Vector field Definition and 403 Threads

Homework Statement This two-part problem is from O'Neill's Elementary Differential Geometry, section 2.5. Let W be a vector field defined on a region containing a regular curve a(t). Then W(a(t)) is a vector field on a(t) called the restriction of W to a(t). 1. Prove that Cov W w.r.t...

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...

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...

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...

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...

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?

I got the answer! thanks for the help

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...

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...

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

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...

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?

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...

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...

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...

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.

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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)...

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...

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) =...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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?

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?

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...

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...

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...

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...

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...

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.

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 =...

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?

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...

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?