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. S

    Mathematica: Illustrate a 3D vector field

    Homework Statement Hi, Firstly, let me apologize if this is not exactly the right topic to ask this questions. But since it is homework and since most of the Physicists for sure illustrate a 3D vector field, I decided to post it here. Attached I have two files with 3D vectors. How does...
  2. S

    Surface integral of vector field

    Homework Statement find ∫E.dS, where E = (Ar^2, Br (sinθ),C cosρ), over the outside conical surface S, given by 1≤r≤2, θ=\pi/3 (this is an open surface, excluding the end faces).Homework Equations The Attempt at a Solution from the context I believe ρ is the plane polar angle on the x-y...
  3. bibo_dvd

    What Does a Unit Vector Field Look Like in Cartesian Coordinates?

    Hello guys ! i have this problems while solving problems on vector calculus .. i solved (a) , (b) as i put P(2,4,3) in the formula of S and i solved it and i solved (b) as a(S)= S/lSl but in (c) i don't how what should i do to solve it ..please help me guys ..Thank you
  4. S

    Vector field (rotors and nabla operators)

    Homework Statement Find ##\alpha ## and ##p## so that ##\nabla \times \vec{A}=0## and ##\nabla \cdot \vec{A}=0##, where in ##\vec{A}=r^{-p}[\vec{n}(\vec{n}\vec{r})-\alpha n^2\vec{r}]## vector ##\vec{n}## is constant. Homework Equations The Attempt at a Solution ##\nabla \times...
  5. J

    Area and volume integral of vector field

    In 2 dimensions given a scalar field f(x,y) is possible to compute the line integral ##\int f ds## and area integral ##\iint f d^2A##. In 3D, given a scalar field f(x,y,z) is possible to compute the surface integral ##\iint f d^2S## and the volume integral too ##\iiint f d^3V##...
  6. J

    'Constant' vector field is equivalent to some scalar field

    To every scalar field s(x,y) there corresponds a 'constant' vector field x = A s(x,y) and y = B s(x,y), where A,B are direction cosines. The vector field is only partially constant since only the directions, and not the magnitudes, which are equal to |f(x,y)|, of the field vectors are constant...
  7. sa1988

    Work done by vector field on straight path

    Homework Statement Homework Equations W= ∫F.dr The Attempt at a Solution I'm fairly sure I've done the right thing, however my lecturer hasn't uploaded any solutions to any of these problems (which is ridiculous - how am I supposed to learn if I don't know when I'm right or...
  8. paulmdrdo1

    MHB Locating Points for Vector Field $F$: $F_x=0$, $F_y=0$, and $|F_x|=1$

    Given Vector Field: $F=2(x+y)\sin\pi za_x-(x^2+y)a_y+\left(\frac{10}{x^2+y^2}\right)a_z$ specify the locus of all points at which a.) $F_x=0$ b.) $F_y=0$ c.) $|F_x|=1$ please help me get started with this. thanks!
  9. C

    Killing vector field => global isomorphisms?

    Suppose we have a vector field ##V## defined everywhere on a manifold ##M##. Consider now point ##p \in M##. As a consequence of the existence and uniqueness theorem of differential equations. this implies that ##V## gives rise to a unique local flow $$\theta:(-\epsilon,\epsilon) \times U \to...
  10. A

    Left-invariant vector field of the additive group of real number

    Hi, I would like to understand the left-invariant vector field of the additive group of real number. The left translation are defined by \begin{equation} L_a : x \mapsto x + a \; , \;\;\; x,a \in G \subseteq \mathbb{R}. \end{equation} The differential map is \begin{equation} L_{a*} =...
  11. C

    Complete vector field X => X defined on the whole manifold?

    According to Isham (Differential Geometry for Physics) at page 115 he claims: "If X is a complete vector field then V can always be chosen to be the entire manifold M" where V is an open subset of a manifold M. He leaves this claim unproved. A complete vector field is a vector field which...
  12. J

    Condition for a vector field be non-linear

    If a vector field ##\vec{v}## is non-divergent, so the identity is satisfied: ##\vec{\nabla}\cdot\vec{v}=0##; if is non-rotational: ##\vec{\nabla}\times\vec{v}=\vec{0}##; but if is "non-linear" Which differential equation the vector ##\vec{v}## satisfies? EDIT: this isn't an arbritrary...
  13. Spinnor

    CMB polarization data plots, almost a vector field, then what?

