Vector field Definition and 382 Threads

In 2 dimensions given a scalar field f(x,y) is possible to compute the line integral ##\int f ds## and area integral ##\iint f d^2A##. In 3D, given a scalar field f(x,y,z) is possible to compute the surface integral ##\iint f d^2S## and the volume integral too ##\iiint f d^3V##...

To every scalar field s(x,y) there corresponds a 'constant' vector field x = A s(x,y) and y = B s(x,y), where A,B are direction cosines. The vector field is only partially constant since only the directions, and not the magnitudes, which are equal to |f(x,y)|, of the field vectors are constant...

Homework Statement Homework Equations W= ∫F.dr The Attempt at a Solution I'm fairly sure I've done the right thing, however my lecturer hasn't uploaded any solutions to any of these problems (which is ridiculous - how am I supposed to learn if I don't know when I'm right or...

Given Vector Field: $F=2(x+y)\sin\pi za_x-(x^2+y)a_y+\left(\frac{10}{x^2+y^2}\right)a_z$ specify the locus of all points at which a.) $F_x=0$ b.) $F_y=0$ c.) $|F_x|=1$ please help me get started with this. thanks!

Suppose we have a vector field ##V## defined everywhere on a manifold ##M##. Consider now point ##p \in M##. As a consequence of the existence and uniqueness theorem of differential equations. this implies that ##V## gives rise to a unique local flow $$\theta:(-\epsilon,\epsilon) \times U \to...

Hi, I would like to understand the left-invariant vector field of the additive group of real number. The left translation are defined by \begin{equation} L_a : x \mapsto x + a \; , \;\;\; x,a \in G \subseteq \mathbb{R}. \end{equation} The differential map is \begin{equation} L_{a*} =...

According to Isham (Differential Geometry for Physics) at page 115 he claims: "If X is a complete vector field then V can always be chosen to be the entire manifold M" where V is an open subset of a manifold M. He leaves this claim unproved. A complete vector field is a vector field which...

If a vector field ##\vec{v}## is non-divergent, so the identity is satisfied: ##\vec{\nabla}\cdot\vec{v}=0##; if is non-rotational: ##\vec{\nabla}\times\vec{v}=\vec{0}##; but if is "non-linear" Which differential equation the vector ##\vec{v}## satisfies? EDIT: this isn't an arbritrary...

The data points of the polarization of the CMB are a magnitude and an orientation that varies between 0° and 180°. What kind of mathematical field is that, not quite a vector field? Thanks for any help!

Hey! :o Determine if the vector field $\overrightarrow{F}=y\hat{i}+(x+z)\hat{j}-y\hat{k}$ is conservative or not.The vector field $\overrightarrow{F}=M\hat{i}+N\hat{j}+P\hat{k}$ is conservative if $$\frac{\partial{M}}{\partial{y}}=\frac{\partial{N}}{\partial{x}}...

Hey! :o I have the following exercise: Apply the divergence theorem to calculate the flux of the vector field $\overrightarrow{F}=(yx-x)\hat{i}+2xyz\hat{j}+y\hat{k}$ at the cube that is bounded by the planes $x= \pm 1, y= \pm 1, z= \pm 1$. I have done the following...Could you tell me if this...

Hello again! :) I am given the following exercise: Find the flux of the vector field $\overrightarrow{F}=zx \hat{i}+ zy \hat{j}+z^2 \hat{k}$ of the surface that consists of the first octant of the sphere $x^2+y^2+z^2=a^2(x,y,z \geq 0).$ That's what I did so far: $\hat{n}=\frac{\nabla{G}}{|...

Hey! :o Apply the divergence theorem over the region $1 \leq x^2+y^2+z^2 \leq 4$ for the vector field $\overrightarrow{F}=-\frac{\hat{i}x+\hat{j}y+\hat{k}z}{p^3}$, where $p=(x^2+y^2+z^2)^\frac{1}{2}$. $\bigtriangledown...

Homework Statement A vector field $$ \vec{u}=(u_1,u_2,u_3) $$ satisfies the equations; $$ \Omega\hat{z} \times \vec{u}=-\nabla p , \nabla \bullet \vec{u}=0$$ where p is a scalar variable, \Omega is a scalar constant. Show that \vec{u} is independant of z. Hint ; how can we remove p from...

Hi, I know that for the electric displacement vector field \oint D.dS=\sum Q_{c} does this mean that I can just use a Gaussian surface to explain why the displacement vector field for a sphere is radial or not without having to talk about the electric field. If not what is the reasoning to...

Homework Statement Can someone guide me through solving a problem involving the circulation of a vector field? The question is as stated for the vector field E = (xy)X^ - (x^2 + 2y^2)Y^ , where the letters next to the parenthesis with the hat mean they x y vector component. I need to find...

