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. H

    Surface Integral Eval: x² + y² + z² = a²

    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...
  2. J

    Surface integral of a parabolic cylinder

    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
  3. E

    How do I use Gauss' theorem to evaluate a surface integral?

    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'...
  4. M

    Surface integral without using Gauss' theorem

    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...
  5. K

    Flux: Surface integral of a sphere.

    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...
  6. B

    Help evaluating a surface integral in polar coordinates

    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...
  7. D

    Spherical coordinates surface integral

    Hi. I have this integral \int_0^{2\pi}\int_0^\pi \mathbf A\cdot\hat r d\theta d\phi where \hat r is the position unit vector in spherical coordinates and \mathbf A is a constant vector. Is it possible to evaluate this integral without calculating the dot product explicitly, i.e. without...
  8. Pengwuino

    Surface Integral in Spherical Coordinates for Arbitrary Vector Field

    Homework Statement I'm looking to do the surface integral of \oint {\vec v \cdot d\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over a} } where v is arbitrary and in spherical coordinates and the surface is the triangle enclosed by the points (0,0,0) -> (0,1,0) ->...
  9. C

    This is the same result as the back of the book. So your solution is correct.

    Homework Statement find the flux of the field F(vector) across the portion of the sphere x^2+y^2+z^2=a^2 in the first octant directed away from the origin Homework Equations F(x,y,z)=zk(hat) The Attempt at a Solution i used Flux=double integral over x-y plane F.n(unit...
  10. C

    Having trouble with a surface integral

    I'm self learning and working my way through the book "Div Grad Curl and All That". On one of the pages (27) the author says \int_{ }^{ } \int_{ }^{ } z^2 dS = \int_{ }^{ } \int_{ }^{ } \sqrt[ ]{ 1 - x^2 - y^2 } dx dy "This is an ordinary double integral, and you should verify that its...
  11. B

    How to Evaluate a Surface Integral on a Portion of a Sphere Above a Plane?

    Homework Statement Evauate Surface integral \int\int_{\sigma}(x^2 + y^2)dS where \sigma is the portion of the sphere x^2 + y^2 + z^2 = 4 above the plane z = 1. Homework Equations \int\int_{\sigma} f(x,y) \sqrt{\frac{\partial z}{\partial x}^2+\frac{\partial z}{\partial y}^2+1}...
  12. C

    Surface integral in spherical coordinates question

    Homework Statement Find the surface area of the portion of the sphere x^2 + y^2 + z^2 = 3c^2 within the paraboloid 2cz = x^2 + y^2 using spherical coordinates. (c is a constant)Homework Equations The Attempt at a Solution I converted all the x's to \rho sin\phi cos\theta, y's to \rho sin\phi...
  13. T

    Surface integral, grad, and stokes theorem

    Hi I'm practicing for my exam but I totally suck at the vector fields stuff. I have three questions: 1. Compute the surface integral \int_{}^{} F \cdot dS F vector is=(x,y,z) dS is the area differential Calculate the integral over a hemispherical cap defined by x ^{2}+y ^{2}+z...
  14. T

    Evaluating Surface Integral with Stokes Theorem

    use the stokes theorem to evaluate the surface integral [curl F dot dS] where F=(x^2+y^2; x; 2xyz) and S is an open surface x^2+y^2+z^2=a^2 for z>=0. So i guess its a hemisphere of radius a lying on x-y plane. I don't see however how to take F dot dr. What is this closed curve dr bounding...
  15. M

    Surface integral from vector calculus

    Homework Statement F = (x^2) i + (y^2) j + (z) k, S is the cone z = (x^2+y^2) ^ (1/2), with x^2 +y^2 <= 1, x >= 0, y >= 0, oriented upward. Homework Equations All of the above The Attempt at a Solution My attempted solution is 0. But other students claim that the answers is...
  16. D

    Surface Integral of Vector Fields

    Evaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = y i + x j + z^2 k S is the helicoid (with upward orientation) with vector...
  17. C

    How Do You Calculate the Surface Integral of a Parametric Surface?

