Vector field Definition and 403 Threads

This is part of a larger question, but this is the part I am having difficulty with. I have had an attempt, but am not sure where I am making a mistake. Any help would be very, very appreciated. 1. Homework Statement Let C2 be the part of an ellipse with centre at (4,0), horizontal semi-axis...

I'm trying to do past exam papers in GR but there are some things I don't yet feel comfortable with, so even though I can do some parts of the question I would be very happy if you could check my solution. Thank you! 1. Homework Statement Spacetime is stationary := there exists a coord chart...

So I have found that everyone conservative vector field is irrotational in a previous problem. Based on the relationship irrotational vector fields and incompressible vector fields have, div(curl*F)=0, does that also imply every conservative vector field is incompressible? Kindly, Shawn

Homework Statement I am to prove (using the equations for gradient, divergence and curl in spherical polar coordinates) that vector field $$\mathbf{w}=w_{\psi}(r,\theta)\hat e_{\psi}$$ is solenoidal, find $$w_{\psi}(r,\theta)$$ when it's irrotational and find a potential in this case. Homework...

Fine the word done in moving a particle in the force field F=<2sin(x)cos(x), 0, 2z> along the path r=<t,t,t2>, 0≤t≤π To do the line integral, I need to find F(r(t)), but I don't understand how to express it. For example I looked at the online notes provided here...

Let's say we have two vector fields, described by 6 functions: Ax, Ay, Az and Bx, By, Bz. We want to divide field A by field B. Do we take Ax/Bx , Ay/By and Az/Bz individually? But in this case we might end up with Three different scalar fields. What's the proper way to do this?

Homework Statement The Lagrangian density for a massive vector field ##C_{\mu}## is given by ##\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}C_{\mu}C^{\mu}## where ##F_{\mu\nu}=\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}##. Derive the equations of motion and show that when ##m...

Homework Statement Evaluate ∫C < −y, x − 1 > dr where C is the closed piecewise continuous curve formed by the line segment joining the point A(− √ 2, √ 2) to the point B( √ 2, − √ 2) followed by the arch of the circle of radius 2, centered at the origin, from B to A. 2. The attempt at a...

I was curious, if you were given a vector field F(x,y,z) = <Fx(x,y,z), Fy(x,y,z), Fz(x,y,z)>, and then some scalar function f(x,y,z), how would you define a function θ(x,y,z) of the angle θ between the scalar function and the vector field at any given point. I know how I would find this at a...

Homework Statement Determine for which real values of the parameter ##\alpha## the vector field given by ##F(x,y) = (\frac{2xy}{y-\alpha}, 2 - \frac{4x^2}{(y-\alpha)^2})## is conservative. For those values of ##\alpha##, calculate the work done along the curve of polar equation: ##\rho =...

Homework Statement Sketch each of the following vector fields. E_5 = \hat \phi r E_6 = \hat r \sin(\phi) I wish to determine the \hat x and \hat y components for the vector fields so that I can plot them using the quiver function in MATLAB. Homework Equations A cylindrical coordinate...

< y^2, 2xy+ e^(3z), 3ye^(3z)> is the vector field. the above vector field is inside an open simply connected domain. the parametric equations all have a continuous first order derivative inside the domain. Lastly, the curl of the vector field is <0, 0, 0> Thus, the vector field is...

I'm using the textbook Electricity and Magnetism by Purcell. In the section about continuous charge distributions I found the following formula \mathbf{E}(x,y,z)= \frac{1}{4\pi\epsilon_0 } \int \frac{\rho(x',y',z')\boldsymbol{\hat r} dx'dy'dz'}{r^{2}} . It's stated that (x,y,z) is fixed...

What does it mean that a Killing vector and a total differential of a certain theory are related to bilinears? In other words, why would bilinears (e.g. of the forms ##<\gamma_0\epsilon, \gamma_5\gamma_{\mu}\epsilon>## and ##<\gamma_0\epsilon, \gamma_{\mu}\epsilon>## tell us anything about...

Hello everyone How is it possible that a n-dimensional spacetime admits m> n INDEPENDENT Killing vectors where m=n(n+1)/2 if the space is maximally symmetric?

In many books of general relativity I have found the following statement: a vector field produces repulsive forces between like charges so can not be used to describe gravity. But they do not show it. How can i show it or where can I find a book or a paper in which it is shown? Please, to avoid...

I know that if a vector field is conserved then there exits a function such that the gradient of this function is equal to the vector field but am just curious to know the reason of it.

Give an example of a formula for a vector field whose graph would closely resemble the one shown. The box for this figure is [−2, 2] x [−2, 2]. Not sure where to start.

