Divergence theorem Definition and 184 Threads

Homework Statement Verify the divergence theorem for F(x,y,z) = (x,y,2z^2) and T is the region bounded by the paraboloid z=x^2+y^2 and the plane z=1. Homework Equations F(ds) = div(F)dV The Attempt at a Solution I have successfully evaluated the integral and come up with an...

Homework Statement F(x,y,z) = (2x-z) i + x2y j + xz2 k and the volume is defined by [0,0,0] and [1,1,1]. Homework Equations flux integral = \int\int\int div F dV The Attempt at a Solution \int\int\int div F dV = \int\int\int (2+x2-2xz)dxdydz = 2 + 1/3 - 1/2 = 11/12 But I...

Homework Statement Evaluate the double integral over M (F \circ dS) where M is the surface of the sphere of radius 3 centered around the origin. (Sorry! I couldn't figure out how to use math symbols!) Homework Equations double integral(F\bulletdS)=triple integral (\nabla\bullet F)dV due...

Let W be the solid bounded by the paraboloid x = y^2 + z^2 and the plane x = 16. Let = 3xi + yj + zk a. Let S1 be the paraboloid surface oriented in the negative x direction. Find the flux of the vector field through the surface S1. b. Let S be the closed boundary of W. Use the Divergence...

Homework Statement See figure attached for problem statement. Homework Equations The Attempt at a Solution See figure attached for my attempt. What I decided to do was add a surface z=0 so that S became a closed surface. Then I preformed the integration using divergence...

Homework Statement what is the divergence of <y,z,x>? Homework Equations The Attempt at a Solution is the answer 0? seems too easy, lol, because the actual question is "compute the surface integral for F dot prod dS over domain T where T is the unit sphere and F = <y,z,x>"...

Homework Statement This problem I have been set is to find real life applications of divergence theorem. I have to show the equivalence between the integral and differential forms of conservation laws using it. 2. The attempt at a solution I have used div theorem to show the equivalence...

Hi. I've been reading PF for quite a while and have decided to ask my first question. Please be gentle. (I'm a retired computer programmer, not a student)... I've been learning Gauss' divergence theorem and I understand what "flux density" is when considering things like fluid transport or...

I've tried to make sense of this conjecture but I can't wrap my head around it. We've been learning about the divergence theorem and the Neumann problem. I came across this question. Use the divergence theorem and the partial differential equation to show that...

Homework Statement This is from a fluid mechanics text. There are no assumptions being made (i.e., no constants): Show that \frac{\partial{}}{\partial{t}}\int_V e\rho \,dV + \int_S e\rho\mathbf{v}\cdot\mathbf{n}\,dA = \rho\frac{De}{Dt}\,dV\qquad(1) where e and \rho are scalar quantities...

In the notes attached to this thread: https://www.physicsforums.com/showthread.php?t=457123 On page 110, how has he gone from equation (369) to eqn (370). He claims to have done it by "integration by parts using the divergence theorem to eliminate derivatives of \delta g_{ab} if present". (The...

Homework Statement Let D be the region x^2 + y^2 + z^2 <=4a^2, x^2 + y^2 >= a^2, and S its boundary (with outward orientation) which consists of the cylindrical part S1 and the spherical part S2. Evaluate the ux of F = (x + yz) i + (y - xz) j + (z -((e^x) sin y)) k through (a) the whole...

Homework Statement Consider the following vector field in cylindrical polar components: F(r) = rz^2 r^ + rz^2 theta^ By directly solving a surface integral, evaluate the flux of F across a cylinder of radius R, height h, centred on the z axis, and with basis lying on the z = 0 plane. Using the...

Homework Statement Use Divergence theorem to determine an alternate formula for \int\int u \nabla^2 u dx dy dz Then use this to prove laplaces equation \nabla^2 u = 0 is unique. u is given on the boundary.Homework Equations u \nabla^2 u = \nabla * (u \nabla u) -(\nabla u)^2 The Attempt at...

Homework Statement The problem statement has been attached with this post. Homework Equations I considered u = ux i + uy j and unit normal n = nx i + ny j. The Attempt at a Solution I used gauss' divergence theorem. Then it came as integral [(dux/dx) d(omega)] + integral...

From Partial Differential Equations: An Introduction, by Walter A. Strauss; Chapter 1.5, no.4 (b). Homework Statement "Consider the Neumann problem (delta) u = f(x,y,z) in D \frac{\partial u}{\partial n}=0 on bdy D." "(b) Use the divergence theorem and the PDE to show that...

Hi everyone, so let me introduce the scalar function \Phi = -(x2+y2+z2)(-1/2) which some of you may recognize as minus one over the radial distance from the origin. When I compute \nabla2\Phi is get 0. Now if I do the following integral on the surface S of the unit sphere x2+y2+z2= 1 ...

