What is Surface integral: Definition and 260 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. Z

    Center of mass of spherical shell inside of cone (Apostol Problem).

    I am asking this question because my solution does not seem to match the solution at the end of the book (Apostol Vol II, section 12.10, problem 9). Here is my attempt to solve this problem. If our coordinate system is chosen such that the z-axis lines up with the axis of the cone then by...
  2. Salmone

    A Calculate a tensor as the sum of gradients and compute a surface integral

    I am trying to compute the stress tensor defined as ##\vec{\Pi}=\eta(\nabla{\vec{u}}+\nabla{\vec{u}}^T)## where ##T## indicates the transpose. The vector field ##\vec{u}## is defined as follows: ##\vec{u}(\vec{r})=(\frac{a}{r})^3(\vec{\omega} \times \vec{r})## with ##a## being a constant...
  3. Addez123

    Calculate surface integral on sphere

    I'm supposed to do the surface integral on A by using spherical coordinates. $$A = (rsin\theta cos\phi, rsin\theta sin\phi, rcos\theta)/r^{3/2}$$ $$dS = h_{\theta}h_{\phi} d_{\theta}d_{\phi} = r^2sin\theta d_{\theta}d_{\phi}$$ Now I'm trying to do $$\iint A dS = (rsin\theta cos\phi, rsin\theta...
  4. WMDhamnekar

    I How to Calculate Surface Integral Using Stokes' Theorem?

    Calculate surface integral ## \displaystyle\iint\limits_S curl F \cdot dS ## where S is the surface, oriented outward in below given figure and F = [ z,2xy,x+y]. How can we answer this question?
  5. 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?
  6. WMDhamnekar

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

    Evaluate the surface integral $\iint\limits_{\sum} f \cdot d\sigma $ where $ f(x,y,z) = x^2\hat{i} + xy\hat{j} + z\hat{k}$ and $\sum$ is the part of the plane 6x +3y +2z =6 with x ≥ 0, y ≥ 0, z ≥ 0 , with the outward unit normal n pointing in the positive z direction. My attempt to answer...
  7. Addez123

    What are the limits for integrating a constrained surface with two variables?

    I start by parametarize the surface with two variables: $$r(u,v) = (u, v, \frac {d -au -bv} c)$$ The I can get the normal vector by $$dr/du \times dr/dv$$ What limits should I use to integrate this only within the elipse? I could redo the whole thing and try write r(u, v) as u being the...
  8. Hamiltonian

    B Doubt on the derivation of an equation for a surface integral

    this method of derivation is approximating the function using a polyhedron. concentrating on one of the surfaces(say the L'th surface which has an area ##\Delta S_l## and let ##(x_l,y_l,z_l)## be the coordinate of the point at which the face is tangent to the surface and let ##\hat n## be the...
  9. K

    Surface integral: Calculate the heat flow from a cylinder

    Hi, I am trying to calculate the heat flow across the boundary of a solid cylinder. The cylinder is described by x^2 + y^2 ≤ 1, 1 ≤ z ≤ 4. The temperature at point (x,y,z) in a region containing the cylinder is T(x,y,z) = (x^2 + y^2)z. The thermal conductivity of the cylinder is 55. The...
  10. M

    Divergence Theorem Verification: Surface Integral

    Hi, I just had a quick question about a step in the method of calculating the surface integral and why it is valid. I have already done the divergence step and it yields the correct result. Method: Let us calculate the normal: ## \nabla (z + x^2 + y^2 - 3) = (2x, 2y, 1) ##. Just to double...
  11. D

    Using a Surface Integral for Mathematical Analysis of the Area of an Island

    I am not clearly understand what the question requests for, is it okay to continue doing like this ? Kindly advise, thanks
  12. Zack K

    Verifying the flux transport theorem

    Let ##S_t## be a uniformly expanding hemisphere described by ##x^2+y^2+z^2=(vt)^2, (z\ge0)## I assume by verify they just want me to calculate this for the surface. I guess that ##\textbf{v}=(x/t,y/t,z/t)## because ##v=\frac{\sqrt{x^2+y^2+z^2}}{t}##. The three terms in the parentheses evaluate...
  13. Z

    Find the electric field on the surface of a sphere using Coulomb's law

    Note that the solution is 5625 V/m in z direction which is found easier using Gauss' law, but I want to find the same result using Coulombs law for confirmation. Lets give the radius 0.04 the variable a = 0.04m. ##\rho## is the charge distribution distributed evenly on the surface of the...
  14. A

    Is this surface integral correct?