Divergence can be defined as the net outward flux per unit volume and can be explained using Gauss' theorem. (I read this in Feynman lectures Vol. 2) In the next page, He derives Stokes' theorem using small squares. The left side of equation represents the total circulation of a vector...

I have the following integral: ## \oint_{S}^{ } f(\theta,\phi) \hat \phi \; ds ##Where s is a sphere of radius R.so ds = ##R^2 Sin(\theta) d\theta d\phi ## Where ds is a scalar surface element. If I was working in Cartesian Coordinates I know the unit vector can be pulled out of integral and...

I know gravity is a conservative force field and can be treated as such for all intents and purposes, but I was just thinking that in order to show that a vector field is conservative that vector field must be defined everywhere (gravitational force field is not defined at r=0). I was thinking...

Homework Statement Suppose v^\mu is a Killing Vector field, the prove that: v^\mu \nabla_\alpha R=0 Homework Equations 1) \nabla_\mu \nabla_\nu v^\beta = R{^\beta_{\mu \nu \alpha}} v^\alpha 2) The second Bianchi Identity. 3) If v^\mu is Killing the it satisfies then Killing equation, viz...

Homework Statement Homework EquationsThe Attempt at a Solution As you can see, the solution is shown just below the question. Essentially, I don't understand how the x, y and z component of the vector field has been separated because the numerator of the vector field's fraction is: (x^2 +...

Homework Statement Find the field line of \vec{E}(\vec{r}) = \frac{m}{4 \pi r^3} (2 \cos\theta, \sin\theta, 0) through the point (a, b, c) (Spherical coordinates). m is a constantI know the answer, but I don't see what I do wrong. The Attempt at a Solution \frac{d\vec{r}}{d \tau} = C...

Hello, I'm having problems understanding what exactly is meant by a conservative vector field. So, allow me to explain how I currently understand it. Let \overrightarrow{F} be some sort of a vector field that is conservative. The stream plot of \overrightarrow{F} is shown below. From my...

Hi all! I am a 13-year-old that is very interested in physics. I am currently studying fields, and I have gotten into some advanced parts of field theory. With that, I have 3 big questions: 1. In vector and higher order tensor fields, what does it mean to "observe the field from point...

I met in several sources (textbooks) phrase «Space can be constructed over any field». But it is always illustrated with linear space over scalar field (or sometimes over ring). Does it make any sense to talk about spaces over vector fields? What kinds of spaces are they? What about tensor...

Sorry if this was addressed in another thread, but I couldn't find a discussion of it in a preliminary search. If it is discussed elsewhere, I'll appreciate being directed to it. Okay, well here's my question. If I take the divergence of the unit radial vector field, I get the result: \vec...

Hello everyone, I would like to know if anyone knows what is the inner product for vector fields ##A_\mu## in curved space-time. Is it just: $$ (A_\mu,A_\mu)=\int d^4x A_\mu A^\mu =\int d^4x g^{\mu\nu}A_\mu A_\nu $$ ? Do I need extra factors of the metric? Thanks!

I just did this following exercise in my text If C is the line segment connecting the point (x_1,y_1) to (x_2,y_2), show that \int_C xdy - ydx = x_1y_2 - x_2y_1 I did, and I also noticed that if we put those points into a matrix with the first column (x_1,y_1) and the second column (x_2,y_2)...

This is a problem from a section on Line Integrals in my Calculus Textbook, I haven't studied any physics relating to E&M yet, and the solutions manual only gives solutions for odd numbered problems. Sorry, if I'm posting in the wrong forum, I hope I'm not. 1. Homework Statement A steady...

Oops. I just realized that this is the physics homework forum... This is actually calculus homework... 1. Homework Statement Homework Equations n/a The Attempt at a Solution I read, analyzed and reread the text, but I am still confused. 1) How was the position vector ##<x,y>##...

Homework Statement [/B] Secondly, is a "conservative vector field" the same thing as a "conservative force" (a force that does not depend on the path taken)? If not what is it? Homework Equations n/a The Attempt at a Solution How was ##f(x,y,z)=\frac{mMG}{\sqrt{x^2+y^2+z^2}}## derived?

Homework Statement Consider the vector field F(r) = Φ^ (a) Calculate ∫ F⋅dl where C is a circle of radius R (oriented counterclockwise) in the xy-plane centered on the origin. Homework Equations maybe Φ^ = -sinΦx^ + cosΦy^ The Attempt at a Solution not really a solution. i am just stuck at...

vector field F = [ -x3 , x, 0] how does this behave for a constant x? Does that mean that I plot the vector field for different values of x?

