Surface integral Definition and 253 Threads

Given that a surface integral of a function, f(x,y,z), is written as \int\int f(x,y,z) dS where dS= |df/dx x df/dy| dA, how can this be generalized into more dimensions? In other words, is it possible to find a way to convert dS into a differential piece of area for more than 3 dimensions? What...

I've been trying to work through this and see whether you can take an "area" in the complex plane, have x,y vary in some interval, and integrate complex functions over that "area." The math doesn't seem to work out; plus intuitively, if you're going to sum up a function in a complex variable...

Homework Statement Find the centroid of the hemisphere, z=sqrt(a-x^2-y^2). Homework Equations z(bar)=(1/m)* Surface integral(z dS) dS= magnitude of the magnitude of normal vector * dA The Attempt at a Solution I use the gradient to the hemisphere to get the magnitude of the normal...

Homework Statement \iint \mathbf{F} \cdot \mathbf{dS} = \iint \mathbf{F} \cdot |\mathbf{r_u} \times \mathbf{r_v} | \mathbf{dA} = \iint \mathbf{F} \cdot |\mathbf{r_u} \times \mathbf{r_v} | \mathbf{\hat{n}}dA = \iint \mathbf{F} \cdot |\mathbf{r_u} \times \mathbf{r_v} | \frac{\mathbf{r_u}...

I am not able to find any good reference to answer my question, so I will post here how does divergence theorem translates to 4 dimensional curved spacetime. I understood how volume integral changes but I am not able to understand how surface integral changes. I will be glad if some one...

Homework Statement Evaluate the surface integral \iint_S y \; dS S is the part of the sphere x^2 + y^2 + z^2 = 1 that lies above the cone z=\sqrt{x^2 + y^2}The Attempt at a Solution I know to use spherical coord so I did r = <\rho cos\theta sin\phi, \rho sin\theta sin\phi, ?> The book did...

Homework Statement My book says proves this formula \iint_S f(x,y,z) dS = \iint_D f(x,y,g(x,y)) \sqrt{\left (\frac{\partial z}{\partial x} \right )^2 + \left (\frac{\partial z}{\partial y} \right )^2 + 1 } \;dA Question How do they know that every parametrization falls nicely...

surface integral - urgent please help Homework Statement Let S be the surface x=z, x^2+y^2<=1, find ∫∫S(x^2+y^2)dS Homework Equations ∫∫SFdS = ∫∫S F(ruxrv The Attempt at a Solution parametrized surface x=rcostheta y=rsintheta z=rcostheta i don't know what to do about the partial...

Homework Statement Calculate the surface integral I = \int\int f dS of the function f(x,y,z) = \sqrt{1/2 + y^{2}} over the surface S given by x^{2} + 2*y^{2} = 1, 0 \leq z \leq x^{2} + y^{2}. (Clue: parametrize the surface.) Homework Equations - The Attempt at a Solution The surface...

Homework Statement See figure attached for problem statement. Homework Equations The Attempt at a Solution See figure attached for my attempt. I tried to solve it without using divergence theorem, just a straight forward surface integral. I got up to this point and got...

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 See figure attached for problem statement. Homework Equations The Attempt at a Solution See figure attached for my attempt so far. I'm confused as to how to do this problem. Either plane I project the surface into (xz or yz) the integral looks pretty nasty...

I'm not sure why, but I'm having issues with these in general. Specifically surface integrals over vector fields. The function is zk. The surface is the paraboloid z=x2+y2 between the planes z=1 and z=2. I parametrized it like so: \vec{r}(u,v)=(u,v,u^{2}+v^{2}) \vec{T_{u}}=(1,0,2u)...

Watching video http://www.khanacademy.org/video/introduction-to-the-surface-integral?playlist=Calculus at 20.10 the guy introduces the concept of what it means for each part of the surface to have a value of a new function f(x,y,z). Could some one explain what this new integral would...

I have to integrate this function: f(x,y,z)=y+x Over the region S which is a tetrahedron defined by points (0,0,0), (2,0,0), (0,2,0), (0,0,2). So after I drew it out I saw that three of the faces were right up against the XZ, YZ, and XY planes. I'm getting stuck on parameterizing the...

Homework Statement Here's a picture of the question: Here's the solution: The Attempt at a Solution I can't really make complete sense of some things around this... Like how did the integral become 36pi based on what is written in the solution? Why is z equal to 0.. What...

Hello everybody! Although this may sound like a homework problem, I can assure you that it isn't. To prove it, I will give you the answer: 40pi. So.. I'm self-studying some electrodynamics. I'm using the third edition of Griffiths, and I have a quick question. For those who own the book and...

