What is Surface integrals: Definition and 90 Discussions

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. WMDhamnekar

    Evaluate the surface integral ##\iint\limits_{\sum} f\cdot d\sigma##

    But the answer provided is ##\frac{15}{4} ## How is that? What is wrong in the above computation of answer?
  2. dRic2

    I Stokes' theorem and surface integrals

    Hi, So my goal is to compute the integral of the "curl" of the vector field ##A_i(x_i)## over a 2-dimensional surface. Following a physics book that I am reading, I introduce the antisymmetric 2-nd rank tensor ##\Omega_{ij}##, defined as: $$\Omega_{ij} = \frac {\partial A_i}{\partial x_j} -...
  3. Isaac0427

    I Confirming my knowledge on surface integrals

    Hi, I want to make sure my understanding of calculating surface integrals of vector fields is accurate. It was never presented this way in a textbook, but I put this together from pieces of knowledge. To my understanding, surface integrals can be calculated in four different ways (depending on...
  4. A

    I Average of the B-field over a volume and surface integrals

    Purcell says that taking the surface integral of the magnetic field ##\textbf{B}## over the surfaces ##S_{1}, S_{2}, S_{3},...## below is a good way of finding the average of the volume integral of ##\textbf{B}## in the neighborhood of these surfaces. More specifically, he says in page...
  5. VVS2000

    Surface integrals to calculate the area of this figure

    I can find the area of the triangles but can't solve the squares for some reason
  6. CaptainX

    I Surface Integrals: Finding Centroid & Inertia of Circle

    How to find the centroid of circle whose surface-density varies as the nth power of the distance from a point O on the circumference. Also it's moments of inertia about the diameter through O. I'm getting x'=2a(n-2)/(n+2) And about diameter -4(a×a)M{something}
  7. A

    I How come surface integrals are single integrals in my book?

    I am currently reading Young & Freedmans textbook on physics as part of a university course, and I've noticed that they repeatedly represent surface integrals (which are double integrals) as single integrals. For instance, they symbolically represent the magnetic flux through a surface as: \int...
  8. superkraken

    Surface integrals and line integrals

    Homework Statement when we calculate the electric field due to a plane sheet or the magnetic field due to a wire,are we calculating it at a single point or the whole field due to the total wire? Homework EquationsThe Attempt at a Solution
  9. Swapnil Das

    Evaluation of Surface Integral in Gauss's Law

    I am a tenth grader, and a newbie to Advanced Calculus. While working out problems sets for Gauss's Law, I encountered the following Surface Integral: I couldn't attempt anything, having no knowledge over surface integration. So please help.
  10. A

    Does Gauss' Law use line integrals or surface integrals?

    In my physics textbook, I see Gauss' Law as https://upload.wikimedia.org/math/0/3/5/035b153014908c0431f00b5ddb60c999.png\ointE [Broken] dA but in other places I see it as...
  11. P

    Surface Integrals: Evaluate A.n dS on 2x+y=6 Plane in 1st Octant

    Homework Statement Evaluate integral A.n dS for A=(y,2x,-z) and S is the surface of the plane 2x+y=6 in the first octant of the plane cut off by z=4 Homework Equations Integral A.n dS The Attempt at a Solution The normal to the plane is (2,1,0) so the unit normal vector is 1/sqrt3 (2,1,0)...
  12. P

    Area of z^2=xy inside Hemisphere: Surface Integrals

    Homework Statement Find the area of the part of z^2=xy that lies inside the hemisphere x^2+y^2+z^2=1, z>0 Homework Equations da= double integral sqrt(1+(dz/dx)^2+(dz/dy)^2))dxdy The Attempt at a Solution (dz/dx)^2=y/2x (dz/dy)^2=x/2y => double integral (x+y)(sqrt(2xy)^-1/5) dxdy Now I'm...
  13. A

    Surface Integrals: Understanding & Examples

    Homework Statement Its more of a general issue of understanding than a specific problem I have to evaluate a few surface integrals and I am not sure about the geometric significance of what I am evaluating or even of what to evaluate. Examples. If n is the unit normal to the surface S...
  14. B

    Surface Integrals and Gauss's Law

    When I learned Integrals in Calc III, the formula looked like this ∫∫ F(r(s,t))⋅(rs x rt)*dA but in physics for Gauss's law it is ∫∫E⋅nhat dA How are these the same basic formula? I know that nhat is a unit vector, so it is n/|n|, but in the actual equation, it is a dot between the cross...
  15. G

    Surface Integrals (Stuck halfway)

    Homework Statement Homework Equations ∫∫D F((r(u,v))⋅(ru x rv) dA The Attempt at a Solution [/B] I got stuck after finding the above, at where the double integrals are. :( May I know how do I find the limits of this? (I always have trouble finding the limits to sub into the integrals...
  16. G

    Solving the Homework Puzzle: Finding Your Error with Surface Integrals

    Homework Statement The problem is given in the attached file. Homework Equations Divergence theorem, flux / surface integral The Attempt at a Solution [/B] As you can see I got the question correct using Divergence theorem. But I wanted to make sure that I could arrive at the same answer...
  17. G