So, this is going to be pretty hard for me to explain, or try to detail out since I only think I know what I'm asking, but I could be asking it with bad wording, so please bear with me and ask questions if need-be. Currently I have a 3D vector field that's being plotted which corresponds to...

Question: Find the outward flux of the vector field F = i-2j-2k across the surface S defined by z = 4-x2-y2 0≤z≤4 At first, I used the Divergence Theorem to solve this problem. I took the divF and got the answer of 0. By definition, integrating 0 three times will still equal 0. Thus, the...

Homework Statement Calculate the flow over the upper surface of sphere ##x^2+y^2+z^2=1## with normal vector pointed away from origin. Vector field is given as ##\vec{F}=(z^2x,\frac{1}{3}y^3+tan(z),x^2z+y^2)##Homework Equations Gaussian law: ##\int \int _{\partial \Sigma }\vec{F}d\vec{S}=\int...

Homework Statement Calculate the flow of ##\vec{F}=(y^2,x^2,x^2y^2)## over surface ##S## defined as ##x^2+y^2+z^2=R^2## for ##z \geq 0## with normal pointed away from the origin.Homework Equations The Attempt at a Solution The easiest was is probably with Gaussian law. I would be really happy...

Homework Statement Integrate vector field ##\vec{F}=(x+y,y+z,z+x)## over surface ##S##, where ##S## is defined as cylinder ##x^2+y^2=1## (without the bottom or top) for ##z\in \left [ 0,h \right ]## where ##h>0##Homework Equations The Attempt at a Solution Since cylinder is not closed (bottom...

Hi, All: Sorry for the length of the post, but I think it is necessary to set things up so that the post is understandable: I'm going through an argument in which we intend to show that a given vector field [ itex]R_ω [/ itex] (actually a Reeb field associated with a contact form ω) is...

I see how our line integral is a method for calculating work along a path by taking infinitesimally small 'slices' of our dot product of Force over our curve (distance). No problem here. Next we look to see if our field is conservative and if so then we know that regardless of the path the...

Homework Statement Evaluate ∫F dot ds Homework Equations F = < 1 - y/ (x^2 + y^2) , 1 + x/(x^2 + y^2) , e^z > C is the curve z = x^2 + y^2 -4 and x + y + z = 100 The Attempt at a Solution I don't think Stokes theorem applies since the vector field is undefined at the origin, so I'm...

Homework Statement By considering its divergence, test whether the following vector field could be a magnetic field: F=(a/r) cos∅ r Where a is constant. NOTE( the 'r' has the hat symbol ontop if it, unit vector i think) Homework Equations You may use that is cylindrical...

Homework Statement Given a vector field F=-y/(x^2+y^2) i +x/(x^2 +y^2) Calculate the curl of it the line integral of it in a unit circle centered at O Homework Equations The Attempt at a Solution I calculated that the curl is 0 but the line integral is 2π. I don't think this...

I need help understanding the significance of curl and divergence. I am nearly at the point where I know how to use Greene's, Stokes and the divergence theorems to convert line, surface, and iterated double and triple integrals. I know how the use the curl and div operators and about...

Say we have two vector fields X and Y and we form the projection of Y, Y' orthogonal to X. Since every vector field is associated with a curve with a corresponding parameter, is there a relation between the parameters of Y and Y'?

Hi all, my friend is writing a sci-fi/fantasy book and for it he asked me for a function that generates a vector field like picture A. So far the closest I've got is i*-(1/x)+1/-yj, which generates B. How would I generate B without using conditions? any help appreciated, thanks.

Homework Statement Assume a vector field:\textbf{F} = \widehat{r} 2r sin\phi + \widehat{\phi} r^2 cos\phi a) Verify the Stokes's theorem over the ABCD contour shown in Fig. 1 . b) Can F be expressed as the gradient of a scalar? Explain My problems results in not being able to verify...

Homework Statement Compute the line integral of \vec{v} = (rcos^{2}\theta)\widehat{r} - (rcos\theta sin\theta)\widehat{\theta} + 3r\widehat{\phi} over the line from (0,1,0) to (0,1,2) (in Cartesian coordinates) The Attempt at a Solution Well, I expressed the path as a...

Homework Statement Explain whether the divergence and curl of each of the vector fields shown below are zero throught the entire region shown. Justify your answer.https://sphotos-a-ord.xx.fbcdn.net/hphotos-prn2/1185774_4956047513788_517908639_n.jpg Homework Equations N/AThe Attempt at a...

Homework Statement Evaluate the divergence of the following vector fields (a) A= XYUx+Y^2Uy-XZUz (b) B= ρZ^2Up+ρsin^2(phi)Uphi+2ρZsin^2(phi)Uz (c) C= rUr+rcos^2(theta)Uphi Homework Equations The Attempt at a Solution Uploaded

Hey guys! So I've been trying to get my head around Divergence of a vector field. I do get the general idea, however I thought of a hypothetical situation I can't get my head around. Look at the second vector field on this page, http://mathinsight.org/divergence_idea it has a negative...