Let's assume that I have a surface defined parametrically by a vector \mathbf{\ r}(r,\theta) Is it acceptable to simplify the Stokes theorum surface integral to: \iint\limits_D\,\nabla \times f \cdot\!(r_r\times\!r_\theta) \,\, \!r \mathrm{d}r\,\mathrm{d}\theta Where r_r and r_theta are...

Homework Statement Compute surface integral. F = <z, x, y> f(x,y) = x + y, 0 <= x <= 1, 0 <= y <= 1.Homework Equations The Attempt at a Solution Well this is what I tried: <z, x, y > * < -fx, -fy, 1> = -z - x + y = -(x+y) - x + y = -2x Then I integrated it using the bounds given and got -1...

Homework Statement Integrate G(x,y,z) = x(y^2+4)^(1/2) over y^2 + 4z = 16 cut by the plane x=0, x=1, and z=0. Homework Equations The Attempt at a Solution How do you parametrize the parabolic cylinder y^2 + 4z = 16? Thanks in advance.

I am agonizing about the following integral identity: \frac{d}{dt} \int \int_{g(x,y) \leq t} f(x,y) dx dy = \int_{g(x,y)=t} f(x,y) \frac{1}{\left| \nabla g(x,y) \right|} ds, where ds is the line element. Clearly, using the Heavisite step function, the condition g(x,y) \leq t is...

Homework Statement Find \iint\limits_S \mathbf{F}\cdot \hat n\, dA Homework Equations \mathbf{F} = [1, 1, a] S: s^2+y^2+4z^2 = 4, z \geq 0 The Attempt at a Solution I parameterized in spherical coordinates x=4\sin{\phi}\cos{\theta} y=4\sin{\phi}\sin{\theta} z=\cos{\phi} Then, I found...

Homework Statement Compute ∫∫SF.dS F(x,y,z)=<y,x2y,exz> over x2+y2=9, -3<=z<=3, outward pointing normal. The Attempt at a Solution I parameterized the surface in cylindrical coordinates: Φ(z,θ)=<3cosθ,3sinθ,z>. The normal vector of this surface is n(z,θ)=<0,0,1>x<-3sinθ,3cosθ,0>=...

Homework Statement Let er be the unit radial vector and r=sqrt(x2+y2+z2). Calculate the integral of F=e-rer over: a. The upper-hemisphere of x2+y2+z2=9, outward pointing normal b. The octant x,y,z>=0 of the unit sphere centered at the origin The Attempt at a Solution...

Hello all, Evaluate \int\int r. da over the whole surface of the cylinder bound by x^{2} + y^{2} = 1, z=0 and z=3. \vec{r} = x \hat{x} + y \hat{y} + z \hat{z} Sorry for the awkward formatting, this site is giving me trouble. Anyways, it seems to me that since I have 3 dimensions...

Homework Statement Find the value of the surface integral \intA \bullet da, where A = xi - yj + zk, over the surface defined by the cylinder c2 = x2 + y2. The height of the cylinder is h. Homework Equations I found the answer quite easily using Gauss's theorem, as the divergence of the...

Homework Statement Find the area integral of the surface z=y^2+2xy-x^2+2 in polar form lying over the annulus \frac{3}{8}\leq x^2+y^2\leq1 Homework Equations The equation in polar form is r^2\sin^2\theta+2r^2\cos\theta\sin\theta-r^2\cos^2\theta+2...

Assume that I know the value of \iint_{S} \overrightarrow{F} \cdot \hat{n} dS over any surface in \mathbb{R}^3, where \overrightarrow{F}(x,y,z) is a vector field in \mathbb{R}^3 and \hat{n} is the normal to the surface at any point considered. Using that I would like to compute...

If we integrate a vector field over a surface, \int_S \vec{F} \cdot \vec{dS}, we get the flux through that surface. What does it mean if the integrad were a vector instead, \int_S \vec{F} dS? I can't picture the Riemann sum.

Homework Statement Evaluate the surface integral, the double integral of zdS if the region is the patch of surface defined by x^2 + y^2 - z^2 = -1 in the first octant with z less than or equal to 4. The attempt at a solution I really don't know where to begin. I believe the equation is...

Apostol page 429, problem 4 Is there a better way to set up this problem or have I made a mistake along the way? (ie easier to integrate by different parameterization) Homework Statement Find the surface area of the surface z^2=2xy lying above the xy plane and bounded by x=2 and y=1...

Homework Statement Seawater has density 1025 kg/m^3 and flows in a velocity field v=yi+xj, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the hemisphere x^2+y^2+z^2=9, z≥0Homework Equations Surface integral of F over...

Homework Statement Find the mass of a spherical surface S of radius R such that at each point (x, y, z) in S the mass density is equal to the distance of (x, y, z) to some fixed point (x_0, y_0, z_0) in S. Homework Equations Integral of a scalar function over a surface. The Attempt at...

