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. Augbrah

    Show that killing vector field satisfies....

    I'm trying to do past exam papers in GR but there are some things I don't yet feel comfortable with, so even though I can do some parts of the question I would be very happy if you could check my solution. Thank you! 1. Homework Statement Spacetime is stationary := there exists a coord chart...
  2. S

    I Is every conservative vector field incompressible?

    So I have found that everyone conservative vector field is irrotational in a previous problem. Based on the relationship irrotational vector fields and incompressible vector fields have, div(curl*F)=0, does that also imply every conservative vector field is incompressible? Kindly, Shawn
  3. Dominika

    Solenoidal and irrotational vector field

    Homework Statement I am to prove (using the equations for gradient, divergence and curl in spherical polar coordinates) that vector field $$\mathbf{w}=w_{\psi}(r,\theta)\hat e_{\psi}$$ is solenoidal, find $$w_{\psi}(r,\theta)$$ when it's irrotational and find a potential in this case. Homework...
  4. S

    How Do You Calculate Work Done in a Vector Field Along a Parametric Path?

    Fine the word done in moving a particle in the force field F=<2sin(x)cos(x), 0, 2z> along the path r=<t,t,t2>, 0≤t≤π To do the line integral, I need to find F(r(t)), but I don't understand how to express it. For example I looked at the online notes provided here...
  5. T

    I How do we divide one vector field by another?

    Let's say we have two vector fields, described by 6 functions: Ax, Ay, Az and Bx, By, Bz. We want to divide field A by field B. Do we take Ax/Bx , Ay/By and Az/Bz individually? But in this case we might end up with Three different scalar fields. What's the proper way to do this?
  6. S

    Equations of motion and Hamiltonian density of a massive vector field

    Homework Statement The Lagrangian density for a massive vector field ##C_{\mu}## is given by ##\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}C_{\mu}C^{\mu}## where ##F_{\mu\nu}=\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}##. Derive the equations of motion and show that when ##m...
  7. S

    Line integral over a vector field

    Homework Statement Evaluate ∫C < −y, x − 1 > dr where C is the closed piecewise continuous curve formed by the line segment joining the point A(− √ 2, √ 2) to the point B( √ 2, − √ 2) followed by the arch of the circle of radius 2, centered at the origin, from B to A. 2. The attempt at a...
  8. K

    I Function of angle between vector field and scalar function

    I was curious, if you were given a vector field F(x,y,z) = <Fx(x,y,z), Fy(x,y,z), Fz(x,y,z)>, and then some scalar function f(x,y,z), how would you define a function θ(x,y,z) of the angle θ between the scalar function and the vector field at any given point. I know how I would find this at a...
  9. G

    Conservative vector field problem

    Homework Statement Determine for which real values of the parameter ##\alpha## the vector field given by ##F(x,y) = (\frac{2xy}{y-\alpha}, 2 - \frac{4x^2}{(y-\alpha)^2})## is conservative. For those values of ##\alpha##, calculate the work done along the curve of polar equation: ##\rho =...
  10. L

    Vector field in cylindrical coordinates

    Homework Statement Sketch each of the following vector fields. E_5 = \hat \phi r E_6 = \hat r \sin(\phi) I wish to determine the \hat x and \hat y components for the vector fields so that I can plot them using the quiver function in MATLAB. Homework Equations A cylindrical coordinate...
  11. R

    An inconsistent conservative vector field

    < y^2, 2xy+ e^(3z), 3ye^(3z)> is the vector field. the above vector field is inside an open simply connected domain. the parametric equations all have a continuous first order derivative inside the domain. Lastly, the curl of the vector field is <0, 0, 0> Thus, the vector field is...
  12. R

    How can a volume integral yield a vector field?

    I'm using the textbook Electricity and Magnetism by Purcell. In the section about continuous charge distributions I found the following formula \mathbf{E}(x,y,z)= \frac{1}{4\pi\epsilon_0 } \int \frac{\rho(x',y',z')\boldsymbol{\hat r} dx'dy'dz'}{r^{2}} . It's stated that (x,y,z) is fixed...
  13. P

    How Do Bilinears Connect to Killing Vectors and Differential Forms?

