Vector field Definition and 382 Threads

Homework Statement I've attached an image with the entire question. Homework Equations Attached an image with relevant equations. Can't use Gauss' Divergence The Attempt at a Solution In the attached image I've also included the start of my calculations, I just need to see if my...

I'm unable to understand this generalization of vectors from a quality having a magnitude and direction, to the more mathematical approach. what is the difference between vector space and vector field? more of an intuitive example?

its components M(x,y) and N(x,y) are differentiable functions that satisfy (∂N(x,y)/∂x) – (∂M(x,y)/∂y) = 1-x. a. is it possible for the vector field to be conservative? Explain. b. Let C be x^2+y^2=1 centered at the origin traced counter clockwise. compute the integral ∫F.dr...

This problem is about Line integral of Vector Field. I believe the equation i need to use is: \intF.dr = \intF.r'dt, with r = r(t) I try to solve it like this: C1: r1= < 1 - t , 3t , 0 > C2: r2= < 0 , 3 - 3t , t > C3: r3= < t , 0 , 1 - t > After some computation, I got stuck at the...

This is a problem from an old final exam in my Calc 3 class. My book is very bad at having examples for these types of problems, and my instructor only went over one or two. Help would be much appreciated. Homework Statement Verify that the Stokes' theorem is true for the vector field...

Homework Statement Please evaluate the line integral \oint dr\cdot\vec{v}, where \vec{v} = (y, 0, 0) along the curve C that is a square in the xy-plane of side length a center at \vec{r} = 0 a) by direct integration b) by Stokes' theoremHomework Equations Stokes' theorem: \oint V \cdot dr =...

Homework Statement I need to analyze these pictures for my homework and find out the curl of the vector field at the point (red) on the picture. Homework Equations http://i1242.photobucket.com/albums/gg525/sjrrkb/ScreenShot2012-11-26at61615PM.png The Attempt at a Solution basically...

Hi, I am experiencing a little bit of trouble grasping the concept of vector fields in fluid mechanics: If the vector V = u i + v j + w k, where i,j,k are unit vectors in the x,y,z directions, then how can u,v,w be functions of x,y,z,t? I.e. if say v is in the y direction, then how can it...

Homework Statement Calculate F=∇V, where V(x,y,z)= xye^z, and computer ∫F"dot"ds, where A)C is any curve from (1,1,0) to (3,e-1) B)C is a the boundary of the square 0≤x≤1, 0≤y≤1... oriented counterclockwise. Homework Equations ∫F"dot"ds= ∫F(c(t)"dot"c'(t) The Attempt at a Solution...

Hi all, I have a question for all of you. I've been wanting to make a 3D vector field that would represent a magnetic field (for fun) around some segment of wire with a constant current flowing through it. I'm assuming I have a parametric equation for the wire segment. The one equation that...

I've attached the problem as a picture. Generally, this would be a simple problem if I were to apply Green's theorem. But I can't use Green because D would not be a simply connected closed region; the vector field isn't defined at the origin. Could someone please give me an idea of where to...

Homework Statement consider the vector field v(x,y,z)=(-h(z)y,h(z)x,g(z)) wherer h:R->R and g:R—>R are differentiable .Let C be a closed curve in the horizontal plane z=z0.show that the circulation of v around C depends only on the area of the reion enclosed by C in the given plane and h(Z0)...

Homework Statement Here is the problem and solution but I am confused as to part B http://gyazo.com/e77d05fc67cb6ac266ff021ef88052dc The Attempt at a Solution I understand the first part, but I am totally lost on how they reached their cartesian answer for part B. Firstly why did they...

Homework Statement \int_C \mathbf F\cdot d \mathbf r where \mathbf F = x^2\vec{i}+e^{\sin^4{y}}\vec{j} and C is the segment of y=x^2 from (-1,1) to (1,1). Homework Equations \int_C \mathbf F\cdot d \mathbf r=\int_a^b \mathbf F( \mathbf r(t))\cdot r'(t) dt=\int_C Pdx+Qdy where \mathbf F =...

Homework Statement Show that the vector fields A = ar(sin2θ)/r2+2aθ(sinθ)/r2 and B = rcosθar+raθ are everywhere parallel to each other. Homework Equations \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos(0) The Attempt at a Solution So, if the dot product equals 1. They should be...

Suppose that F and G are vector fields and that F-G = ▽μ for some real-valued function μ(x,y). Prove that ∫ F.dx = ∫ G.dx for all piecewise smooth curves C in the xy-plane I just need some help in getting started really. Thanks

I was wondering if such an approximation is possible and plausible... The first term would have to look sth like this: \vec{f}(\vec{x_{0}}) + \textbf{J}_{\vec{f}}(\vec{x_{0}})\cdot(\vec{x}-\vec{x_{0}}) No clue about the second term though... We would have to calculate the Jacobian of the...