    Homework Statement For the parametrically defined surface S given by r(u,v) = <cos(u+v), sin(u+v), uv>, find the following differential: In double integral over S of f(x, y, z)dS, dS = Homework Equations Above The Attempt at a Solution I thought I needed to put x, y, and...
  18. B

    Finding da_x in Spherical Coordinates

    I want to integrate something in spherical coordinates I have da=R^{2}sin(g)dgdh \hat{r} with g and h angles and \hat{r}=sin(g)cos(h) \hat{i}+sin(g)sin(h) \hat{j}+cos(g) \hat{k} But what is now da_{x}=dydz \hat{i} in spherical coordinates? So I have the expression in ordinary...
  19. C

    Solving Surface Integral Questions w/Check Solutions

    Surface Integral Question and Solution Check Hi everyone, this is my first post and I was hoping someone could help me check my solution to this problem (which could be completely wrong) and help me get unstuck at part 3. Any help would be greatly appreciated. Homework Statement...
  20. B

    Surface Integral of a Square Pyramid

    Homework Statement Prove that the following surface integral for the four slanted faces of a square pyramid with a square base in the xy plane with corners at (0,0) (0,2) (2,0) (2,2) and a top vertex at (1,1,2) is equal to 4 by evaluating it as it stands: \int\int (\nabla \times V)\cdot n dS...
  21. K

    Volume Integral of f over Sphere: Find Solution

    Homework Statement Find the volume integral of the function f=x^{2}+y^{2}+z^{2} over the region inside a sphere of radius R, centered on the origin. Homework Equations Spherical polars x=rsin(\theta)cos(\phi), y=rsin(\theta)sin(\phi), z=rcos(\theta) Jacobian in spherical polars =...
  22. B

    Stokes' Theorem ( Surface Integral )

    [SOLVED] Stokes' Theorem ( Surface Integral ) Homework Statement Use stokes' theorem to find the value of the surface integral \int\int (curl f) dot n) dS over the surface S: Let S by the part of the plane z=y+1 above the disk x^2+y^2<=1, and let f=(2z,-x,x). Homework Equations...
  23. G

    Surface integral with differential forms

    Hi, I'm trying to solve a problem in David Bachman's Geometric Approach to Differential Forms (teaching myself.) The problem is to integrate the scalar function f(x,y,z) = z^2 over the top half of the unit sphere centered at the origin, parameterized by \phi(r,\theta) = (rcos\theta, rsin\theta...
  24. J

    Surface integral of half sphere

    Homework Statement I am trying to sort out surface integrals in my head, and have become more confused when attempting to calculate the surface integral of a hemisphere. I am getting confused about which values to use as boundaries. Homework Equations da=R^2 sinθdθdφ The...
  25. Saladsamurai

    Surface Integral Homework: Compute Int Int S F•n dS

    Homework Statement Let S be the part of the paraboloid z=1+x^2+y^2 lying above the rectangle x between 0 and 1; y between -1 and 0 and oriented by the upward normal. Compute \int\int_SF\cdot n\,dS where F=<xz, xy, yz> So I have Parametrized the surface S as r(x,y,z)=<x, y, 1+x2+y2>...
  26. Saladsamurai

    Surface Integral using Divergence Theorem

    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...
  27. Saladsamurai

    Solve Surface Integral: \vec{F}\cdot\vec{n}\, dS

    Homework Statement Evaluate the surface integral \vec{F}\cdot\vec{n}\, dS where \vec{F}=<-y,x,0> and S is the part of the plane z=8x-4y-5 that lies below the triangle with vertices at (0,0,0,), (0,1,0,) and (1,0,0). The orientation of S is given by the upward normal vector. answer: 2...
  28. M

    Surface Integral of Vector Field

    Homework Statement Find \int\int_{S} F dS where S is determined by z=0, 0\leqx\leq1, 0\leqy\leq1 and F (x,y,z) = xi+x2j-yzk. Homework Equations Tu=\frac{\partial(x)}{\partial(u)}(u,v)i+\frac{\partial(y)}{\partial(u)}(u,v)j+\frac{\partial(z)}{\partial(u)}(u,v)k...
  29. W

