Vector fields Definition and 171 Threads

Homework Statement This problem is in Introduction to Eletrodynamics, of Griffiths, 3rd edition, p.20, problem 1.19. He asks a vector function v(x,y,z), other than the constant, that has: \nabla\cdot\vec{v}=0 \mbox{ and } \nabla\times\vec{v}=0 Homework Equations I hope you know them...

Hi, Is there a specific product rule or something one must follow when applying the lie bracket/ commutator to two vector fields such that one of them is multiplied by a function and added to another vector field? This is the expression given in my textbook but I don't see how: [fX+Z,Y] =...

Hi, I don't understand a particular coordinate expansion of the commutator of 2 vector fields: [X, Y ]f = X(Y f) − Y (Xf) = X_be_b(Y _ae_af) − Y _be_b(X_ae_af) = (X_b(e_bY_ a) − Y _b(e_bX_a))e_af + X_aY _b[e_a, e_b]f X,Y = Vector fields f = function X_i = Components of X and...

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 Trying to get my head around the physical interpretation of pathlines and the math that describes them. The physical explanation is simple enough, they are just the path a particle of fluid will follow through the vector field. Homework Equations In the vector field...

Are all vector fields invariants or is this a particular characteristic to some fields? For example, suppose the vector field E = x^2 x + xy y. If I write it in terms of covariant and contravariant basis through the polar coordinates I get the same results, or in other words \vec{E} =...

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 Let S be the ellipsoid where a,b, and c are all positive constants. x2/(a+1)+y2/(b2)+z2/(c2) = 1 → → → → → Let F = (r - ai) / ||r - ai|| [* r and i are vectors = I tried inserting the arrows] a)Where...

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...

Take a tangent vector field with isolated singularities on a compact smooth Riemannian surface ( 2 dimensional manifold without boundary). Divide v by its norm to get a field of unit vectors with isolated discontinuities. Around each singularity chose a small open disc. The tangent circle...

Need urgent assistance(Gussian curvature and differentiable vector fields) Hi I have a very difficult problem where I know some of the dots but can't connect them :( So therefore I hope that there is someone who can assist me (hopefully :)) Homework Statement Let S be a surface with...

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"...

Homework Statement Let G be a Lie group with unit e. For every g in G let Lg: G -> G, h-> gh be left multiplication. a) Prove that the map G x TeG -> TG, (g,v) -> De Lg (v) is a diffeomorphism, and conclude G has a basis of vector fields. b) Prove for every v in TeG the is a unique...

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...

Homework Statement Let (M,g) be a spacetime. (a) Let A and A' be vector fields on M such that g(A,B)=g(A',B) for any future-pointing timelike vector field Y. Show that X=X'. (b) Let w and w' be two two-forms on M. Suppose that i¬A w = i¬A w' for any future -pointing timelike vector field A...

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...