Homework Statement Check the divergence theorem using the function: \mathbf{v} = y^2\mathbf{\hat{x}} + (2xy + z^2) \mathbf{\hat{y}} + (2yz)\mathbf{\hat{z}} Homework Equations \int_\script{v} (\mathbf{\nabla . v }) d\tau = \oint_\script{S} \mathbf{v} . d\mathbf{a} The Attempt at a Solution...

Homework Statement Let the surface, G, be the paraboloid z = x^2 + y^2 be capped by the disk x^2 + y^2 \leq 1 in the plane z = 1. Verify the Divergence Theorem for \textbf{F}(x,y,z) = 2x\textbf{i} - yz\textbf{j} + z^2\textbf{k} Homework Equations I have solved the problem using the...

Homework Statement 1. Consider a cube with vertices at A=(0,0,0) B=(2,0,0) C=(2,2,0) D=(0,2,0) E=(0,0,2) F=(2,0,2) G=(2,2,2) H=(0,2,2) A)Calculate the flux of the vector fieldF=xi through each face of the cube by taking the normal vectors pointing outwards. B)Verify Gauss's divergence theorem...

Homework Statement Verify Gauss Divergence Theorem ∭∇.F dxdydz=∬F. (N)dA Where the closed surface S is the sphere x^2+y^2+z^2=9 and the vector field F = xz^2i+x^2yj+y^2zk The Attempt at a Solution I have tried to solve the left hand side which appear to be (972*pi)/5 However, I...

Another question of a practice test. How do I use the Divergence theorem to find the outward flux of the field F = (x3,x2y,xy) out through the surface of the solid U = (x,y,z): 0 < y < 5-z, 0 < z < 4-x2. The answer is 4608/35.

I have been contemplating my confusion about my intuition regarding GR and believe I have tracked down the primary source of confusion. The classical theories I have been taught assumed flat space with independent time and used the divergence theorem to derive inverse squared laws for fields...

Note: I've attached images of my work at the bottem of this post. I've calculated the flux through a given surface by using The Divergence Theorem and by using the regular flux method. These methods give different results, however. This leads me to assume one of the following is...

Use the divergence theorem to show that \oint\oints (nXF)dS = \int\int\intR (\nablaXF)dV. The divergence theorem states: \oint\oints (n.F)dS = \int\int\intR (\nabla.F)dV. The difference is switching from dot product to cross product. I have no idea how to start. Can someone please point...

Verify the divergence theorem when F=xi+yj+zk and sigma is the closed surface bounded by the cylindrical surface x^2+y^2=1 and the planes z=0, z=1. I've done the triple integral side of the equation and got 3pi but don't know how to solve the flux side of the equation \oint\ointF.ds. Any...

Alright so I found div F=3x2+3y2+3z2 The integral then becomes the triple integral of the divergence of F times the derivative of the volume. Changing into spherical coordinates, the integral becomes 3\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{1}p^{4}sin{\phi}dpd{\phi}d{\theta} which ends up...

Hi, I'm having some trouble understanding this theorem in Lang's book, (pp. 497) "Fundamentals of Differential Geometry." It goes as follows: \int_{M} \mathcal{L}_X(\Omega)= \int_{\partial M} \langle X, N \rangle \omega where N is the unit outward normal vector to \partial M , X...

Homework Statement Use the divergence theorem to find the outward flux of a vector field F=\sqrt{x^2+y^2+z^2}(x\hat{i}+y\hat{j}+z\hat{k}) across the boundary of the region 1\leq x^2+y^2+z^2 \leq4 Homework Equations The Gauss Divergence Theorem states \int_D dV \nabla \bullet...

Homework Statement Let E be the solid region defined by 0 \leq z \leq 9+x^2+y^2 and x^2+y^2 \leq 16. Let S be the boundary surface of E, with positive (outward) orientation. Also, consider the vector field F(x,y,z)=<x,y,x^4+y^4+z> There are five parts to the problem A) Compute the...

Homework Statement the problem is to calculate \int (\nabla \cdot \vec{F}) d\tau over the region x^2 + y^2 + x^2 \leq 25 where \vec{F} = (x^2 + y^2 + x^2)(x\hat{i} +y\hat{j} + z\hat{k}) in the simplest manner possible.Homework Equations divergence theorem!The Attempt at a Solution...

In h.m. schey, div grad curl and all that, II-25: Use the divergence theorem to show that \int\int_S \hat{\mathbf{n}}\,dS=0, where S is a closed surface and \hat{\mathbf{n}} the unit vector normal to the surface S. How should I understand the l.h.s. ? Coordinatewise? The r.h.s. is not...

http://img60.imageshack.us/img60/9696/21249035.jpg I am stuck at the last step. Can anyone give some hints? Thanks in advanced.