Homework Statement Homework EquationsThe Attempt at a Solution I solved #2,4 but I don't understand what #1,3 need to me. I know that scalar field is a function of points associating scalar value. But how can I prove some function is scalar field or vector field?

Homework Statement Let ν(x,y,z) = (xi + yj + zk)rk where v, i, j, k are vectors The k in rk∈ℝ and r=√(x2+y2+z2). Show that ∇.v=λrk except at r=0 and find λ in terms of k. Homework Equations As far as I understand it, ∇.v=∂/∂x i + ∂/∂y j + ∂/∂z k, but this may very well be wrong. The Attempt...

Some context for my question: If you have a smooth manifold \mathcal{M} you can define tangent vectors to parametrized paths in the following way: If \mathcal{P}(s) is a parametrized path, then \frac{d}{ds} \mathcal{P}(s) = V where V is the differential operator that acts on scalar fields...

Homework Statement }[/B] Find the divergence and curl of the vector field \vec{V}=x^2y \hat{i} + xy^2 \hat{j} + xyz \hat{k} then for both, evaluate them at the point \bar{r} = (1,1,1) Homework Equations div(\vec{F})= \nabla \cdot \vec{F} \\ curl(\vec{F})= \nabla \times\vec{F} The Attempt...

Homework Statement Working in Cartesian coordinates (x,y,z) and given that the function g is independent of x, find the functions f and g such that: v=coszi+f(x,y,z)j+g(y,z)k is a Beltrami field. Homework Equations From wolfram alpha a Beltrami field is defined as v x (curl v)=0 The Attempt...

Im doing some revision of vector calculus and came across the following problem Q: Calculate the work done by the force field F = 3xyi - 2j in moving from A: (1,0,0) to D: (2,0,0) and then from D: (2,0,0) to B: (2,sqrt(3),0) I got stuck and decided to look at the answers. In the answers (part...

Homework Statement I'm solving for the magnetic potential vector field given a time varying electrical field. It might seem silly, but I don't really understand how to approach the problem since the field varies on both space and time. I'm given the motion of a charged particle. Can someone...

Hello, I'm having some difficulty with a conceptual question on a practice test I was using to study. I have the answer but not the solution unfortunately. 1. Homework Statement "For every differentiable function f = f(x,y,z) and differentiable 3-dimensional vector field F=F(x,y,z), the...

Homework Statement Recall that a flow line, c(t), of a vector field F has c'(t)=F(c(t)) at all times t. Show all work below. a.) Let c(t) be the flow line of a particle moving in a conservative force field F=-grad(f), where f:R^3->R, f(x,y,z) >=0 for all (x,y,z), represents the potential...

Homework Statement ##\iint_S (x^2+y^2)dS##, ##S##is the surface with vector equation ##r(u, v)## = ##(2uv, u^2-v^2, u^2+v^2)##, ##u^2+v^2 \leq 1## Homework Equations Surface Integral. ##\iint_S f(x, y, z)dS = \iint f(r(u, v))\left | r_u \times r_v \right |dA##, The Attempt at a Solution...

Homework Statement Find the constant vector field F giving the flux of 5 trough the surface S, a disk of radius 4 perpendicular to both F and the y-axis, and oriented away from the origin. Homework EquationsThe Attempt at a Solution I have gone through several articles on the web and searched...

Homework Statement I have to explain why this vector field is not conservative. Homework Equations Maybe it is: if ##\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}## then F(x, y) = p(x, y)i + Q(x, y)j is a conservative field. I tried to figure out what P and Q is, but that...

Homework Statement Homework Equations ∫∫D F((r(u,v))⋅(ru x rv) dA The Attempt at a Solution [/B] I got stuck after finding the above, at where the double integrals are. :( May I know how do I find the limits of this? (I always have trouble finding the limits to sub into the integrals...

Homework Statement [/B] Find the outward flux of the vector field ## \vec F = y^2e^{z^2+y^2} i + x^2 e^{z^2+x^2} j + z^2 e^{x^2+y^2} k## across that part of the ellipsoid $$ x^2 + y^2 + 4z^2 = 8$$ which lies in the region ##0 ≤ z ≤ 1## (Note: The two “horizontal discs” at the top and bottom are...

Homework Statement Hello I was given the vector field: \vec A (\vec r) =(−y(x^2+y^2),x(x^2+y^2),xyz) and had to calculate the line integral \oint \vec A \cdot d \vec r over a circle centered at the origin in the xy-plane, with radius R , by using the theorem of Stokes. Another thing, when...