    The data points of the polarization of the CMB are a magnitude and an orientation that varies between 0° and 180°. What kind of mathematical field is that, not quite a vector field? Thanks for any help!
  14. M

    MHB Determine if the vector field is conservative or not

    Hey! :o Determine if the vector field $\overrightarrow{F}=y\hat{i}+(x+z)\hat{j}-y\hat{k}$ is conservative or not.The vector field $\overrightarrow{F}=M\hat{i}+N\hat{j}+P\hat{k}$ is conservative if $$\frac{\partial{M}}{\partial{y}}=\frac{\partial{N}}{\partial{x}}...
  15. M

    MHB Apply the divergence theorem to calculate the flux of the vector field

    Hey! :o I have the following exercise: Apply the divergence theorem to calculate the flux of the vector field $\overrightarrow{F}=(yx-x)\hat{i}+2xyz\hat{j}+y\hat{k}$ at the cube that is bounded by the planes $x= \pm 1, y= \pm 1, z= \pm 1$. I have done the following...Could you tell me if this...
  16. evinda

    MHB Calculating Flux of Vector Field on a Spherical Surface

    Hello again! :) I am given the following exercise: Find the flux of the vector field $\overrightarrow{F}=zx \hat{i}+ zy \hat{j}+z^2 \hat{k}$ of the surface that consists of the first octant of the sphere $x^2+y^2+z^2=a^2(x,y,z \geq 0).$ That's what I did so far: $\hat{n}=\frac{\nabla{G}}{|...
  17. M

    MHB Apply the divergence theorem for the vector field F

    Hey! :o Apply the divergence theorem over the region $1 \leq x^2+y^2+z^2 \leq 4$ for the vector field $\overrightarrow{F}=-\frac{\hat{i}x+\hat{j}y+\hat{k}z}{p^3}$, where $p=(x^2+y^2+z^2)^\frac{1}{2}$. $\bigtriangledown...
  18. Matt atkinson

    Solving Vector Field for Independence of z

    Homework Statement A vector field $$ \vec{u}=(u_1,u_2,u_3) $$ satisfies the equations; $$ \Omega\hat{z} \times \vec{u}=-\nabla p , \nabla \bullet \vec{u}=0$$ where p is a scalar variable, \Omega is a scalar constant. Show that \vec{u} is independant of z. Hint ; how can we remove p from...
  19. P

    The electric displacement vector field and Gauss' law?

    Hi, I know that for the electric displacement vector field \oint D.dS=\sum Q_{c} does this mean that I can just use a Gaussian surface to explain why the displacement vector field for a sphere is radial or not without having to talk about the electric field. If not what is the reasoning to...
  20. P

    Finding the circulation of a vector field

    Homework Statement Can someone guide me through solving a problem involving the circulation of a vector field? The question is as stated for the vector field E = (xy)X^ - (x^2 + 2y^2)Y^ , where the letters next to the parenthesis with the hat mean they x y vector component. I need to find...
  21. P

    How Can MATLAB Simulate Particle Motion in a 3D Vector Field?

    So, this is going to be pretty hard for me to explain, or try to detail out since I only think I know what I'm asking, but I could be asking it with bad wording, so please bear with me and ask questions if need-be. Currently I have a 3D vector field that's being plotted which corresponds to...
  22. D

    Outward Flux of Vector Field F across Surface S

    Question: Find the outward flux of the vector field F = i-2j-2k across the surface S defined by z = 4-x2-y2 0≤z≤4 At first, I used the Divergence Theorem to solve this problem. I took the divF and got the answer of 0. By definition, integrating 0 three times will still equal 0. Thus, the...
  23. S

    Vector field flow over upper surface of sphere

    Homework Statement Calculate the flow over the upper surface of sphere ##x^2+y^2+z^2=1## with normal vector pointed away from origin. Vector field is given as ##\vec{F}=(z^2x,\frac{1}{3}y^3+tan(z),x^2z+y^2)##Homework Equations Gaussian law: ##\int \int _{\partial \Sigma }\vec{F}d\vec{S}=\int...
  24. S

    Vector field flow over surface in 3D

    Homework Statement Calculate the flow of ##\vec{F}=(y^2,x^2,x^2y^2)## over surface ##S## defined as ##x^2+y^2+z^2=R^2## for ##z \geq 0## with normal pointed away from the origin.Homework Equations The Attempt at a Solution The easiest was is probably with Gaussian law. I would be really happy...
  25. S

    Calculate the vector field over surface S using Gassian Law

    Homework Statement Integrate vector field ##\vec{F}=(x+y,y+z,z+x)## over surface ##S##, where ##S## is defined as cylinder ##x^2+y^2=1## (without the bottom or top) for ##z\in \left [ 0,h \right ]## where ##h>0##Homework Equations The Attempt at a Solution Since cylinder is not closed (bottom...
  26. W

    Transversality of a Vector Field in terms of Forms (Open Books)

    Hi, All: Sorry for the length of the post, but I think it is necessary to set things up so that the post is understandable: I'm going through an argument in which we intend to show that a given vector field [ itex]R_ω [/ itex] (actually a Reeb field associated with a contact form ω) is...
  27. mesa

    What is a conservative vector field?

