculate the \oint \vec{v}.d\vec{A} of a object with the following parameters...
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