Vector fields Definition and 161 Threads

Hi, this is a very simple question about the curl theorem. It says in my book: " If F is a vector field defined on all R3 whose component functions have continuous partial derivatives and curl F = 0 , then F is a conservative vector field" I might sound stupid, but what exactly does...

Homework Equations Hey guys I had a slight problem trying to find divergence of vector fields for the following equation: F(x,y,z)=(yzi-xzj-xyk)/(x^2 + y^2 + z^2) So I want to know if its possible of substitute (x^2 + y^2 + z^2) for 1 since that is the equation of a sphere? If not...

I just came across the term, and was hoping someone could tell me what they are and any general form of them. Thank you!

F(x,y,z)=ax P(x,y,z)+ay Q(x,y,z)+az R(x,y,z) F is vectoral field. ax , ay and az are unit vectors. P , Q ,R are scalar functions. The question is this: If F is non-conservative vectoral field ; what are the characteristics of P Q and R? thanks in advance. have a nice day

Homework Statement Let X,Y be vector fields and x(t) be a curve satisfying \dot x(t) = X(x(t)) + u(t) Y(x(t)), u(t) \in \mathbb R [/itex] and assume there exists p(t) an adjoint curve satisfying \dot p(t) = -p(t) \left( \frac{\partial X}{\partial x}(x(t)) + u(t) \frac{\partial Y}{\partial x}...

Homework Statement F=-ysin(x)i+cos(x)j Homework Equations Can the Curl test be applied to this vector field and state three facts you can deduce after applying the curl test. The Attempt at a Solution

I have done relatively few in physics courses and I need to re-learn how to do some flux integrals for constant vector fields through rectangular and circular surfaces. If anyone has any direction to some great resources, or themselves could be a great resource for help please post what are...

I was wondering, if you have a non-conservative vector field (so that the line integral of each path from point A to point B isn't the same) that represents some sort of force, then is there a method to find the path that requires the least amount of work from a designated point A to point B...

Homework Statement Given two vector fields: i) A \frac{\vec{r}}{r^{n_{1}}} ii) Ae_{z} \times \frac{\vec{r}}{r^{n_{2}}} where A is a constant and n_{1} \neq 3 and n_{2} \neq 2 find \int \vec{F} dS through surface of a sphere of radius R Homework Equations \int \vec{F} r^{2}...

I am reading the Wikipedia entry http://en.wikipedia.org/wiki/Vector_fields_in_cylindrical_and_spherical_coordinates" . There, in particular I see this: Time derivative of a vector field To find out how the vector field A changes in time we calculate the time derivatives. In cartesian...

What is the relation between the flux through a given surface by a vector field? And how does stokes theorem relate to the line integral around a surface in that field

Given the two vector fields: \vec E and \vec B Where the first is the electric vector field and the second is the magnetic vector field, we have the following identity: curl(\vec E) = -\frac{\partial \vec B } { \partial t } and further that: curl(curl(\vec E)) =...

If C is the curve given by r(t)=<1+3sin(t), 1+5sin^2(t), 1+5sin^3(t)>, 0≤t≤π/2 and F is the radial vector field F(x, y, z)=<x, y, z>, compute the work done by F on a particle moving along C. Work= int (F dot dr) If F is the potential function(?), do I integrate F with respect to each...

Online integrator: http://www.wolframalpha.com/ \int \int (x,y,z) \times (x,y,z) \times (x,y,z) \int_{L1} \int_{L2} (dl1,0,0) \times (dl2,0,0) \times (0,-1,0) What would be correct syntax to evaluate this double integral? I tried these, but they produce wrong result: try...

Hey - I'm stuck on a concept: Are ALL vector fields expressable as the product of a scalar field \varphi and a constant vector \vec{c}? i.e. Is there always a \varphi such that \vec{A} = \varphi \vec{c} ? for ANY field \vec{A}? I ask because there are some derivations from...

i need a vector field on S^2 that vanishes at 1 point. there was a thread like this here, but the answer was ((v1/1+x^2),(v2/1+y^2)) and i really don't see how this vanishes at a point although i do get it intuitively. My professor hinted that i should take a non zero vector fiels in S^2 x R^2...

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

Homework Statement \int delxA dv = -\oint Axds where A is a vector field Left hand side is integral over volume. Right hand side is integral over closed surface. Homework Equations The Attempt at a Solution Can't understand what Axds means.

