Surface integral Definition and 261 Threads

In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate a scalar field (that is, a function of position which returns a scalar as a value) over the surface, or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a surface as shown in the illustration.
Surface integrals have applications in physics, particularly with the theories of classical electromagnetism.

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  1. D

    Surface integral of normal vector

    Hi. Does anyone know how to prove that \int \int dS \hat \mathbf n = \int \mathbf r \times d\mathbf r i.e., the surface integral of the unit normal vector equals the line integral on the r.h.s. ?
  2. M

    Surface Integral over Half Shell

    From Div,Grad, Rot and all that Disclaimer: Sorry, why do the Latex tags not work? Homework Statement "An electrostatic field is given by \vec{E} = \lambda (\hat{\vec{i}}yz + \hat{\vec{j}}xz + \hat{\vec{k}}xy), where \lambda is a constant. Use Gauss' law to find the total charge enclosed by a...
  3. A

    Surface Integral: Solving for Unknown Functions | Homework Help

    Homework Statement Homework Equations The Attempt at a Solution How to integrate this function?
  4. S

    Generalization of Surface Integral

    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...
  5. P

    Is there any parallel in Complex Analysis to a surface integral?

    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...
  6. W

    Centroid of a Hemisphere in Spherical Coordinates

    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...
  7. F

    How do you remember the order for the cross product in this surface integral?

    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}...
  8. V

    Divergence theorem/ Surface integral

    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...
  9. F

    Surface integral parametrization

    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...
  10. F

    Surface integral problem formula question

    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...
  11. K

    Surface Integral of x^2+y^2 over Parametrized Surface x=z, x^2+y^2<=1

    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...
  12. P

    Parametrize a surface and calculating a surface integral

    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...
  13. jegues

    Tricky Surface Integral Homework | Solved Step by Step"

    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...
  14. jegues

    Surface Integral (Divergence Theorem?)

    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...
  15. jegues

    How to do this surface integral

    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...
  16. L

    Troubleshooting Surface Integral Issues

    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)...
  17. A

    What does the new integral in surface integral theory represent?

    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...
  18. L

    Surface Integral Over Tetrahedron

    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...
  19. F

    Surface integral over a cylinder

    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...
  20. S

    Surface integral in cylindrical coordinates

    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...
  21. W

    Stokes Theorum Surface integral

    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...
  22. B

    Surface Integral Homework: Compute F = <z,x,y> f(x,y)

    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...
  23. M

    Surface Integral: Integrating G(x, y, z) over Parabolic Cylinder

    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.
  24. D

    Surface integral to line integral

    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...
  25. TheFerruccio

    Surface Integral: Evaluating with Spherical Coordinates

    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...
  26. K

    Surface Integral of F on Cone S 0≤z≤4

    I'm working on this problem, Let S be the cone described by z=\sqrt{x^2+y^2} where 0\leq z \leq 4 If \textbf{F}(x,y,z)=y\textbf{i}-x\textbf{j}+z^2\textbf{k} find the surface integral \int\int_S \textbf{F} \bullet d\textbf{S} where the orientation of S is given by the inner normal...
  27. M

    How do I evaluate a vector surface integral over a cylindrical surface?

    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>=...
  28. M

    Calculating Vector Surface Integrals in Spherical Coordinates

    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...
  29. M

    Surface Integral of quarter-cylinder help

    Homework Statement I am taking the surface integral over a quarter cylinder. Everything is fine and I can get the correct answer, it's just a conceptual problem that I need help with. Homework Equations The da for the "curved" outer surface is da=sd\phi dz\hat{s} The da for the bottom...
  30. M

    Surface Integral of quarter-cylinder

    Homework Statement I am taking the surface integral over a quarter cylinder. Everything is fine and I can get the correct answer, it's just a conceptual problem that I need help with. Homework Equations The da for the "curved" outer surface is da=sd\phi dz\hat{s} The da for the bottom...
  31. D

    Surface Integral (and Greens theorem) confusion

    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...
  32. M

    Surface integral without using Gauss's Theorem

    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...
  33. B

    Surface integral over an annulus

    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...
  34. S

    How to inverse surface integral of a vector field

    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...
  35. T

    Surface integral with vector integrand

    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.
  36. B

    Surface Integral: Evaluating zdS in Hyperboloid of Two Sheets

    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...
  37. T

    Could Surface Integration of z^2=2xy Be Simplified?

    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...
  38. N

    Surface Integral over a Hemisphere (Work check please I end up with zero)

    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...
  39. M

    Surface integral of scalar function

    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...
  40. maverick280857

    MATLAB 2D surface integral in MATLAB for Finite Element Calculation

    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...
  41. S

    Surface Integral: Finding K = $\int\int_S z/2 dA$

    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.
  42. L

    Surface Integrals: Why Dot ds with Normal?

    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
  43. N

    Line Integral vs. Surface Integral: Range of t?

    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?
  44. B

    Surface Integral - or Line Integral?

    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)...
  45. B

    Surface Integral Homework: A.n dS in Plane 2x+y=6, z=4

    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...
  46. J

    Surface Integral Help: Area of Sphere Inside Paraboloid (No Quotation Marks)

    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...
  47. G

    Surface Integral: Calculating Outward-Pointing Normal & Integral Limits

    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...
  48. T

    Surface Integral for Curl of Vector Field V over a Parametrized Triangle

    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...
  49. N

    Surface Integral Homework: Solving with Variable Substitution

    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...
  50. H

    How Do You Calculate the Surface Integral of a Cylinder?

    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...
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