Homework Statement My problem is, I have a scalar field and I take the gradient of this field. It is known that the gradient of a scalar function is a conservative vector field; but I need to run a procedure in this field that will modify the vector field; the modified vector field could be a...

I thought i had a strong understanding of parameterizing curves and sketching vector fields. However when I was going through my practice test I came across this problem which I don't full grasp Let F^ ⃗=xi ^⃗+(x+y) j ^⃗+(x-y+z)k^⃗ . a) Find a point at which F ⃗ is parallel to the...

Homework Statement I know I posted a question yesterday also, but this homework is getting on my nerves. My prof isn't the best out there. So basically the question is as follows: http://img812.imageshack.us/img812/8130/problem6.png Homework Equations Not sure, I sort of answered the...

Homework Statement Show that \nabla_a(\sqrt{-det\;h}S^a)=\partial_a(\sqrt{-det\;h}S^a) where h is the metric and S^a a vector. Homework Equations \nabla_a V^b = \partial_a V^b+\Gamma^b_{ac}V^c \Gamma^a_{ab} = \frac{1}{2det\;h}\partial_b\sqrt{det\;h} \nabla_a\sqrt{-det\;h} (is that...

I'm on the last chapter of a 1200 page calc book, I'm really psyched. Homework Statement The Attempt at a Solution The method I learned for finding the unit normal of a vector field, n, is take the derivative of the equation and divide that by the magnitude of the...

Homework Statement Compute the flux of the vector field F(x,y,z)=(z,y,x) across the unit sphere x2+y2+z2=1 Homework Equations I believe the forumla is ∫∫D F(I(u,v))*n dudv I do not know how to do the parameterization of the sphere and then I keep getting messed up with the normal vector.*...

Homework Statement http://gyazo.com/94783c14f2d2d05e62e479ab33c73830 Homework Equations I know the dot product and cross product, but even for the first one I don't see how either helps. The Attempt at a Solution 1. the gradient of the 2 scalars multiplied together (not crossed...

Homework Statement confirm that the given function is apotential for the given vector field ln(x^{2} + y^{2}) for \frac{2x}{\sqrt{x^{2}+y^{2}}} \vec{i} + \frac{2y}{\sqrt{x^{2}+y^{2}}} \vec{j} Homework Equations The Attempt at a Solution the first thing i did was let my equation...

Homework Statement F(x,y,z) = (-x+y)i + (y+z)j + (-z+x)k Find divergence Homework Equations The Attempt at a Solution The gradient is -i + j + -k Dotting that with F, I get x - y + y + z + z - x = 2z My book lists the answer as -1. What the heck are they talking...

Hey all, I have a vector field described by a complex potential function (so I have potential lines and streamlines). I am looking for a way to express its curvature at every point, but I can't find such a formula in my books. I have searched in wikipedia and I read that the way to define it...

Homework Statement Consider the intersection,R, between two circles : x2+y2=2 and (x-2)2+y2=2 a) Find a 2-Dimensional vector field F=(M(x,y),N(x,y)) such that ∂N/∂x - ∂M/∂y=1 Homework Equations none. The Attempt at a Solution There are other parts to the main question but I don't think I will...

I have a bad habit of deciding how I would solve the concepts presented to us in lecture before the instructor covers the method. I try my methods immediately and use the textbook method if I hit a road block. This leads to problems for me. Right now we are covering line integrals over...

Homework Statement Find a vector field \vec{A}(\vec{r}) in ℝ3 such that: \vec{\nabla} \times \vec{A} = y2cos(y)e-y\hat{i} + xsin(x)e-x2\hat{j} The Attempt at a Solution I broke it down into a series of PDE's that would be the result of \vec{\nabla} \times \vec{A}: ∂A3/∂y - ∂A2/∂z...

Homework Statement Let \vec{E}(\vec{r}) = \vec{r}/r2, r = |\vec{r}|, \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} be a vector field in ℝ3. Show that \vec{E} is conservative and find its scalar potential. Homework Equations All of the above. The Attempt at a Solution \vec{\nabla}...

Problem statement attached. The correct way to do this seems to plug in your given x, y, z into F then integrate the dot product of F and <x',y',z'> dp from 0 to 1, however, this results in way too messy of an integral. Answer is 3/e...