    I see how our line integral is a method for calculating work along a path by taking infinitesimally small 'slices' of our dot product of Force over our curve (distance). No problem here. Next we look to see if our field is conservative and if so then we know that regardless of the path the...
  28. C

    A challenging vector field path integral

    Homework Statement Evaluate ∫F dot ds Homework Equations F = < 1 - y/ (x^2 + y^2) , 1 + x/(x^2 + y^2) , e^z > C is the curve z = x^2 + y^2 -4 and x + y + z = 100 The Attempt at a Solution I don't think Stokes theorem applies since the vector field is undefined at the origin, so I'm...
  29. M

    Testing to see if the vector field could be a magnetic field.

    Homework Statement By considering its divergence, test whether the following vector field could be a magnetic field: F=(a/r) cos∅ r Where a is constant. NOTE( the 'r' has the hat symbol ontop if it, unit vector i think) Homework Equations You may use that is cylindrical...
  30. P

    Curl & Line Integral of Vector Field: Calculations & Results

    Homework Statement Given a vector field F=-y/(x^2+y^2) i +x/(x^2 +y^2) Calculate the curl of it the line integral of it in a unit circle centered at O Homework Equations The Attempt at a Solution I calculated that the curl is 0 but the line integral is 2π. I don't think this...
  31. jssamp

    What is the significance of curl of of a vector field.

    I need help understanding the significance of curl and divergence. I am nearly at the point where I know how to use Greene's, Stokes and the divergence theorems to convert line, surface, and iterated double and triple integrals. I know how the use the curl and div operators and about...
  32. C

    Relation between parameters of a vector field and it's projection

    Say we have two vector fields X and Y and we form the projection of Y, Y' orthogonal to X. Since every vector field is associated with a curve with a corresponding parameter, is there a relation between the parameters of Y and Y'?
  33. Superposed_Cat

    Generating square vector field

    Hi all, my friend is writing a sci-fi/fantasy book and for it he asked me for a function that generates a vector field like picture A. So far the closest I've got is i*-(1/x)+1/-yj, which generates B. How would I generate B without using conditions? any help appreciated, thanks.
  34. X

    Can F be expressed as the gradient of a scalar?

    Homework Statement Assume a vector field:\textbf{F} = \widehat{r} 2r sin\phi + \widehat{\phi} r^2 cos\phi a) Verify the Stokes's theorem over the ABCD contour shown in Fig. 1 . b) Can F be expressed as the gradient of a scalar? Explain My problems results in not being able to verify...
  35. PeteyCoco

    Line integral of a spherical vector field over cartesian path

    Homework Statement Compute the line integral of \vec{v} = (rcos^{2}\theta)\widehat{r} - (rcos\theta sin\theta)\widehat{\theta} + 3r\widehat{\phi} over the line from (0,1,0) to (0,1,2) (in Cartesian coordinates) The Attempt at a Solution Well, I expressed the path as a...
  36. M

    'Eyeballing' non-zero divergence and curl from vector field diagrams

    Homework Statement Explain whether the divergence and curl of each of the vector fields shown below are zero throught the entire region shown. Justify your answer.https://sphotos-a-ord.xx.fbcdn.net/hphotos-prn2/1185774_4956047513788_517908639_n.jpg Homework Equations N/AThe Attempt at a...
  37. D

    Evaluate the divergence of the vector field

    Homework Statement Evaluate the divergence of the following vector fields (a) A= XYUx+Y^2Uy-XZUz (b) B= ρZ^2Up+ρsin^2(phi)Uphi+2ρZsin^2(phi)Uz (c) C= rUr+rcos^2(theta)Uphi Homework Equations The Attempt at a Solution Uploaded
  38. D

    What Happens to Divergence When Field Lines Change Length?

    Hey guys! So I've been trying to get my head around Divergence of a vector field. I do get the general idea, however I thought of a hypothetical situation I can't get my head around. Look at the second vector field on this page, http://mathinsight.org/divergence_idea it has a negative...
  39. W

    Is the curl of a div. free vector field perpendicular to the field?