    Surface Integrals of first octant

    Homework Statement Evaluate ∫∫ F⋅dS, where F = yi+x2j+z2k and S is the portion of the plane 3x+2y+z = 6 in the first octant. The orientation of S is given by the upward normal vector. Homework Equations ∫∫S F⋅dS = ∫∫D F(r(u,v))⋅||ru x rv|| dA, dA=dudv The Attempt at a Solution [/B] Since...
  18. M

    Find a Rigorous Calculus of Multiple Variables Book for Advanced Learners

    Hi! I am looking for a very rigorous book on some of the topics covered in Calculus of Multiple Variables. My University uses the last part of Adams "Calculus: a complete course" and I found the presentation therein more fit for people needing to know enough to perform the calculations than for...
  19. S

    Stokes's Theorem showing 2 surface integrals are equal

    Homework Statement Let F = <z,x,y>. The plane D1: z = 2x +2y-1 and the paraboloid D2: z = x^2 + y^2 intersect in a closed curve. Stoke's Theorem implies that the surface integrals of the of either surface is equal since they share a boundary (provided that the orientations match)...
  20. B

    Surface integrals to derive area of sphere

    Given a sphere x^2 + y^2 + z^2 = a^2 how would I derive the surface area by using surface integrals? The method I've tried is as follows: dA = sec\ \gamma \ dxdy where gamma is the angle between the tangent plane at dA and the xy plane. sec \gamma = \frac{|\nabla \varphi|}{\partial \varphi...
  21. Feodalherren

    Surface integrals - parametrizing a part of a sphere

    Homework Statement Find the area of the part of the sphere x^2 + y^2 + z^2 = 4z that lies inside the paraboloid x^2 + y^2 = z Homework Equations The Attempt at a Solution I solved for the intercepts and found that they are z=0 and z=3. The sphere is centered two units in the z-direction above...
  22. J

    The true TFC for surface integrals

    The true FTC for surface integrals Let's say that ##\vec{f}## is an exact one-form, so we have that ##\vec{f}=\vec{\nabla}f##, and ##\vec{F}## is an exact two-form, so we have that ##\vec{F}=\vec{\nabla}\times \vec{f}##. The fundamental theorem of calculus for line integral says that...
  23. S

    Fortran 90, surface integrals

    I am dumfounded on how one would perform surface integrals in Fortran 90 over a platelet, or a rectangular box. I can do single and double integrals but I have no idea on how to do surface integrals Thanks in advance!
  24. K

    Multivariable Calculus - Surface integrals

    1. Homework Statement ∫∫S xz dS where S is the boundary region enclosed by the cylinder y2 + z2 = 9 and the planes x = 0 and x + y = 5. 2. Relevant equation∫∫Sf(x,y,z)dS = ∫∫Df(r(u,v)) * |ru χ rv|dA 3. The Attempt at a Solution I think I have broken this up into 3 surfaces. The...
  25. B

    Area element vector for parametric surface integrals

    When doing surface integrals of surfaces described parametrically, we use the area element dA = ndS = (rv x rw)dvdw Where dS is the surface area element and v and w are the parameters. I'm fine with the derivation of this (I think) but I don't understand why it's necessary to have n and dS...
  26. J

    Fundamental theorem of calculus for surface integrals?

    Hellow! A simple question: if exist the fundamental theorem of calculus for line integrals not should exist too a fundamental theorem of calculus for surface integrals? I was searching about in google but I found nothing... What do you think? Such theorem make sense?
  27. B

    Surface Integrals in Gauss's Theorem of Charge in Motion

    gauss's theorem is also applicable to charge in motion.but how the surface integral has to be taken??
  28. T

    Surface integrals and parametrization

    An area A in the xy-plane is defined by the y-axis and by the parabola with the equation x=6-y^2. Furthermore a surface S is given by that part of the graph for the function h(x,y)=6-x-y^2 that satisfies x>=0 and z>=0. I have to parametrisize A and S. Could this be a...
  29. S

    Difference Between Surface Integrals and Surface area using double integrals .

    Hi all, Thanks for response :) I Dont really understand what is surface integrals ?? and its difference with Surface Area using double integrals. Can anyone help ? thanks a lot...
  30. C

    Vector Calculus Question about Surface Integrals

    Why is it that when the force field is z^2 and you take the surface integral over a sphere of radius a using spherical coordinates, that yields the flux to be (4pi a^3 )/ 3 BUT in a calculus book, the force field is z instead of z^2 evaluated using polar coordinates and it yields the same...
  31. T

    Surface integrals evaluation problem

    Homework Statement Evaluate ∫∫F.nds where F=2yi-zj+x^2k and S is the surface of the parabolic cylinder y^2=8x in the first octant bounded by the line y=4, z=6 Homework Equations We were told that the projection is supposed to be taken in the yz plane but how?? and i have a feeling that...
  32. C

    Difference between double and surface integrals? Purpose of surface integrals?