    Problem Statement: Requesting for re check Relevant Equations: Requesting for re check In this eq.A4 putting ##v=Hr+u## the first integrand in eq.A5 is coming as ##H(r(\nabla•u)-(r•\nabla)u+2u)\ne\nabla×(r×u)## Am I right?? Can I request anyone to please recheck it... using this the author...
  15. M

    I Why do we ignore the contribution to a surface integral from the point r=0?

    Let ##V'## be the volume of dipole distribution and ##S'## be the boundary. The potential of a dipole distribution at a point ##P## is: ##\displaystyle\psi=-k \int_{V'} \dfrac{\vec{\nabla'}.\vec{M'}}{r}dV' +k \oint_{S'}\dfrac{\vec{M'}.\hat{n}}{r}dS'## If ##P\in V'## and ##P\in S'##, the...
  16. JD_PM

    What are the limits of integration for this surface integral?

    I want to compute: $$\oint_{c} F \cdot dr$$ I have done the following: $$\iint_{R} (\nabla \times v) \cdot n \frac{dxdy}{|n \cdot k|} = \iint (9z-1) dxdy$$ I don't know what limits the surface integral will have. Actually, I am not sure what's the surface. May you shed some light...
  17. JD_PM

    Understanding the argument of the surface area integral

    Homework Statement Find ##\iint_S ydS##, where ##s## is the part of the cone ##z = \sqrt{2(x^2 + y^2)}## that lies below the plane ##z = 1 + y## Homework EquationsThe Attempt at a Solution [/B] I have already posted this question on MSE...
  18. M

    I Why is this volume/surface integration unaffected by a singularity?

    ##\mathbf{M'}## is a vector field in volume ##V'## and ##P## be any point on the surface of ##V'## with position vector ##\mathbf {r}## Now by Gauss divergence theorem: \begin{align} \iiint_{V'} \left[ \nabla' . \left( \dfrac{\mathbf{M'}}{\left| \mathbf{r}-\mathbf{r'} \right|}...
  19. L

    Flux Through a Cube's Face with our Point Charge at a Corner

    Homework Statement A charge q is placed at one corner of a cube. What is the value of the flux of the charge's electric field through one of its faces? Homework Equations The flux surface integral of an electric field is equal to the value of the charge enclosed divided by the epsilon_naught...
  20. Hawkingo

    How to find the limits of a volume integral?

    Homework Statement If ##\vec { F } = x \hat { i } + y \hat { j } + z \hat { k }## then find the value of ##\int \int _ { S } \vec { F } \cdot \hat { n } d s## where S is the sphere ##x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4##. The Attempt at a Solution From gauss divergence theorem we know ##\int...
  21. Boltzman Oscillation

    How would I perform this surface integral?

    Homework Statement ∫∫ F ⋅ ndτ over the spherical region x^2 + y^2 + z^2 = 25 given F = r^3 r i already converted the cartesian coordinates to spherical in FHomework Equations n = r[/B]The Attempt at a Solution I know I can plug in F into the equation and then dot it with r to get the...
  22. D

    I A surface integral over infinite space

    Hi. If a function f is normalizable ,ie f→0 as | x | → infinity or r→ infinity then I presume the following surface integral f dS over infinite space is zero ? But I thought about this again and it seems like a case of zero x infinity. The function is zero at the infinite surface but the area...
  23. T

    How to evaluate a surface integral with three points?

    Homework Statement Let G=x^2i+xyj+zk And let S be the surface with points connecting (0,0,0) , (1,1,0) and (2,2,2) Find ∬GdS. (over S) Homework EquationsThe Attempt at a Solution I parametrised the surface and found 0=2x-2y. I’m not sure if this is correct. And I’m also uncertain about...
  24. M

    Calculate the given surface integral [Mathematical physics]

    Homework Statement Calculate \int_{S} \vec{F} \cdot d\vec{S} where \vec{F} = z \hat{z} - \frac{x\hat{x} + y \hat{y} }{ x^2 + y^2 } And S is part of the Ellipsoid x^2 + y^2 + 2z^2 = 4 , z > 0 and the normal directed such that \vec{n} \cdot \hat{z} > 0 Homework Equations All the...
  25. D