    What does it mean that a Killing vector and a total differential of a certain theory are related to bilinears? In other words, why would bilinears (e.g. of the forms ##<\gamma_0\epsilon, \gamma_5\gamma_{\mu}\epsilon>## and ##<\gamma_0\epsilon, \gamma_{\mu}\epsilon>## tell us anything about...
  14. A

    Independent Killing vector field

    Hello everyone How is it possible that a n-dimensional spacetime admits m> n INDEPENDENT Killing vectors where m=n(n+1)/2 if the space is maximally symmetric?
  15. C

    Why a vector field can not be used to describe gravity?

    In many books of general relativity I have found the following statement: a vector field produces repulsive forces between like charges so can not be used to describe gravity. But they do not show it. How can i show it or where can I find a book or a paper in which it is shown? Please, to avoid...
  16. Z

    Why is a conserved vector field a gradient of a certain func

    I know that if a vector field is conserved then there exits a function such that the gradient of this function is equal to the vector field but am just curious to know the reason of it.
  17. C

    MHB Vector Field Formula for Graph in [-2,2] x [-2,2]

    Give an example of a formula for a vector field whose graph would closely resemble the one shown. The box for this figure is [−2, 2] x [−2, 2]. Not sure where to start.
  18. Titan97

    Meaning of Curl from stokes' theorem

    Divergence can be defined as the net outward flux per unit volume and can be explained using Gauss' theorem. (I read this in Feynman lectures Vol. 2) In the next page, He derives Stokes' theorem using small squares. The left side of equation represents the total circulation of a vector...
  19. A

    Integrate a vector field in spherical coordinates

    I have the following integral: ## \oint_{S}^{ } f(\theta,\phi) \hat \phi \; ds ##Where s is a sphere of radius R.so ds = ##R^2 Sin(\theta) d\theta d\phi ## Where ds is a scalar surface element. If I was working in Cartesian Coordinates I know the unit vector can be pulled out of integral and...
  20. chi_rho

    Is Gravitational Force conserved at the origin (r=0)?

    I know gravity is a conservative force field and can be treated as such for all intents and purposes, but I was just thinking that in order to show that a vector field is conservative that vector field must be defined everywhere (gravitational force field is not defined at r=0). I was thinking...
  21. loops496

    Can a Killing Vector Field Prove v^\mu \nabla_\alpha R=0?

    Homework Statement Suppose v^\mu is a Killing Vector field, the prove that: v^\mu \nabla_\alpha R=0 Homework Equations 1) \nabla_\mu \nabla_\nu v^\beta = R{^\beta_{\mu \nu \alpha}} v^\alpha 2) The second Bianchi Identity. 3) If v^\mu is Killing the it satisfies then Killing equation, viz...
  22. Q

    Showing a vector field is imcompressible

    Homework Statement Homework EquationsThe Attempt at a Solution As you can see, the solution is shown just below the question. Essentially, I don't understand how the x, y and z component of the vector field has been separated because the numerator of the vector field's fraction is: (x^2 +...
  23. S

    Finding a field line of a vector field

    Homework Statement Find the field line of \vec{E}(\vec{r}) = \frac{m}{4 \pi r^3} (2 \cos\theta, \sin\theta, 0) through the point (a, b, c) (Spherical coordinates). m is a constantI know the answer, but I don't see what I do wrong. The Attempt at a Solution \frac{d\vec{r}}{d \tau} = C...
  24. A

    Understanding Conservative Vector Fields: Explanation and Illustration

    Hello, I'm having problems understanding what exactly is meant by a conservative vector field. So, allow me to explain how I currently understand it. Let \overrightarrow{F} be some sort of a vector field that is conservative. The stream plot of \overrightarrow{F} is shown below. From my...
  25. I

    What are some common misconceptions about fields in physics?

    Hi all! I am a 13-year-old that is very interested in physics. I am currently studying fields, and I have gotten into some advanced parts of field theory. With that, I have 3 big questions: 1. In vector and higher order tensor fields, what does it mean to "observe the field from point...
  26. S

    Does phrase «Space over vector field» make any sense?