Homework Statement Use the divergence theorem to evaluate \int\int_{\sigma}F . n ds Where n is the outer unit normal to \sigma we have F(x,y,z)=2x i + 2y j +2z k and \sigma is the sphere x^2 + y^2 +z^2=9 Homework Equations \int\int_{s}F . dA = \int\int\int_{R}divF dV The...

Homework Statement Water in an irrigation ditch of width w = 3.0 m and depth d = 2.0 m flows with a speed of 0.40 m/s. For each case, sketch the situation, then find the mass flux through the surface: (a) a surface of area wd, entirely in the water, perpendicular to the flow; (b) a surface...

Homework Statement use the divergence theorem to evaluate the integral F dot dA F = (2x-z)i + x2yj + xz2k s is the surface enclosing the unit cube and oriented outward Homework Equations The Attempt at a Solution is the the region from -1 to 1 for x y and z div F = x2 +...

Use the divergence theorem to compute the surface integral F dot dS , where F=(xy^2, 2y^2, xy^3) over closed cylindrical surface bounded by x^2+z^2=4 and y is from -1 to 1. I've tried doing it and got 32pi/3 (i guess its wrong, so how to do it?) Is it ok to compute Div F in terms of xyz and...

Evaluate the flux integral using the Divergence Theorem if F(x,y,z)=2xi+3yj+4zk and S is the sphere x^2+y^2+z^2=9 answer is 324pi so far i took the partial derivitavs of i j k for x y z and added them to get 9. so i have the triple integral of 9 dzdxdy i think u have to use polar...

Homework Statement F = xi + yj + zk, s = x^2 + y^2 + z^2 Homework Equations The Attempt at a Solution div F = 1+1+1=3 area of sphere = 4pi i can just multiply them to get 12pi as an answer right?

Given F = xyz i + (y^2 + 1) j + z^3 k Let S be the surface of the unit cube 0 ≤ x, y, z ≤ 1. Evaluate the surface integral ∫∫(∇xF).n dS using a) the divergence theorem b) using Stokes' theorem --- Since the divergence theorem involves a dot product rather than a curl,how would it...

Homework Statement http://img16.imageshack.us/img16/88/fluxm.th.jpg Homework Equations The Attempt at a Solution I've tried to find the divergence of F and I got 3x^2 + 3y^2 + 3z^2 and as this is a variable I need to set up the integral... how do I set the integral

Homework Statement By using divergence theorem find the flux of vector F out of the surface of the paraboloid z = x^2 + y^2, z<=9, when F = (y^3)i + (x^3)j + (3z^2)kHomework Equations Divergence theorem equation stated in the attempt partThe Attempt at a Solution

The question I was given asks to verify the divergence theorem by showing that both sides of the theorem show the same result. With the divergence theorem obviously being \iint_S\mathbf{F}\cdot\mathbf{n}\,dS = \iiint_V \nabla\cdot\mathbf{F}\,dV . The vector field is...

Homework Statement Use the divergence theorem in three dimensions \int\int\int\nabla\bullet V d\tau= \int \int V \bullet n d \sigma to evaluate the flux of the vector field V= (3x-2y)i + x4zj + (1-2z)k through the hemisphere bounded by the spherical surface x2+y2+z2=a2 (for z>0)...

I don't know whether it was proved or can be prove. I don't know whether it is useful. maybe it can be used in string theory or some other things. any comment or address will be appreciated.

I need to show that, using Gauss' Theorem (Divergence Theorem), i.e. integration by parts, that: \int_V dV e^{-r} \nabla \cdot (\frac{\vec{\hat{r}}}{r^2}) = \int_V dV \frac{e^{-r}}{r^2} any ideas on where to start?

This may well be the wrong place to post this so apologies for that if it's the case. Anyway, I'm stuck on this question, any help appreciated Use Gauss' Theorem to show that: (i) If \psi($\mathbf{r}$) ~ \frac{1}{r} as r \rightarrow \infty , then, \int_V {\psi \nabla^{2} \psi}...

Homework Statement Check the Divergence Theorem \int_V(\nabla\cdot\bold{v})\,d\tau=\oint_S\bold{v}\cdot d\bold{a} using the function \bold{v}=<y^2, 2xy+z^2, 2yz> and the unit cube below. Now when I calculate the divergence I get (\nabla\cdot\bold{v})=2y+2x+2y but Griffith's...

I was told this problem could be done with divergence theorem, instead of as a surface integral, by adding the unit disc on the bottom, doing the calculation, then subtracting it again. Homework Statement Homework Equations The Attempt at a Solution for del . f I get i + j =...

Homework Statement Evaluate the surface Integral I=\int\int_S\vec{F}\cdot\vec{n}\,dS where \vec{F}=<z^2+xy^2,x^100e^x, y+x^2z> and S is the surface bounded by the paraboloid z=x^2+y^2 and the plane z=1; oriented by the outward normal.The Attempt at a Solution...