Hi PF-members. My intuition tells me that: Given a divergence free vector field \mathbf{F} , then the curl of the field will be perpendicular to field. But I'm having a hard time proving this to my self. I'know that : \nabla\cdot\mathbf{F} = 0 \hspace{3mm} \Rightarrow \hspace{3mm}...

Homework Statement Given a vector field F(x,y,z) = (yz + 3x^{2})\hat{i} + xz\hat{j} + xy\hat{k} Calculate the line integral ∫_{A}^{B}F\bullet dl where A = (0,1,3) and B = (1,2,2) Homework Equations Right, first of all, what is dl ? I've gone over all my course notes and...

Let's say we have some time-independent scalar field \phi. Obviously \phi\left(\mathbf{q}\right)-\phi\left(\mathbf{p}\right) = \int_{\gamma[\mathbf{p},\,\mathbf{q}]} \nabla\phi(\mathbf{x})\cdot d\mathbf{x}. This is of course still true if the path \gamma is the trajectory of a particle moving...

I am trying to understand why in the definition of a stationary spacetime the Killing vector field has to be timelike. It is required that the metric is time independent, i.e. the time translations x^0 \to x^0 + \epsilon leave the metric unchanged. So the Killing vector is...

Hi everyone! I've been having a hard time figuring this one out for a while, so any help will be appreciated! Homework Statement \textbf{F}= <(2zx)/(x^2+e^z*y^2), (2ze^z*y)/(x^2+e^z*y^2), log(x^2+e^z*y^2) + (ze^z*y^2)/(x^2+e^z*y^2)> (a) Where is the following vector field defined? (b) Is this...

Homework Statement "Consider the Vector field F(x,y)=<cos(sin(x)+y)cos(x)+e^x, cos(sin(x)+y)+y>. Compute the work done as you traverse the Archimedes spiral (r=θ) from (x,y)=(0,0) to (x,y)=(2∏,0). (Hint: check to see if the vector field is conservative) Homework Equations 1) F(x,y)=<P,Q>...

Homework Statement Find The Divergence Of The Vector Field: < ex2 -2xy, sin(y^2), 3yz-2x> Homework Equations I know that divergence is ∇ dot F. The Attempt at a Solution When I did it by hand I got 2xex2 + 2ycos(y2) + 3y However wolfram alpha says it should be 2xex2 +...

Homework Statement Three vector fields are listed below. Determine whether each of them is electrostatic field or magnetic field.Homework Equations F1(x, y, z) = A (9yz ex + xz ey + xy ez) F2(r,∅,z) = A [(cosx/r)er + (sinx/r) e∅] F3(r,θ,∅) = Ar2 e(-r/a) erThe Attempt at a Solution Used matrix...

Homework Statement n/a Homework Equations ∇ x F = 0 ∂Q/∂x = ∂P/∂y The Attempt at a Solution n/a Given that no sketch of the vector field is given; Is determining the curl of a vector field the most fail proof of determining whether it is conservative? I'm just...

Homework Statement Is it possible to find the vector field line expression without the use of differential equations? Say I've sketched the field and found the shape to be parabolas, how would I find the general expression by just using the points I've been given?Homework Equations The Attempt...

Given is vector field \overrightarrow{C}(\overrightarrow{r})=r calculate flux \Phi =\int_{S} \overrightarrow{C}\cdot d\overrightarrow{A} through sphere S with beginning in [0,0,0] and r=1

Let's say there is a cube sitting in the first octant. Our F(x,y,z): <ax , by, cz> and Each face of the cube is oriented to outward pointing normal. Can I just calculate the the flux of one face and then multiply this by the number of faces to get the total flux? Will flux in a cube always be...

Hello there, I've got a vector field which you can see here: Sketch of the vector field . It is: \vec{v} = \cos(x)\,\sin(y)\vec{i}-\sin(x)\,\cos(y)\vec{j} Say I want to find the circulation around the square formed by -\frac{\pi}{2} \, \leq x \leq \, \frac{\pi}{2} and -\frac{\pi}{2} \...

Homework Statement Define the vector field F = sec(x) i + k (a) Express the flow lines of F in equations form. (b) Express in equations form the particular flow line through the point (0, 3, 2). My next question is a bonus question. I'm just reading up on this now but if someone could...

Hello there, What is wrong with my way of finding stream lines of a vector field? Say I have this vector field: \vec{v} = x\,y\,\vec{i} + y\,\vec{j} You can see a plot here: http://kevinmehall.net/p/equationexplorer/vectorfield.html#xyi+yj%7C%5B-10,10,-10,10%5D It appears as if the stream...

Simple question. It came out of lecture, so it's not homework or anything. My professor said that the curl of a vector field is always perpendicular to itself. The example he gave is that the magnetic vector potential A is always perpendicular to the direction of the magnetic field B. (I haven't...