    A Determine the flux of the vector field trough the surface

    From my drawings it seems to be half of hemisphere. Am I right? How can I solve this task? Determine the flux of the vector field $$ f=(x,(z+y)e^x,-xz^2)^T$$ through the surface $Q(u,w)$, which is defined in the follwoing way: 1) the two boundaries are given by $$\delta...
  26. E

    Determining between direct evaluation or vector theorems

    So the main thing I'm wondering is given a question how do we determine whether to use one of the fundamentals theorems of vector calculus or just directly evaluate the integral, and if usage of one of the theorems is required how do we determine which one to use in the situation? Examples are...
  27. Mr Davis 97

    Solving a surface integral

    Homework Statement Solve the surface integral ##\displaystyle \iint_S z^2 \, dS##, where ##S## is the part of the paraboloid ##x=y^2+z^2## given by ##0 \le x \le 1##. Homework EquationsThe Attempt at a Solution First, we make the parametrization ##x=u^2+v^2, \, y=u, \, z = v##, so let...
  28. X

    Surface Integral of Outward Normal Vector over a Spherical Surface

    Homework Statement Let n be the unit outward normal of a spherical surface of Radius R, let the surface of the sphere be denoted by S. Evalute Surface integral of nndS Homework EquationsThe Attempt at a Solution I have evaluated the surface integral of ndS and found it to be 0. but am not...
  29. J

    How can I find this surface integral in cylindrical coordina

    Homework Statement A vector field $\vec F$ is defined in cylindrical polar coordinates $\rho , \theta , z$ by $\vec F = F_0(\frac{xcos (\lambda z)}{a}\hat i \ + \frac{ycos(\lambda z)}{a}\hat j \ + sin(\lambda z)\hat k) \ \equiv \frac{F_0 \rho}{a}cos(\lambda z)\hat \rho \ + F_0sin(\lambda...
  30. R

    MHB Surface Integral of $F$ Over Region V

    Let V be the region bounded by the hemisphere z=1-sqrt(1-x^2-y^2) and the plane z=1, and let S be the surface enclosing V. consider the vector field $F= x(z-1)\hat{\imath}+y(z-1)\hat{\jmath}-xy\hat{k}$.
  31. F

    Surface Integral Homework: Is the Author's Solution Wrong?

    Homework Statement Is the solution provided by the author wrong ? Stokes theorem is used to calculate the line integral of vector filed , am i right ? Homework EquationsThe Attempt at a Solution To find the surface integral of many different planes in a solid , we need to use Gauss theorem ...
  32. maxhersch

    Estimate Vector Field Surface Integral

    I assume this is a simple summation of the normal components of the vector fields at the given points multiplied by dA which in this case would be 1/4. This is not being accepted as the correct answer. Not sure where I am going wrong. My textbook doesn't discuss estimating surface integrals...
  33. R

    Surface Integral (Integral Setup)

    Homework Statement I'm just required to setup the integral for the question posted below Homework EquationsThe Attempt at a Solution So solving for phi @ the intersection of the sphere and the plane z=2: z = pcos(phi) 2 = 3cos(phi) phi = arccos(2/3) so my limits for phi would go from 0 to...
  34. E

    Find the Surface integral of a Paraboloid using Stoke's Theorem

    Homework Statement Let S be the portion of the paraboloid ##z = 4 - x^2 - y^2 ## that lies above the plane ##z = 0## and let ##\vec F = < z-y, x+z, -e^{ xyz }cos y >##. Use Stoke's Theorem to find the surface integral ##\iint_S (\nabla × \vec F) ⋅ \vec n \,dS##. Homework Equations ##\iint_S...
  35. JulienB

    Surface integral of vector fields (sphere)

    Homework Statement Hi everybody! I'm currently training at surface integrals of vector fields, and I'd like to check if my results are correct AND if there is any shortcut possible in the method I use. I'm preparing for an exam, and I found that it takes me way too much time to solve it. I...
  36. radji

    Evaluating a Surface Integral: How to Solve a Tricky Integration Problem?