    I met in several sources (textbooks) phrase «Space can be constructed over any field». But it is always illustrated with linear space over scalar field (or sometimes over ring). Does it make any sense to talk about spaces over vector fields? What kinds of spaces are they? What about tensor...
  27. T

    Divergence of radial unit vector field

    Sorry if this was addressed in another thread, but I couldn't find a discussion of it in a preliminary search. If it is discussed elsewhere, I'll appreciate being directed to it. Okay, well here's my question. If I take the divergence of the unit radial vector field, I get the result: \vec...
  28. Einj

    Inner product for vector field in curved background

    Hello everyone, I would like to know if anyone knows what is the inner product for vector fields ##A_\mu## in curved space-time. Is it just: $$ (A_\mu,A_\mu)=\int d^4x A_\mu A^\mu =\int d^4x g^{\mu\nu}A_\mu A_\nu $$ ? Do I need extra factors of the metric? Thanks!
  29. kostoglotov

    Insight into determinants and certain line integrals

    I just did this following exercise in my text If C is the line segment connecting the point (x_1,y_1) to (x_2,y_2), show that \int_C xdy - ydx = x_1y_2 - x_2y_1 I did, and I also noticed that if we put those points into a matrix with the first column (x_1,y_1) and the second column (x_2,y_2)...
  30. kostoglotov

    Line Integral/Ampere's Law: is my logic valid?

    This is a problem from a section on Line Integrals in my Calculus Textbook, I haven't studied any physics relating to E&M yet, and the solutions manual only gives solutions for odd numbered problems. Sorry, if I'm posting in the wrong forum, I hope I'm not. 1. Homework Statement A steady...
  31. Calpalned

    Understanding Vector Fields and Their Relationships to Circles

    Oops. I just realized that this is the physics homework forum... This is actually calculus homework... 1. Homework Statement Homework Equations n/a The Attempt at a Solution I read, analyzed and reread the text, but I am still confused. 1) How was the position vector ##<x,y>##...
  32. Calpalned

    What is a Conservative Vector Field in Relation to Conservative Forces?

    Homework Statement [/B] Secondly, is a "conservative vector field" the same thing as a "conservative force" (a force that does not depend on the path taken)? If not what is it? Homework Equations n/a The Attempt at a Solution How was ##f(x,y,z)=\frac{mMG}{\sqrt{x^2+y^2+z^2}}## derived?
  33. nmsurobert

    Line integral of a vector field

    Homework Statement Consider the vector field F(r) = Φ^ (a) Calculate ∫ F⋅dl where C is a circle of radius R (oriented counterclockwise) in the xy-plane centered on the origin. Homework Equations maybe Φ^ = -sinΦx^ + cosΦy^ The Attempt at a Solution not really a solution. i am just stuck at...
  34. U

    Exploring the Behavior of a Vector Field with Constant x Values

    vector field F = [ -x3 , x, 0] how does this behave for a constant x? Does that mean that I plot the vector field for different values of x?
  35. F

    How to prove some functions are scalar field or vector field

    Homework Statement Homework EquationsThe Attempt at a Solution I solved #2,4 but I don't understand what #1,3 need to me. I know that scalar field is a function of points associating scalar value. But how can I prove some function is scalar field or vector field?
  36. W

    Divergence of vector field: Del operator/nabla

    Homework Statement Let ν(x,y,z) = (xi + yj + zk)rk where v, i, j, k are vectors The k in rk∈ℝ and r=√(x2+y2+z2). Show that ∇.v=λrk except at r=0 and find λ in terms of k. Homework Equations As far as I understand it, ∇.v=∂/∂x i + ∂/∂y j + ∂/∂z k, but this may very well be wrong. The Attempt...
  37. stevendaryl

    Derivative of Vector Field: Connection Needed?

    Some context for my question: If you have a smooth manifold \mathcal{M} you can define tangent vectors to parametrized paths in the following way: If \mathcal{P}(s) is a parametrized path, then \frac{d}{ds} \mathcal{P}(s) = V where V is the differential operator that acts on scalar fields...
  38. F

    Finding Div and Curl of a Vector Field then evaluating a point

    Homework Statement }[/B] Find the divergence and curl of the vector field \vec{V}=x^2y \hat{i} + xy^2 \hat{j} + xyz \hat{k} then for both, evaluate them at the point \bar{r} = (1,1,1) Homework Equations div(\vec{F})= \nabla \cdot \vec{F} \\ curl(\vec{F})= \nabla \times\vec{F} The Attempt...
  39. S