Homework Statement Let F = ( -y/(x2+y2) , x/(x2+y2) ) Show that this vector field is irrotational on ℝ2 - {0}, the real plane less the origin. Then calculate directly the line integral of F around a circle of radius 1.Homework Equations The Attempt at a Solution To show F is irrotational we...

I was wondering what the orbits of a Killing vector field are. Do you have any good sources or reading material for this?

Show a flowline of a vector field?? Homework Statement Consider the vector field F(x,y,z)=(8y,8x,2z). Show that r(t)=(e8t+e−8t, e8t−e−8t, e2t) is a flowline for the vector field F. r'(t)=F(r(t)) = (_,_,_) Now consider the curve r(t)=(cos(8t), sin(8t), e2t) . It is not a flowline of...

\nabla\timesgrad(f) is always the zero vector. Can anyone in terms of physical concepts make it intuitive for me, why that is so. I get that the curl is a measure of the tendency of a vector field to rotate or something like that, but couldn't really assemble an understanding just from that.

Homework Statement Consider the surface S with the graph z = 1-x^{2}-y^{2} with z≥0, and also the unit disc in the xy plane. Give this surface an outer normal. Compute: \int\int_{S}\vec{F}\bulletd\vec{S} where \vec{F}(x,y,z) = (2x,2y,z)Homework Equations \int\int_{S}\vec{F}\bulletd\vec{S} =...

Homework Statement Given the vector field \vec{v} = (-y\hat{x} + x\hat{y})/(x^2+y^2) Show that \oint \vec{dl}\cdot\vec{v} = 2\pi\oint dl for any closed path, where dl is the line integral around the path.Homework Equations Stokes' Theorem: \oint_{\delta R} \vec{dl}\cdot\vec{v} = \int_R...

Hi, I'm studying calculus 3 and am currently learning about conservative vector fields. ============================= Fundamental Theorem for Line Integrals ============================= Let F be a a continuous vector field on an open connected region R in ℝ^{2} (or D in ℝ^{3}). There exists...

Homework Statement First a thanks for the existence of this site, i find it quite useful but had no need to actually post till now. I am stuck on the following problem in "introduction to physics" We should calculate the \oint \vec{v}.d\vec{A} of a object with the following parameters...

Let f(x,y) = ( \frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2}) with f : D \subset \mathbb{R}^2 \to \mathbb{R}^2 I know if I take D = D_1 = \mathbb{R}^2 - \{ (0,0) \} the vector field is not conservative, for the circulation over a circunference centered at the origin does not equal zero. But...

Homework Statement Vector field F = (2siny-cosz, 2cosx+3sinz, cosy-2sinx) Compute the flux of F through a sphere of radius one centered at the origin with respect to the outer unit normal. Homework Equations Divergence theorem/Gauss's Law The Attempt at a Solution The only...

What if when I'm finding the repeated terms from integrating the vector field with respect to x and then y, i come across two terms that are same except one is negative and the other positive. What does that mean, and how do i represent that in the overall potential function I'm finding?

I have a technical question and at the time being I can't ask it to a professor. So, I'm here: If I try to quantize the vector field in the Coulomb gauge (radiation gauge) A_0(x)=0,\quad \vec\nabla\cdot\vec A=0. by imposing the equal-time commutation relation...

Homework Statement If a, b, and c are any three vector fields in locally Minkowskain 4-manifold, show that the field ε_{ijkl}a^{i}b^{k}c^{l} is orthogonal to \vec{a}, \vec{b}, and \vec{c}. Homework Equations The Attempt at a Solution I know I have to show that multiplying the...

I have a question about Bernard Shutz's derivation of the Lie derivative of a vector field in his book Geometrical Methods for Mathematical Physics. I will try to reproduce part of his argument here. Essentially, we have 2 vector fields V and U which are represented by \frac{d}{d\lambda} and...

Hello all, I was hoping someone would be able to clarify this issue I am having with the Lie Derivative of a vector field. We define the lie derivative of a vector field Y with respect to a vector field X to be L_X Y :=\operatorname{\frac{d}{dt}} |_{t=0} (\phi_t^*Y), where \phi_t is the...

Homework Statement "A gradient of a vector field is symmetric if and only if this vector field is a gradient of a function" Pure Strain Deformations of Surfaces Marek L. Szwabowicz J Elasticity (2008) 92:255–275 DOI 10.1007/s10659-008-9161-5 f=5x^3+3xy-15y^3 So the gradient of this function...

Let's assume that a compact Lie group and left invariant vector filed X are given. I wonder why the divergence (with respect to Haar measure) of this field has to be equall 0. I found such result in one paper but I don't know how to prove it. Any suggestions?

Homework Statement Homework Equations The Attempt at a Solution What's surprise? And is my work correct?