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. I

    Why is spin-1 field described by a vector field?

    It's a famous claim that spin-0, spin-1 and spin-2 fields are described by scalar, vector and second-rank tensor, respectively. My question is: why not other objects? For example, consider spin-1 field, we can use a field that carries two left spinor indexes. From the group-theoretic relation we...
  2. J

    Massive vector field from CG coefficients

    I'm currently reading Weinberg, see http://books.google.com/books?id=3ws6RJzqisQC&lpg=PA207&ots=Cu9twmTMTE&pg=PA207#v=onepage&q&f=false" for the relevant section. In §5.3, the coefficient functions at zero momentum (or polarisation vectors) are unique up to a Lorentz transformation and scale...
  3. 0

    Lie derivative and vector field notion.

    Here is an approach for lie derivative. And i would like to know how wrong is it. Assuming lie derivative of a vector field measures change of a vector field along a vector field, take a coordinate system, xi , and the vector field fi along which Ti is being changed. I go this way, i take the...
  4. P

    Flux of a vector field over an elliptical region

    Homework Statement Find the Flux of the Vector Field <-1, -1, -y> where the surface is the part of the plane region z + x = 1 that is on the ellipsoid {x}^{2}+2\,{y}^{2}+{z}^{2}=1 (oriented in the +ve z direction)Homework Equations Surface Integral The Attempt at a Solution Parametrize the...
  5. G

    How does a one form relate to a vector field under a change of coordinates?

    One form <-----> vector field How exactly does having a one form yield a vector field in a smooth way? I understand it's a duality relationship, but can anyone give me some more insight into this?
  6. P

    Vector Field Problem Answer: Thanks for Help

    I got the answer! thanks for the help
  7. quasar987

    Question about smoothness of a vector field (Reeb).

    Hello, I am simply looking for an argument proving the smoothness of the Reeb vector field of a given contact form. If you don't know the relevant definitions, the problem is simply this: Let M be a manifold of odd dimension 2n+1 and let \alpha be a 1-form on M such that 1) \alpha is...
  8. L

    Another proof that a vector field on the sphere must have a zero?

    The proofs I have seen that a vector field on the 2-sphere must have a zero rely on the general theorem that the index of any vector field on a manifold equals the manifold's Euler characteristic. How about this for a proof that does not appeal to this general theorem? The tangent circle...
  9. M

    Line Integral of Vector Field: Is 0 a Meaningful Value?

    Can line integral of a vector field ever be zero? If can, what is the interpretation of this value (0) ? Thanks.
  10. R

    Flux integral on a radial vector field

    Homework Statement (The S1 after the double integral is supposed to be underneath them btw, I just can't seem to do it right using LaTeX right now so bear with me please.) Suppose F is a radial force field, S1 is a sphere of radius 9 centered at the origin, and the flux integral...
  11. S

    Flux of the vector field F = (1,1,1)

    I'm studying for a test. How do I find the flux of the vector field F = (1,1,1) down through the surface \sigma, given by z = \sqrt{x^2+y^2} and 1 < z < 2. The answer is 3pi but have no idea how to get it. I got it down to\int\int_R x+y/\sqrt{x^2+y^2} +1 dA. Now what?
  12. fluidistic

    Vector field, cylindrical coordinates

    Homework Statement Describe the following vector field: \bold v (\bold x)=\frac{\bold a \times \bold x}{(\bold a \times \bold x)(\bold a \times \bold x)} with \bold a = \text{constant}. Calculate its divergence and curl. In what region is there a potential for \bold v? Calculate it. Hint...
  13. M

    Divergence of a vector field is a scalar field?