    Surface Integral of a Sphere (non-divergence)

    Homework Statement Evaluate: \int\intG(r)dA Where G = z S: x2 + y2 + z2 = 9 z \geq 0 Homework Equations Parameterization x = r sinu cosv y = r sinu sin v z = r cos u The Attempt at a Solution r(u,v) = (r sinu cosv)i + (r...
  30. D

    Surface Integral: Calculating Max Area from Circular Log

    Homework Statement A log of wood which is approximately circular in cross section has diameter equal to 0.5m. Calculate the maximum area of the rectangular timber section that can be obtained from the log. Homework Equations The Attempt at a Solution No idea of solution
  31. F

    How do I evaluate a surface integral with parametric equations?

    Homework Statement Evaluate the double integral of yz dS. S is the surface with parametric equations x=(u^2), y=usinv, z=ucosv, 0<u<1, 0<v<(pi/2) (all the "less than" signs signify "less than or equal to" here) Homework Equations double integral of dot product of (F) and normal...
  32. C

    Simple Surface Integral of a Cylinder: Homework Statement and Solution Attempt

    Homework Statement This is annoying me because I am so clearly being a muppet somewhere. I need to integrate the vector field (x,-y,z).(vector)ds over the surface of a cyliner x^2 + y^2 < 4 (or equal to) and z is between 0 and 1. The Attempt at a Solution I have to do it both with and...
  33. E

    Surface Integral Help: Solving a Paraboloid and Cylinder Intersection Problem

    Homework Statement \int\int{\frac {x}{\sqrt {1+4\,{x}^{2}+4\,{y}^{2}}}}dS Where S is the parabaloid z = 25 - x^{2} -y^{2} that lies within the cylinder x^{2}+(y-1)^{2}=1 The Attempt at a Solution First i use the following: {\it dS}=\sqrt {1+{\frac {{{\it df}}^{2}}{{{\it...
  34. rohanprabhu

    Electric Flux question [Surface integral]

    Homework Statement Q] A charge 'Q' is kept over a non-conducting square plate of side 'l' at a height l/2 over the center of the plate. Find the electric flux through the square plate surface. Neglect any induction that may occur. Homework Equations \phi = \int \overrightarrow{E}\cdot...
  35. A

    Solve Surface Integral for xy-Plane Projection

    Homework Statement I'm not sure how to convert this surface integral into a double integral for evaluation. \iint_S \frac{1}{1 + 4(x^2 + y^2)} dS S is the portion of the paraboloid z = x^2 + y^2 between z = 0 and z = 1. The Attempt at a Solution How do you project this onto the xy-plane...
  36. C

    Surface Integral: Transform, Calculate and Add Up Results?

    As you know surface integrals are integrated with respect to dS. We then tranform the integral into one in dxdy. Is this the end of the problem or must we calculate it for dxdz and dydz as well and if so do you add up all results at the end!?
  37. P

    Surface Integral: Right Side = Left Side?

    May be this should have been in math section but since this came out while studying Electrodynamics i put it here we have \boxed{\int_{S} \nabla \times \vec{B}.d\vec{a}=\oint \vec{B}.d\vec{l}} Q.well there are many areas with the same boundary which one to choose from? well if we know the...
  38. M

    Surface Integral: Compute g = xy over Triangle x+y+z=1, x,y,z>=0

    Homework Statement (Q) Compute the surface integral g = xy over the triangle x+y+z=1, x,y,z>=0. Homework Equations The Attempt at a Solution The triangular region basically means that the region in consideration is a plane and not a sphere, cylinder etc... Therefore, we can...
  39. Simfish