Hi everyone, As part of a project, I am required to numerically compute the expression b_{i}^{e} &=& \frac{E_{0}^{i}k_0^2(\epsilon_r-1/\mu_r)}{2\Delta^e}\left[\iint\limits_{\Omega^e}(a_i^e + b_i^e x + c_i^e y)e^{-jk_0 x} dx dy\right] \nonumber\\&&- \frac{jk_0 E_0^i r'}{2\Delta^e...

Let S be a parametrised surface given by (x, y, z) = R(u, v) := (u2, v2, u + v), for 0 \leq u \leq 1 and 0 \leq v \leq 1. How do I find the integral K := \int\int_S z/2 dA.

Homework Statement This is not a HW prob. Just a question. When doing surface integrals, why should the area element ds be dotted with the normal. I don't get it . \ointA.n ds Homework Equations The Attempt at a Solution

what is the different between line integral and surface integral? If we parameterize curve by x=t , y=t , what is the range of t ? Is it 0<= t <=1? why?

Homework Statement Air is flowing with a speed of 0.4m/s in the direction of the vector (-1, -1, 1). Calculate the volume of air flowing per second through the loop which consists of straight lines joining, in turn, the following (1,1,0), (1,0,0), (0,0,0), (0,1,1), (1,1,1) and (1,1,0)...

Homework Statement So trying to find the Integral of A.n dS where A is (y,2x,-z) and S is the surface of the plane 2x+y = 6 in the first octant cut off by the plane z=4 Homework Equations The Attempt at a Solution So i always solve these by projection...but I am a bit confused...

Homework Statement What is the area of the portion of the sphere x^{2}+y^{2}+(z-a)^{2}=a^{2} that is inside the paraboloid z=x^{2}+y^{2} Homework Equations \int\int_{S} dS The Attempt at a Solution I used this \int\int_{S} dS=\int\int_{R}\sqrt{f^{2}_{x}+f^{2}_{y}+1}dx dy...

Homework Statement Let S be the boundary of the region {(x,y,z) : 0<z<h , a^(2)<x^(2)+y^(2)<b^2 , and a<b F is defined at the point with position vector r=(x,y,z) by F(r)=exp (x^2+y^2)r Evaluate the surface integral \int F.n dS Where n is the outward pointing unit normal to...

I asked for some help on how to do surface integrals, and this is what I understood from that applied to a question, which i am getting the wrong answer to, so can someone please let me know which part of this I am doing wrong? I need to find the integral of the curl of V, over the triangle...

Homework Statement I am to use a substitution of variables u = x, v = x + 2y to evaluate the surface integral int(0,1/2)int(0,1-y) exp(x/(x+y))dxdy where int(a,b) means integral sign with lower limit a and upper limit b. Homework Equations The Attempt at a Solution I...

Homework Statement I’m trying to integrate the surface of a cylinder. I know when integrating the surface of a cylinder the surface element is: ρdØdz Where ρ² + z² = r² And for a sphere it is: r²sinθdθdØ In a sphere r=ρ But in a cylinder when I’m integrating its surface...

Homework Statement Evaluate ∫∫ z² dS where S is the part of the surface of the sphere x² + y² + z² = a² with z>= 0The Attempt at a Solution I get: ∫∫ (a² -x² - y²) . (1 + 4a²) a. da.dθ where dS = a. da.dθ I think I'm making a mistake somewhere, perhaps I'm getting confused with a² and...

Homework Statement Find the flux of the vector field F=xi+yj+2k through the surface of the inclined parametric cylinder shown below. Assume that the surface is oriented outward. Homework Equations The Attempt at a Solution

Find and evaluate numerically x^10 + y^10 + z^10 dSx^2 + y^2 + z^2 =4It says you're supposed to use gauss' divergence thm to convert surface integral to volume integral, then integrate volume integral by converting to spherical coordinates... I can do the second part but how do i use gauss'...

Homework Statement Calculate §§ A.n dS if A= 2y(x^2)i-(y^2)j + 4xzk over the region in the first octant bounded by (y^2)+(z^2) = 9 and x = 2 Homework Equations The Attempt at a Solution Let n = (yj + zk) / 3 then A.n = [-(y^3) +4xz^3] / 3...

Homework Statement Find the surface integral of \vec{r} over a surface of a sphere of radius a and center at the origin. Also find the volume integral of \nabla \bullet \vec{r}.Homework Equations Divergence theorem.The Attempt at a Solution First I did the volume integral part of the divergence...

Homework Statement I have to evaluate the surface integral of the following function over the top hemisphere of a sphere.Homework Equations \sigma (x,y,z) = \frac{\sigma_0 (x^2+y^2)}{r^2} z = \sqrt{r^2-x^2-y^2} \iint G[x,y, f(x,y)] \sqrt{1+ \frac{\partial f}{\partial x}+ \frac{\partial...