    Hello. How can I show the Divergence of a vector field is a scalar field(in E^{3}) ? Should I show that Div is invariant under rotation? x^{i'}=a^{ij}x^{j},V^{'}_{i}(\stackrel{\rightarrow}{x})=a_{ij}v_{j}(\stackrel{\rightarrow}{x}) then \frac{\partial...
  14. A

    Scalar potential and line integral of a vector field

    Homework Statement Homework Equations Given above. The Attempt at a Solution I attempted this problem first without looking at the hint. I've defined F(r) as (B+A)/2 + t(B-A)/2, with dr as (B-A)/2 dt . Thus F(r)dr = ((B+A)/2)*((B-A)/2)+((B-A)/2)^2 dt When I integrate this from -1 to 1 I...
  15. P

    Continuum Mechanics Homework - Vector Field in Polar Coordinates

    Hi, so I scanned an image of the problem statement and my attempt at the solution. I don't know if I am headed in the right direction and need some guidance. This is my first post ever and I hope I am doing this properly. Thank you for any help you guys can provide.
  16. T

    Vector Field everywhere parallel to

    Homework Statement The Vector field B(x) is everywhere parallel to the normals to a family of surfaces f(x) = constant. Show that B \bullet ( \nabla \times B ) = 0 Homework Equations The Attempt at a Solution Clearly B(x) is orthogonal to the tangent planes at each points...
  17. L

    Vector Laplacian: Exploring Vector Fields

    Can all vector fields be described as the vector Laplacian of another vector field?
  18. P

    Flux of a Vector Field on a Sphere

    Homework Statement Consider the vector field: F = r/r3 where r = xi + yj + zk Compute the flux of F out of a sphere of radius a centred at the origin. Homework Equations The Attempt at a Solution Hi everyone, I have: flux = \intF.dA I can't use Gauss' Law, because the...
  19. J

    Flux of vector field through box

    Homework Statement Consider the vector field \vec F=\frac{\vec r}{r^3} with \vec r =x\hat{i} +y\hat{j} +z\hat{k} Compute the flux of F out of the box 1\leq x \leq 2, 0\leq y \leq 1, 0\leq z \leq 1 Homework Equations I can't use the Gauss divergence theorem since the divergence of...
  20. J

    Flux of vector field proportional to 1/r^3 through sphere

    Homework Statement Consider the vector field \vec F=\frac{\vec r}{r^3} with \vec r=x\hat{i}+y\hat{j}+z\hat{k} . Compute the flux of \vec F out of a sphere of radius "a" centred at the origin. Homework Equations The Gauss Divergence Theorem \int_D dV \nabla \bullet F=\int_S F\bullet dA...
  21. R

    Why Do Velocity Components Depend on All Spatial Variables in Fluid Mechanics?

    HI I was a under a little confusion about vector field. Consider velocity field of fluid flow: V = u i + v j + w k here V is vector and consider a cap over i, j, k (since they represent x,y,z directions) now we know that u,v,w are functions of x,y,z,t. This is where i am confused...
  22. S

    Gradient Tensor of a vector field

    Hi, I'm trying to compute the gradient tensor of a vector field and I must say I'm quite confused. In other words I have a vector field which is given in spherical coordinates as: \vec{F}=\begin{bmatrix} 0 \\ \frac{1}{\sin\theta}A \\ -B \end{bmatrix} , with A,B some scalar potentials and I...
  23. J

    Converting a Vector Field from Cartesian to Cylindrical Coordinates

    Homework Statement I have a rather complicated vector field given in cartesian coordinates that I need to evaluate the line integral of over a unit square. I know to use Stoke's Theorem to do this, and I suspect that the integral would be greatly simplified if it were in cylindrical...
  24. U

    Conservative vector field conditions

    My calculus book states that a vector field is conservative if and only if the curl of the vector field is the zero vector. And, as far as I can tell a conservative vector field is the same as a path-independent vector field. The thing is, I came across this...
  25. W

    Flux of a Vector field of a square on a plane x+y+z=20

    I have been trying this problem for multiple hours now, and cannot figure out what I am doing wrong. --Calculate the flux of the vector field F(vector)= 5i + 8j through a square of side 2 lying in the plane x + y + z = 20 oriented away from the origin. I realize that I need the integral...
  26. S

    Line integral of a conservative vector field

    Homework Statement This is an example in my book, and this is the information in the question. Find the work done by thr force field F(x,y) = (1/2)xy[B] i + (1/4)x^2 j (with i and j vectors) on a particle that moves from (0,0) to (1,1) along each path (graph shows a x=y^2 curve from (0,0)...
  27. K