    Question on surface integral of curl

    Homework Statement Let F be F = ( x^2 z^2 ) i + (sin xyz) j + (e^x z) k.Find \int\int \nabla \times F \cdot n dS where the region E is above the cone z^2 = x^2 + y^2 and inside the sphere centered at (0,0,1) and with radius 1. (so it is x^2 + y^2 + (z-1)^2 = 1).. I know that they intersect at...
  40. T

    Multivariable surface integral

    a) Find the area of the part of the surface S = {x^2+ y^2+ (z-1)^2 = 4, 0 ≤ z ≤ 1}. Note that this is part of the sphere of radius 2 with center (0,0,1).
  41. A

    Surface Integral: Evaluating Double Integral of f.n ds on Sphere

    Homework Statement Evaluate [double integral]f.n ds where f=xi+yj-2zk and S is the surface of the sphere x^2+y^2+z^2=a^2 above x-y plane. The Attempt at a Solution I know that the sphere's orthogonal projection has to be taken on the x-y plane,but I'm having trouble with the...
  42. A

    Evaluate Surface Integral f.n ds for Sphere x^2+y^2+z^2=a^2

    Problem : Evaluate [double integral]f.n ds where f=xi+yj-2zk and S is the surface of the sphere x^2+y^2+z^2=a^2 above x-y plane. My effort:: I know that the sphere's orthogonal projection has to be taken on the x-y plane,but I'm having trouble with the integration.Please help!
  43. N

    Understanding Stoke's Theorem: Does Surface Integral Depend on Shape?

    i'm trying to understand stoke's theorem and am having trouble seeing whether the surface integral for a given surface changes with any change in its shape, or if it only changes depending on the cross sectional area perpendicular to the direction of the vector field. can anybody help me out?
  44. S

    How Do I Solve This Challenging Surface Integral Problem?

    Okay - I thought that I figured this stuff out, but I didn't. The Problem When G(x, y, z) = (1-x^2-y^2)^{3/2}, and z = \sqrt{1-x^2-y^2}, evaluate the surface integral. My Work I keep trying this but I end up with the following integral that I cannot evaluate: \int_{-1}^{1} \!\!\...
  45. S

    How Do I Evaluate This Surface Integral on a Paraboloid?

    The Problem Evaluate the surface integral of G(x, y, z) = \frac{1}{1 + 4(x^2+y^2)} where z is the paraboloid defined by z = x^2 + y^2 , from z = 0 to z = 1. My Work I rewrote G(x, y, z) as \frac{1}{1+4z}. Then, I evaluated the surface integral (I'm skipping a few steps in the...
  46. B

    How do I parameterize a triangle for a surface integral?

    Hi, I'm having problems evaluating a surface integral. \int\limits_{}^{} {\int\limits_S^{} {xdS} } where S is the triangle with vertices (1,0,0), (0,2,0) and (0,1,1). I need to parameterise the triangle but I don't know how to. I tried (x,y,z) = (1,0,0) + u[(0,2,0)-(1,0,0)] +...
  47. G

    Solving Surface Integral: x(C - x^2/3)^3/2 dx

    \int x {(C - x^{2/3})}^{3/2} dx Any ideas?
  48. denian

    How to Calculate the Surface Integral of a Sphere?

    [PLAIN]http://www.mrnerdy.com/forum_img/dont%20know.JPG what do i need to do next to find the surface integral btw the planes z=1 and z=2? or is there anything wrong that i did?
  49. M

    Calculate Surface Integral F.ndS on Sphere at Origin

    I am really struggling with this one: Calculate \Int F.ndS , where F = a * x^3 * i + b*y^3*j + c*z^3*k where a,b and c are constants, over the surface of a sphere of radius a, centred at the origin. note that F and n are vectors (sorry, tried to type them in bold...but it...
  50. Reshma

    How Do You Calculate Surface Integrals for Vector Fields?

    I have two problems on surface integrals. 1] I have a constant vector \vec v = v_0\hat k. I have to evaluate the flux of this vector field through a curved hemispherical surface defined by x^2 + y^2 + z^2 = r^2, for z>0. The question says use Stoke's theorem. Stoke's theorem suggests...
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