    Finding Beltrami field in Cartesian coordinates

    Homework Statement Working in Cartesian coordinates (x,y,z) and given that the function g is independent of x, find the functions f and g such that: v=coszi+f(x,y,z)j+g(y,z)k is a Beltrami field. Homework Equations From wolfram alpha a Beltrami field is defined as v x (curl v)=0 The Attempt...
  40. J

    Vector field question + reasoning

    Im doing some revision of vector calculus and came across the following problem Q: Calculate the work done by the force field F = 3xyi - 2j in moving from A: (1,0,0) to D: (2,0,0) and then from D: (2,0,0) to B: (2,sqrt(3),0) I got stuck and decided to look at the answers. In the answers (part...
  41. S

    How to represent a time varying vector field

    Homework Statement I'm solving for the magnetic potential vector field given a time varying electrical field. It might seem silly, but I don't really understand how to approach the problem since the field varies on both space and time. I'm given the motion of a charged particle. Can someone...
  42. A

    Curl of a function and vector field

    Hello, I'm having some difficulty with a conceptual question on a practice test I was using to study. I have the answer but not the solution unfortunately. 1. Homework Statement "For every differentiable function f = f(x,y,z) and differentiable 3-dimensional vector field F=F(x,y,z), the...
  43. T

    Flow line in conservative vector field

    Homework Statement Recall that a flow line, c(t), of a vector field F has c'(t)=F(c(t)) at all times t. Show all work below. a.) Let c(t) be the flow line of a particle moving in a conservative force field F=-grad(f), where f:R^3->R, f(x,y,z) >=0 for all (x,y,z), represents the potential...
  44. A

    Surface integral problem: ##\iint_S (x^2+y^2)dS##

    Homework Statement ##\iint_S (x^2+y^2)dS##, ##S##is the surface with vector equation ##r(u, v)## = ##(2uv, u^2-v^2, u^2+v^2)##, ##u^2+v^2 \leq 1## Homework Equations Surface Integral. ##\iint_S f(x, y, z)dS = \iint f(r(u, v))\left | r_u \times r_v \right |dA##, The Attempt at a Solution...
  45. T

    Calculating Flux with a Constant Vector Field on a Disk of Radius 4

    Homework Statement Find the constant vector field F giving the flux of 5 trough the surface S, a disk of radius 4 perpendicular to both F and the y-axis, and oriented away from the origin. Homework EquationsThe Attempt at a Solution I have gone through several articles on the web and searched...
  46. A

    How can I explain that this vector field is not conservative?

    Homework Statement I have to explain why this vector field is not conservative. Homework Equations Maybe it is: if ##\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}## then F(x, y) = p(x, y)i + Q(x, y)j is a conservative field. I tried to figure out what P and Q is, but that...
  47. G

    How Do You Determine the Limits for Surface Integrals?

    Homework Statement Homework Equations ∫∫D F((r(u,v))⋅(ru x rv) dA The Attempt at a Solution [/B] I got stuck after finding the above, at where the double integrals are. :( May I know how do I find the limits of this? (I always have trouble finding the limits to sub into the integrals...
  48. 1

    Find the outward flux of a vector field across an ellipsoid

    Homework Statement [/B] Find the outward flux of the vector field ## \vec F = y^2e^{z^2+y^2} i + x^2 e^{z^2+x^2} j + z^2 e^{x^2+y^2} k## across that part of the ellipsoid $$ x^2 + y^2 + 4z^2 = 8$$ which lies in the region ##0 ≤ z ≤ 1## (Note: The two “horizontal discs” at the top and bottom are...
  49. AwesomeTrains

    Line Integral - Stokes theorem

    Homework Statement Hello I was given the vector field: \vec A (\vec r) =(−y(x^2+y^2),x(x^2+y^2),xyz) and had to calculate the line integral \oint \vec A \cdot d \vec r over a circle centered at the origin in the xy-plane, with radius R , by using the theorem of Stokes. Another thing, when...
  50. L

    Line integral over vector field

    Homework Statement Let F be a vector field F = [-y3, x3+e-y,0] The path in space is x2 + y2 = 25, z = 2. My parametrization is r(t) = [5cos(t),5sin(t),2] Homework Equations Line integral is the integral of F(r(t)) * r'(t) dt, where here the asterisk * is for the dot product, not normal...
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