    Calculating Outward Flux of Vector Field on Ball Boundary

    Homework Statement Calculate the outward flux of the two dimensional vector field f:\Re^{2}\rightarrow\Re^{2} , f(x,y)=(x/2 + y\sqrt{x^{2}+y^{2}},y/2 + x\sqrt{x^{2}+y^{2}}) through the boundary of the ball \Omega = {(x,y)\in\Re^{2} \left| x^{2}+y^{2} \leq R^{2}} \subset\Re^{2}, R>0...
  28. P

    What is the approach to calculating line integrals in a vector field?

    First I want to greet everyone because I am new here. I have attended to applied electromagnetic course which seems to be pretty hard to understand and issues came up at very first time after I went at calculations. I try to explain this as good as possible. 1. Vectorfield F(x,y,z) =...
  29. J

    Verify Stokes's theorem with the given surface and vector field

    Homework Statement verify Stokes's theorem for the given surface and vector field. S is defined by x^2 + y^2 + z^2 = 4, z <= 4, oriented by downward normal; F = (2y-z, x + y^2 - z, 4y - 3x) Homework Equations double integral over S of the curl F ds = integral over S' of F ds...
  30. B

    Plotting a circular vector field

    Homework Statement I'm supposed to sketch the vector field and verify that all the vectors of the following equation have the same length.Homework Equations G(x,y) = \frac{-iy + jx}{\sqrt{x^2+y^2}}The Attempt at a Solution If I start plugging in numbers, for example the point (1,1) into...
  31. P

    Showing a vector field is conservative.

    Ok so I'm new to vector analysis, just started about a week or 2 ago. I'm using Paul C. Matthews' book, "Vector Calculus". This is an example problem from it which I have difficulty understanding because of integration with partial derivatives. The problem is solved, I just have trouble...
  32. T

    Line Integral over Vector Field?

    Not exactly a homework problem, a problem from a sample test. I'm boning up for my qualifying exam. Homework Statement Consider the vector field: F = (ax + by)i + (cx + dy)j where a, b, c, d are constants. Let C be the circle of radius r centered at the origin and going around...
  33. A

    Need help with 3d plotting Vector Field.

    I am in real need for a graphical application with 3d plotting capabilities. I need to plot some particles given their space coordinate. This has been well managed using VMD. But i am clueless how to plot associated velocity vector with particles. So basicall i am looking for to plot velocity...
  34. H

    Gauss' Theorum and curl of a vector field

    Two problems one that I have some idea about solving, the other I have no idea at all about where to start. 1. Find the surface integral of E . dS where E is a vector field given; E = yi - xj + 1/3 z3 and S is the surface x2 + z2 < r2 and 0 < y < b Well Gauss' theorum would be the place...
  35. E

    Unit vector field and gradient

    Homework Statement http://img245.imageshack.us/img245/2353/87006064.th.jpg I need to find the unit vector in the direction of \vec{F} at the point (1, 2, -2). Homework Equations The Attempt at a Solution well first of all I need to find what F is right, which is gradf.. how can I get...
  36. E

    Calculating Flux Through Cylinder w/ Vector Field

    Homework Statement Find the flux of the vector field through the surface of the closed cylinder of radius c and height c, centered on the z-axis with base on the xy-plane. Homework Equations The Attempt at a Solution Can I just use the divergence theorem here? Find the...
  37. E

    Flux of Vector Field \vec{F} through Surface S

    Homework Statement Question is: Compute the flux of the vector field, \vec{F} , through the surface, S. \vec{F} = 7\vec{r} and S is the part of the surface z = x^2 + y^2 above the disk x^2 + y^2 \leq 4 oriented downward. Homework Equations The Attempt at a Solution...
  38. E

    Calculating Flux of a Vector Field

    Homework Statement http://img16.imageshack.us/img16/9926/fluxs.th.jpg Homework Equations The Attempt at a Solution I really don't know how to solve this, can anyone help me please?
  39. E