Graphics of one variable functions are two dimensional lines. Graphics of two variable functions are three dimensional surfaces. Three variable functions cannot be plotted. But can I think of the usual 3D representations of vector and scalar fields as manners of visualizing a three variable...

Homework Statement F = < z^2/x, z^2/y, 2zlog(xy)> F = \nabla f, where f = z^2log(xy) Homework Equations Evaluate \int F \cdot ds for any path c from P = (1/2, 4, 2) to Q = (2, 3, 3) contained in the region x > 0, y > 0, z > 0 Why is it necessary to specify that the path lie in the...

this is a very quick question. my teacher wants me to prove that ((kq)/(sqroot(x^2+y^2+z^2))) (x,y,z) is conservative. by this does it mean that F=((kq)/(sqroot(x^2+y^2+z^2)))i+((kq)/(sqroot(x^2+y^2+z^2)))j+((kq)/(sqroot(x^2+y^2+z^2)))k or does it she mean that...

Homework Statement Determine if the following is conservative. F(x, y, z) = (4xy + z^2)i + (2x^2 + 6yz)j + (2xz)k Homework Equations The Attempt at a Solution I'm not entirely sure I'm doing this correctly. I've taken the partial of M with respect to y and got 4x. I then took the...

Homework Statement Show that the vector field given is conservative and find its potential function. F(x, y, z)= (6x^2+4z^3)i +(4x^3y+4e^3z)j + (12xz^2+12ye^3z)k. Homework Equations The Attempt at a Solution When I take partial derivative with respect of y for...

Evaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = y i + x j + z^2 k S is the helicoid (with upward orientation) with vector...

Given the curl and divergence of a vector field, how would one solve for that vector field? In the particular case I would like to solve, divergence is zero at all coordinates.

We are told that the force field F=(\muz + y -x)i + (x-\lambdaz)j + (z+(\lambda-2)y - \mux)k Having already calculated in the previous parts of the question 1:the line integral (F.dx) along the straight line from (0,0,0) to (1,1,1) 2:the line integral (F.dx) along the path x(t)=(t,t,t^2)...

Homework Statement I am asked to sketch the following vector field in the xy-plane (a) F(r) = 2r (b) F(r) = -r/||r||3 (c) F(x,y) = 4xi + xj Homework Equations The Attempt at a Solution Can someone please give me some hints on how to proceed

Given an equation describing the curl of a vector field, is it possible to derive an equation for the originating vector field? The divergence of the field is known to be zero at all points

Hi, everyone: I am trying to produce a V.Field that is 0 only at one point of S^2. I have been thinking of using the homeo. between S^2-{pt.} and R^2 to do this. Please tell me if this works: We take a V.Field on R^2 that is nowhere zero, but goes to 0 as (x,y) grows (in the...

Homework Statement Calculate the (1) divergence and (2) curl of the following vector fields. (a) \widehat{E}(\widehat{x}) = r^{n}\widehat{x} (b) \widehat{E}(\widehat{x}) = r^{n}\widehat{a} (c) \widehat{E}(\widehat{x}) = r^{n}*(\widehat{a} X \widehat{x} where r = |\widehat{x}| and...

Homework Statement F=<0,3,x^2> computer the surface integral over the hemisphere x^2 + y^2 + z^2 = 9 z greater than or equal to 0, outward pointing normal. Homework Equations The Attempt at a Solution I don't know why I keep getting this problem wrong. The general formula for...

Homework Statement The original problem was x'=(-2 1; 1 -2)*x and I needed to find two linearly independent solutions. Homework Equations The Attempt at a Solution I found that x1=(1;1)e^(-t) and x2=(1;-1)e^(-3t). Now I am trying to plot a vector field of this. Is there an easy way...

Homework Statement Find the work done by the force field F(x, y, z) = 3xi +3yj + 7k on a particle that moves along the helix r(t) = 4 cos(t)i + 4 sin(t)j + 4tk, 0 ≤ t ≤ 2π (As in the previous problem, recall that the work of the force F on the helix corresponds to the circulation of this...