    I'm preparing for a vector calculus course in the fall and I've been self studying some topics. I've taken multivariable calculus and I'm familiar with using double integrals, how to solve them and how to use them to find volume. From what I've read so far, I'm familiar with how to SOLVE a...
  33. H

    Confusion on sign convention for surface integrals

    Homework Statement Compute the surface integral for F = [3x^2, y^22, 0] and S being a portion of the plane r(u,v)=[u,v,2u+3v], 0≤u≤2, −1≤v≤1.The Attempt at a Solution I managed to get the correct answer, because with some luck I defined the normal in the correct direction. I am just confused...
  34. C

    Surface integrals of vector fields

    The integral for calculating the flux of a vector field through a surface S with parametrization r(u,v) can be written as: \int\int_{D}F\bullet(r_{u}\times r_{v})dA But what's to stop one from multiplying the normal vector r_{u}\times r_{v} by a scalar, which would result in a different...
  35. K

    Question about orientation and surface integrals

    Homework Statement I'm a bit confused as to how to determine which component must be positive or negative if the question gives you a surface and says the normal vector is pointing outward or inward. Some surfaces have it so that the z component is positive if n is pointing outward and...
  36. H

    Surface Integrals: Solving Q6 & Q7

    Ok, so for Q6, I first said that z = 3 - 3x - 1.5y Using (∂z/∂x)^2 = 9, (∂z/∂y)^2 = 9/4 I then did a double integral of (x + y + (3 - 3x - 1.5y)) * sqrt(9 + 9/4 + 1) dA Letting y and x be bounded below by 0 as stated, and x bounded above by 1 - 0.5y and y bounded above by 2, I went...
  37. Y

    Evaluate the integral over the helicoid [Surface integrals]

    Homework Statement Evaluate the integral \int\int_S \sqrt{1+x^{2}+y^{2}}dS where S:{ r(u,v) = (ucos(v),usin(v),v) | 0\leq u\leq 4,0\leq v\leq 4\pi } 2. The attempt at a solution Here is my attempt, I am fairly sure I am right, but it is an online assignment and it keeps telling me I am...
  38. T

    Taking Multiple Surface Integrals

    Homework Statement http://img687.imageshack.us/img687/1158/skjermbilde20111204kl85.png [Broken] The Attempt at a SolutionI thought this was pretty hard and involved a number of different parts. Here's my work: Let x=cosθ and z=sinθ, also let 0≤y≤2-x=2-cosθ. I parametrize Q1, which I...
  39. B

    Surface Integrals: Clearing Up Misunderstanding

    Hi, I understand that from my EM class there exist a surface integral which is actually a way of summing infinitesimally small surface elements ds. But then I ran into some theorems on internet and I saw the denotation of double integral, over a surface S. And they called that a surface...
  40. T

    Calculating flux using surface integrals.

    This isn't homework. I've been restudying vector calculus from the beginning to end on my free time and got stuck on this problem. I am not sure what I'm doing wrong, but it may be a calculation error since it has so much calculation involved. Homework Statement Evaluate the surface integral...
  41. S

    Kinematics of surface integrals question

    This is a continuum mechanics/fluid dynamics question concerning the time rate of change of a surface integral of a vector field, where the surface is flowing along in a velocity field (like in a fluid). (Gauss's law is for fixed surfaces.) This integral goes by various names in different...
  42. W

    Surface integrals of vectors: Need check of my work- answer doesn't look right

    Homework Statement Find surface integral of vector field F=<x,y,x+y> over the surface z=x^2+y^2 where x^2+y^2 less than 1. Use outward pointing normals Homework Equations The Attempt at a Solution So I did the whole thing and got a zero which doesn't look right to me. My algebra...
  43. C

    What are line and surface integrals?

    You can say an integral is the area under a curve and the derivative is the slope. What are the equivalents for line and surface integrals? I've tried google and wikipedia but I can't find a dumbed down version. :cry: I know that line integrals are related in some way to arc length since a line...
  44. N

    Surface Integrals: Solving for S_1 & S_2

    Homework Statement Consider the surface S_1 described by the equations x = (1-w)^3cos(u), y = (1-w)^3sin(u), z = w, 0 <= u < 2\pi, 0 <= w < 1 The first few parts of the question were quite simple. Firstly we had to calculate dS and then compute the surface integral for the vector field...
  45. H

    Divergence theorem and surface integrals

    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...
  46. K

    General question about surface integrals

    hi in my engineering mathematics class, we are going over surface integrals again. i have some general question about this subject. sorry for not using the template. say that i have a problem that goes like this. "evaluate \int(v*dS) (where the * means dot) where v= (3y,2x^{2},z^{3}) and S is...
  47. M

    Surface Integrals and Average Surface Temperature of a Torus

    Homework Statement A torus is a surface obtained by rotating a circle about a straight line. (It looks like a doughnut.) If the z-axis is the axis of rotation and the circle has radius b, centre (0, a, 0) with a > b, and lies in y − z plane, the torus obtained has the parametric form r(u, v) =...
  48. X

    Difference between these two surface integrals?

    In my Calculus book, in the chapter that introduces multiple integration, it has a chapter on integrals that calculate the surface area of a function in space. They define the integral to be.. \int \int dS = \int \int \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial...
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