    Hi PF-members. My intuition tells me that: Given a divergence free vector field \mathbf{F} , then the curl of the field will be perpendicular to field. But I'm having a hard time proving this to my self. I'know that : \nabla\cdot\mathbf{F} = 0 \hspace{3mm} \Rightarrow \hspace{3mm}...
  40. S

    Line integral over a Vector Field

    Homework Statement Given a vector field F(x,y,z) = (yz + 3x^{2})\hat{i} + xz\hat{j} + xy\hat{k} Calculate the line integral ∫_{A}^{B}F\bullet dl where A = (0,1,3) and B = (1,2,2) Homework Equations Right, first of all, what is dl ? I've gone over all my course notes and...
  41. D

    Gradient theorem for time-dependent vector field

    Let's say we have some time-independent scalar field \phi. Obviously \phi\left(\mathbf{q}\right)-\phi\left(\mathbf{p}\right) = \int_{\gamma[\mathbf{p},\,\mathbf{q}]} \nabla\phi(\mathbf{x})\cdot d\mathbf{x}. This is of course still true if the path \gamma is the trajectory of a particle moving...
  42. M

    Timelike Killing vector field and stationary spacetime

    I am trying to understand why in the definition of a stationary spacetime the Killing vector field has to be timelike. It is required that the metric is time independent, i.e. the time translations x^0 \to x^0 + \epsilon leave the metric unchanged. So the Killing vector is...
  43. N

    Finding a Potential Function for a Vector Field

    Hi everyone! I've been having a hard time figuring this one out for a while, so any help will be appreciated! Homework Statement \textbf{F}= <(2zx)/(x^2+e^z*y^2), (2ze^z*y)/(x^2+e^z*y^2), log(x^2+e^z*y^2) + (ze^z*y^2)/(x^2+e^z*y^2)> (a) Where is the following vector field defined? (b) Is this...
  44. S

    (Line integral) Compute work through vector field

    Homework Statement "Consider the Vector field F(x,y)=<cos(sin(x)+y)cos(x)+e^x, cos(sin(x)+y)+y>. Compute the work done as you traverse the Archimedes spiral (r=θ) from (x,y)=(0,0) to (x,y)=(2∏,0). (Hint: check to see if the vector field is conservative) Homework Equations 1) F(x,y)=<P,Q>...
  45. B

    Finding The Divergence Of A Vector Field

    Homework Statement Find The Divergence Of The Vector Field: < ex2 -2xy, sin(y^2), 3yz-2x> Homework Equations I know that divergence is ∇ dot F. The Attempt at a Solution When I did it by hand I got 2xex2 + 2ycos(y2) + 3y However wolfram alpha says it should be 2xex2 +...
  46. Roodles01

    Determine whether vector field is magnetic or electrostatic

    Homework Statement Three vector fields are listed below. Determine whether each of them is electrostatic field or magnetic field.Homework Equations F1(x, y, z) = A (9yz ex + xz ey + xy ez) F2(r,∅,z) = A [(cosx/r)er + (sinx/r) e∅] F3(r,θ,∅) = Ar2 e(-r/a) erThe Attempt at a Solution Used matrix...
  47. B

    Vector Analysis - Determining whether a vector field is conservative

    Homework Statement n/a Homework Equations ∇ x F = 0 ∂Q/∂x = ∂P/∂y The Attempt at a Solution n/a Given that no sketch of the vector field is given; Is determining the curl of a vector field the most fail proof of determining whether it is conservative? I'm just...
  48. D

    Deriving vector field line equations from sketches?

    Homework Statement Is it possible to find the vector field line expression without the use of differential equations? Say I've sketched the field and found the shape to be parabolas, how would I find the general expression by just using the points I've been given?Homework Equations The Attempt...
  49. M

    Calculate Vector Field Flux Through Sphere S of Radius 1

    Given is vector field \overrightarrow{C}(\overrightarrow{r})=r calculate flux \Phi =\int_{S} \overrightarrow{C}\cdot d\overrightarrow{A} through sphere S with beginning in [0,0,0] and r=1
  50. M

    Calculute the flux resulting from a certain vector field in a cube.

    Let's say there is a cube sitting in the first octant. Our F(x,y,z): <ax , by, cz> and Each face of the cube is oriented to outward pointing normal. Can I just calculate the the flux of one face and then multiply this by the number of faces to get the total flux? Will flux in a cube always be...
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