Divergence theorem Definition and 176 Threads

Can anyone tell me whether or not the divergence theorem requires a conservative vector field? On a practice exam my professor gave a vector field that was nonconservative (I checked the curl) and proceeded to perform the divergence theorem to find the flux. On one of my homework problems I...

S\int\int F*Nds F(x,y,z) = (xy^2 + cosz)i + (x^2*y + sinz)j + e^(z)*k s: z = 1/2\sqrt{x^2 + y^2} , z = 8 divF = y^2 + x^2 +e^z Q\int\int\int (y^2 + x^2 + e^k)dV This is as far as I got, I have no idea how to do the limits for this triple integral thanks in advance guys.

Homework Statement Show divergence theorem works For the vector field E = \hat{r}10e^{-r}-\hat{z}3z Homework Equations \int_{v}\nabla \cdot E dv = \oint_{s} E \cdot ds The Attempt at a Solution \nabla \cdot E = 1/r \frac{d}{dr}(rAr)+1/r\frac{dA\phi}{d\phi}+\frac{dAz}{dz}...

i having a some trouble verifying the Divergence theorem for A=y^2zex-2x^3yey+xyz^2ez with respect to V being a unit cube

The fundamental theorem of calculus is basically the divergence theorem but dealing with a ball in R^1 instead of a ball in R^3. The fundamental theorem of Calculus relates the stuff inside the ball to its boundary, just like how the divergence theorem relates the stuff inside a volume with its...

Are there versions of the divergence theorem that don't require a compact domain? In my reference literature the divergence theorem is proved under the assumption that the domain is compact.

Question Evaluate both sides of the divergence theorem for V =(x)i +(y)j over a circle of radius R Correct answer The answer should be 2(pi)(R^2) My Answer the divergence theorem is **integral** (V . n ) d(sigma) = **double intergral** DivV d(tau) in 2D. Where (sigma)...

Homework Statement Find \int_{s} \vec{A} \cdot d\vec{a} given \vec{A} = ( x\hat{i} + y\hat{j} + z\hat{k} ) ( x^2 + y^2 + z^2 ) and the surface S is defined by the sphere R^2 = x^2 + y^2 + z^2 directly and by Gauss's theorem. Homework Equations \int_{s} \vec{A} \cdot d\vec{a} =...

[SIZE="5"]The Background: I'm trying to construct a rigorous proof for the divergence theorem, but I'm far from my goal. So far, I have constructed a basic proof, but it is filled with errors, assumptions, non-rigorousness, etc. I want to make it rigorous; in so doing, I will learn how to...

Consider the volume V bounded below by the x-y plane and above by the upper half-sphere x^2 + y^2 + z^2 = 4 and inside the cylinder x^2 + y^2 = 1 Given vector field: A = xi + yj + zk Use the divergence theorem to calculate the flux of A out of V through the spherical cap on the cylinder...

This question is throwing me for a loop. Q: If u = x^2 in the square S = \{ -1<x,y<1\} , verify the divergence theorem when \vec w = \Nabla u : \int\int_S div\,grad\,u\,dx\,dy = \int_C \hat n \cdot grad\,u\,ds If a different u satisfies Laplace's equation in S , what is the net flow...

Ok I am stuck yet again. Below is a synopsis of everything I have done. D. J. Griffiths, 3rd ed., Intro. to Electrodynamics, pg. 45, Problem #1.42(a) and (b): (a) Find the divergence of the vector function: \vec{v} = s(2 + sin^2\phi)\hat{s} + s \cdot sin\phi \cdot cos\phi \hat{\phi} +...

Let \vec{v} = r^{2} \hat{r}. Show that the divergence theorm is correct using 0 <= r <= R , 0 <= \theta <= \pi , and 0 <= \phi <= 2\pi . $ \int \nabla \cdot \vec{v} d \tau = \int \vec{v} \cdot d \vec{a} $ First the divergence of \vec{v}. \nabla \cdot \vec{v} = 2r. Then the volume...