    Homework Statement It is evaluating a surface integral. Homework Equations ∫s∫ f(x,y,z) dS = ∫R∫ f[x,y,g(x,y)]√(1+[gx(x,y)]2+[gy(x,y)]2) dA The Attempt at a Solution I set z=g(x) and found my partial derivatives to be gx=√x, and gy=0. I then inserted them back into the radical and came up...
  37. F

    Normal vector in surface integral of vector field

    Homework Statement when the normal vector n is oriented upward , why the dz/dx and dz/dy is negative ? shouldn't the k = positive , while the dz/dx and dz/dy is also positive? Homework EquationsThe Attempt at a Solution is the author wrong ? [/B]
  38. C

    I Can Mass be Found Using Surface Integral and Density?

    in part b , we can find mass by density x area ? is it because of the thin plate, so, the thickness of plate can be ignored?
  39. 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.
  40. W

    Surface Integral Limits: Solving for u and v

    Homework Statement Problem is in image uploaded Homework Equations n/a The Attempt at a Solution x = u, y = v and z = 1 - u - v ∂r/∂u × ∂r/∂v = i + j + k F dot N = u^2 + 3v^2 ∫∫(u^2 + 3v^2 )dudv My problem is I'm not sure what I should take as the limits? Should I flip around the order of...
  41. W

    Simple Surface Integral - Heat Flow on Surface of Star

    Homework Statement I have this problem in an online assignment. Someone told me the answer, so I already got it right, but I don't know why my logic leads me to the wrong answer. The problem: The temperature u of a star of conductivity 1 is defined by u = \frac{1}{sqrt(x^2+y^2+z^2)}. If the...
  42. 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)...
  43. M

    MHB Calculation of a surface integral

    Hey! :o I want to calculate the surface integral of $$F(x,y,z)=(0,0,z)$$ on the unit sphere with parametrization $$x=\sin u \cos v, \ y=\sin u \sin v , \ z=\cos u \\ 0\leq u\leq \pi, \ 0\leq v\leq 2\pi$$ with positive direction the direction of $T_u\times T_v$. Could you give some hints how...
  44. 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...
  45. Odious Suspect

    Curl as the limit vol->0 of a surface integral

    Joos asserts on page 31 https://books.google.com/books?id=btrCAgAAQBAJ&lpg=PP1&pg=PA31#v=onepage&q&f=false that $$\nabla \times \mathfrak{v} = \lim_{\Delta \tau \to 0} \frac{1}{\Delta \tau }\oint d\mathfrak{S}\times \mathfrak{v}$$ I tried to demonstrate this, and neglected to place the surface...
  46. Odious Suspect

    Divergence as the limit of a surface integral a volume->0

    The following is my interpretation of the development of the divergence of a vector field given by Joos: $$dy dz dv_x=dy dz\left(v_x(dx)-v_x(0)\right)=dy dz\left(v_x(0)+dx\frac{\partial v_x}{\partial x}(0)- v_x(0)\right)$$ $$=dy dz dx\frac{\partial v_x}{\partial x}(0)=d\tau \frac{\partial...
  47. I

    Surface integral for line current

    Homework Statement Calculate the integral ##\oint_C \vec F \cdot d\vec S##, where ##C## is the closed curve constructed by the intersection of the surfaces ##z = \frac{x^2+y^2}{4a}## and ##x^2+y^2+z^2=9a^2##, and ##\vec F## is the field ##\vec F = F_0\left( \frac{a}{\rho}+\frac{\rho^2}{a^2}...
  48. B

    Spherical coordinates path integral and stokes theorem

    Homework Statement Homework Equations The path integral equation, Stokes Theorem, the curl The Attempt at a Solution [/B] sorry to put it in like this but it seemed easier than typing it all out. I have a couple of questions regarding this problem that I hope can be answered. First...
  49. I

    Force on a superconducting cube

    Hi everyone, I need some help to look if I did these calculations right.Let us assume a three dimensional magnetic field: ##\vec{B}(x,y,z) = B_x(x,y,z)\hat{x} + B_y(x,y,z)\hat{y} + B_z(x,y,z)\hat{z}## The equation for the force on a superconducting particle in a magnetic field is given by...
  50. B3NR4Y

    What is the purpose of a surface integral and how is it calculated?

    I'm beyond multi-variable calculus, where this is taught, but I still don't know what the hell a surface integral is. I understand that d\sigma is the surface element, and | \frac{\partial \vec{r}}{\partial u}du \times \frac{\partial \vec{r}}{\partial v}dv | = d\sigma = |\frac{\partial...
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