Consider this quote from Mandl and Shaw, p. 237 ...this interaction coulpes the field W_{\alpha}(x) to the leptonic vector current. Hence it must be a vector field, and the W particles are vector bosons with spin 1. Could someone explain this for me? I do not understand the "hence"...

Let M be a Riemmanian manifold. Prove that a parallel vector field along a curve c(t) preserves the length of the parallel transported vector. Furthermore if M is an oriented manifold, prove that P preserves the orientation. So I want to prove that d/dt <P, P> = 0, so <P, P> = constant...

can anyone please give me an example of how to calculate the homothetic vector field of say a Bianchi Type I exact solution of the Einstein field equation (refer to dynamical systems in cosmology by Wainwright and Ellis chapter 9) note : I know already how to calculate the...

I am considering two vector fields in the spacetime of General Relativity. One is spacelike, the other is timelike, they are normalized and orthogonal: U.U = -1 V.V = +1 U.V = 0 where dot denotes scalar product. In addition, it is known the integral curves of U and V always remain in...

Q: Let F= (-y/(x2+y2), x/(x2+y2), z) be a vector field and let U be the interior of the torus obtained by rotating the circle (x-2)2 + z2 = 1, y=0 about the z-axis. a) Show that curl F=0 but ∫ F . dx = 2pi where dx=(dx,dy,dz) C and C (contained in U) is the circle x2+y2=4, z=0. Therefore F...

Suppose that normal derivative = \nablag . n = dg/dn, then f \nablag . n = f dg/dn [I used . for dot product] But how is this possible? For f \nablag . n, I would interpret it as (f \nablag) . n But f dg/dn = f (\nablag . n) which is DIFFERENT (note the location of brackets) I...

Conservative Vector Fields -- Is this right? Homework Statement G = <(1 + x)e^{x+y}, xe^{x+y}+2z, -2y> Evaluate \int_{C}G.dR where C is the path given by: x = (1 - t)e^{t}, y = t, z = 2t, 1=>t>=0 Homework Equations The Attempt at a Solution First, i noticed that there is a scalar potential...

Homework Statement Is it possible to find a vector field whose curl is yi? xi? Homework Equations The Attempt at a Solution I found that if F = <0,0,y^2/2>, its curl will be yi. However, I cannot figure out a vector field whose curl will be xi. I tried using exponentials and just...

I have a question regarding conservative vectorfields and singularities. Suppose we have a vectorfield who is defined everywhere in R^2 except at the origin where it has a singularity, and suppose it's curl is zero. We then have that it is conservative in every open, simply connected subset in...

Are the W+,W- and Z0 the field quanta of the massive charged vector fields? ie Proca fields

From a mathematical standpoint I have no trouble understanding the difference between a conservative vector field and a non-conservative vector field. It's rather simple. The conservative field can be reduced to some functions gradient vector, doesn't care what path you decide to take, and...

Hi: I am going over Lee's Riemm. mflds, and there is an exercise that asks: Let M<M' (< is subset) be an embedded submanifold. Show that any vector field X on M can be extended to a vector field on M'. Now, I don't know if he means that X can be extended to the _whole_ of M'...

Supposing we have as 2 vector fields: F = x^2i + 2zj +3k and G = r^2e_r + 2\cos\Theta e_{\Theta} + 3\sin(2\phi) e_\phi how do i perform the following operations on them? - F\cdot r - F\times r - |G| - G\cdot r

Homework Statement describe what field lines are (7 marks) Homework Equations The Attempt at a Solution A field line is a locus that is defined by a vector field and a starting location within the field. A vector field defines a direction at all points in space; a field line may...

Task: Find a potential function for the conservative vector field (y,x,1) My work: (Df/Dx, Df/Dy and Df/Dz are the partial derivatives) Df/Dx=1, Df/Dy=x, Df/Dz=1 I take the first eq. and integrate, so that I get f(x,y,z) = yx + C I then derivate with respect to y: Df/Dy=...

Can a killing vector field generate a diffeomorphism that only shifts points inside a small part of the manifold and preserves points outside of it? Rotation preserves the metric of a sphere but shifts every points on the sphere, I'd to find out if there is a killing vector field that...

What are the general rules that one should use in graphing vector fields. I'm having a lot of trouble doing this and don't really know where to start. If you take F(x,y) = -yi + xj What should be the next step in terms of graphing? They have it drawn in our book as a bunch of vectors...