This problem is either really easy, or I'm really dumb, and since there are no answers to check my work I figured someone here might want to help :) Q: w=(x,y,z) what is the flux \int \int w \cdot n\,\, dS out of a unit cube and a unit sphere? Compute both sides in the divergence theorem...

question says answer in whichever would be easier, the surface integral or the triple integral, then gives me(I'm in a mad hurry, excuse the lack of formatting...stuff) the triple integral of del F over the region x^2+y^2+z^2>=25 F=((x^2+y^2+z^2)(xi+yj+zk)), so del F would be 3(x^2+y^2+z^2)...

Suppose D \subset \Re^3 is a bounded, smooth domain with boundary \partial D having outer unit normal n = (n_1, n_2, n_3) . Suppose f: \Re^3 \rightarrow \Re is a given smooth function. Use the divergence theorem to prove that \int_{D} f_{y}(x, y, z)dxdydz = \int_{\partial D} f(x, y...

Let Q denote the unit cube in \Re^3 (that is the unite cube with 0<x,y,z<1). Let G(x,y,z) = (y, xe^z+3y, y^3*sinx). Verify the validity of the divergence theorem. \int_{Q} \bigtriangledown} \cdot G dxdydz = \int_{\partial Q} G \cdot n dS I am not sure how to evaluate the right side. Any...

Hi everyone! I am having some trouble with this particular problem on Vector Calculus from Griffith's book. The question is: Check the divergence theorem for the vector function(in spherical coordinates) \vec v = r^2\cos\theta\hat r + r^2\cos\phi \hat \theta - r^2\cos\theta\sin\phi\hat...

Im having a bit of a problem understanding the crucial part of the divergence theorem from Electromagnetic Fields and Waves by Lorrain and Corson. Ill try descibe the set up of the problem 1st and see if anyone can help me in any way before i continue with the electromagnetism course I am doing...

so we know that the divergence theorm, was proved by Gauss, and also proved by ostrogradsky. but infact, the divergence theorm was discovred by Lagrange...correct? now, did these 3 guys prove it differently? I'm sure it couldn't be exactly the same way right? basically, I've been...

Hi, I'm having trouble proving the following result: \int_{V} (\nabla\times\vec{A}) dV = -\int_{S} (\vec{A}\times\vec{n}) dS I'm not sure how I should Stokes' and/or the Divergence Theorem in proving this, or if you should use them at all. Thanks in advance.

ok this probley seems simple but i just need to see how to do it, ok well how do u evaluate this... find the flux of the vector field... \vec{F}=<x,y,z> throught this surface above the xy-plane.. z = 4-x^2-y^2 how do u evaluate this with surface integrals method and the divergence...

Hi! We are nearing the end of our course --- culminating in Stokes and Divergence Theorems for surface integrals, and I am having some difficulty with the following 1. F(x,y,z) = <x^3y, -x^2y^2, -x^2yz> where S is the solid bounded by the hyperboloid x^2 + y^2 - z^2 =1 and the planes z...

I was wondering if someone could give me a hand here with 2b) on the following link. http://www.am.qub.ac.uk/users/j.mccann/teaching/ama102/2003/assignments/assign_8.pdf For part a) I got it to be equal to 3x^2+3y^2+3z^2+2y-2xy, and I'm hoping that's right! However, for part b) I can't...

I need help evaluating both sides of the divergence theorem if V=xi+yj+zk and the surface S is the sphere x^2+y^2+z^2=1, and so verify the divergence theorem for this case. Is the divergence theorem the triple integral over V (div V) dxdydz= the double integral over S (V dot normal)dS? If so...

I need help evaluating both sides of the divergence theorem if V=xi+yj+zk and the surface S is the sphere x^2+y^2+z^2=1, and so verify the divergence theorem for this case. Is the divergence theorem the triple integral over V (div V) dxdydz= the double integral over S (V dot normal)dS? If so...