    Finding the Curl of a Vector Field

    Homework Statement http://img5.imageshack.us/img5/8295/capturewmw.th.jpg Homework Equations The Attempt at a Solution I tried to find the curl first and what i got is y - 3 and then I multiply that by the area of the circle which is 4pi.. am I doing something wrong?
  40. B

    Vector field change of variables

    Homework Statement I just need to be able to change a vector field from spherical to cartesian The question is about verifying stokes theorem (curl theorem) for a given vector field within and on a given path. It says not to use spherical coordinates, but the vector field is given in...
  41. P

    Integration of a solenoidal vector field over a volume

    Homework Statement div(J)=0 in volume V, and J.n=0 on surface S enclosing V, where n is the normal vector to the surface. Show that the integral over V of J dV is zero. Homework Equations The Attempt at a Solution I can't get anywhere with it! The divergence theorem doesn't...
  42. I

    Non-vanishing vector field on S1xS2

    Homework Statement Hi everyone, This is a question on my tensor analysis/differential geometry homework due tomorrow, and I'm just not sure of the answer. The problem is to define a non-vanishing vector field V on S1 x S2. Part b of the question is to "sketch a nonvanishing vector field on...
  43. M

    Given a vector field, show flux across all paths is the same.

    Homework Statement Given the vector field F=3x^2i-y^3j, show that the flux over any two curves C1 and C2 going from the x to the y axes are the same. Homework Equations Flux = int(F dot n ds) = int(Mdy - Ndx) divF = Ny + Mx The Attempt at a Solution We can show the divergence of...
  44. G

    What are the operations on vector field A and how do I simplify the results?

    Hello. I am stuck trying to find an understandable answer to this online: Carry out the following operations on the vector field A reducing the results to their simplest forms: a. (d/dx i + d/dy j + d/dz k) . (Ax i + Ay j + Ax k) b. (d/dx i + d/dy j + d/dz k) x (Ax i + Ay j + Ax k) I...
  45. P

    Integrating a Vector Field Over a Circular Disk

    Hi, How do integrate this? I wish to see it step by step and I'm glad for any help i can get. \int_{ \vec{r}\in{A}} \frac{ \vec{v}+ \vec{\omega}\times\vec{r}}{| \vec{v}+ \vec{\omega}\times\vec{r}|}d^{2}r where A is area of disk with radius R.
  46. H

    Conservative Vector Field: Finding the Value of 'a

    Homework Statement For what value(s) of the scalar 'a' is the vector field F(x,y,z)= 2xz i + ay^3 j + (x^2 + y^4) k conservative The Attempt at a Solution F1=2xz F2=ay^3z F3=(x^2 + y^4) I used 3D curl test?? 1)(partial F2)/(partial dx) - (partial F1)/ (partial dy)= 0-0 =...
  47. J

    Prove if a vector field is conservative or not

    How do you prove if a vector field is conservative or if it isn't conservative? For example, if we have the vector field F(x, y, z) = x^2yz ı + y  + x^2 k, how do we find out if it is conservative or not conservative?
  48. M

    Surface Integral of Vector Field

    Homework Statement Find \int\int_{S} F dS where S is determined by z=0, 0\leqx\leq1, 0\leqy\leq1 and F (x,y,z) = xi+x2j-yzk. Homework Equations Tu=\frac{\partial(x)}{\partial(u)}(u,v)i+\frac{\partial(y)}{\partial(u)}(u,v)j+\frac{\partial(z)}{\partial(u)}(u,v)k...
  49. Fredrik

    Lie derivative of vector field = commutator

    Can somone remind me how to see that the Lie derivative of a vector field, defined as (L_XY)_p=\lim_{t\rightarrow 0}\frac{\phi_{-t}_*Y_{\phi_t(p)}-Y_p}{t} is actually equal to [X,Y]_p?
  50. J

    Line integral of a vector field.

    Hi all, I'm new to the forums so if i do something stupid don't hesitate to tell me. Anyway I'm struggling with this problem: I could do part a ok, but part b has me stumped, I am in the second year of a physics degree and this is a from a maths problem sheet, i haven't done line...
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