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.

View More On Wikipedia.org
Recent contents Top users
  1. V

    MHB Differential form surface integral

    Question: Evaluate the surface integral $$J = 2xzdy \land dz+2yzdz \land dx-{z}^{2}dx \land dy$$ where S \subset {\Bbb{R}}^{3} is the rectangle parametrised by: $$x(u,v) = 1-u,\ y(u,v) = u,\ z(u,v) = v,\ \ 0\le u, v \le 1$$ so far I have: \begin{array}{}x = u\cos v, &dx = \cos v\, du -...
  2. samjohnny

    Surface integral in Spherical Polar

    Homework Statement Attached. Homework Equations The Attempt at a Solution Hi, Ok, so for the first part of this question it asks to evaluate the integral of the dot product between A and dS. The magnitude of dS is as shown above, and it is in the radial direction in spherical polar...
  3. ZARATHUSTRA

    Surface integral for scalar quantities

    so for surface integral for scalar quantities. Why do we use cross product not dot product in the integral? but can we just add an unit normal vector n to make the direction the same? My question seems really stupid too a lot people, but this is really my confusion to surface integral. please...
  4. B

    Surface Integral, Polar Coordinates

    Homework Statement Express f(x,y) = \frac{1}{\sqrt{x^{2} + y^{2}}}\frac{y}{\sqrt{x^{2} + y^{2}}}e^{-2\sqrt{x^2 + y^2}} in terms of the polar coordinates \rho and \phi and then evaluate the integral of f(x,y) over a circle of radius 1 centered at the origin. Homework Equations y = \rho...
  5. 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...
  6. L

    Question on Surface Integral and Flux

    Homework Statement I have a coordinate system, (x,y,z). There is a uniform-magnetic-field of 2.0 T that exists along the direction of the y-axis. There is a rectangular plane bounded by the points (3,0,0),(0,1,0),(0,1,1),(3,0,1). Calculate how much flux is traveling through the rectangular...
  7. P

    Understanding Vector Fields on a Sphere

    Homework Statement find the values of the integral \int_{S} \vec A\cdot\ d\vec a where, \vec A\ = (x^2+y^2+z^2)(x\hat e_{1}+y\hat e_{2}+z\hat e_{3}) and the surface S is defined by the sphere R^2=x^2+y^2+z^2 Homework Equations first i must evaluate the integral directly, so i don't...
  8. M

    Flux Through a Non-Concentric Sphere

    Homework Statement Assume I want to calculate the electric flux through a spherical surface centred at point P with radius R which contains a point charge Q, that is not concentric with the spherical surface. Here, I can no longer assume that ∫∫sEdA = E.A, and I have to calculate the value of...
  9. A

    Surface integral problem: ##\iint_S (x^2+y^2)dS##

    Homework Statement ##\iint_S (x^2+y^2)dS##, ##S##is the surface with vector equation ##r(u, v)## = ##(2uv, u^2-v^2, u^2+v^2)##, ##u^2+v^2 \leq 1## Homework Equations Surface Integral. ##\iint_S f(x, y, z)dS = \iint f(r(u, v))\left | r_u \times r_v \right |dA##, The Attempt at a Solution...
  10. 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...
  11. PhysicsKid0123

    Surface integrals/Surface areas of arbitrary domain regions

    I'm having trouble evaluating this surface integral. This would be very simple to solve if the parameter domain of the variables u and u was a square region. However, that isn't the case here. I've tried using a change of variables and saying that u = r cos x, and v = r sin x. Where 0 < x < 2pi...
  12. NuclearMeerkat

    Verification of Stoke's Theorem for a Cylinder

    Homework Statement Homework Equations Stoke's Theorem: The Attempt at a Solution ∇×A = (3x,-y,-2(z+y)) I have parametric equation for wall and bottom: Wall: x(θ,z) = acosθ ; y(θ,z) = asinθ ; z(θ,z) = z [0≤θ≤2π],[0≤z≤h] Bottom: x(θ,r) = rcosθ ; y(θ,r) = rsinθ ; z(θ,r) = 0 [0≤θ≤2π],[0≤r≤a]...
  13. N

    Understanding Surface Integrals: Scalar vs. Vector Functions Explained

    Hello, We know that surface integrals come to the form of a surface integral of a scalar function over a surface and a vector field over a surface. First one is \oint F(x,y,z)d{S} and the second one is \oint \vec{F}(x,y,z)\cdot d\vec{S}=\oint \vec{F}(x,y,z)\cdot \vec{n}dS, where n is the unit...
  14. S

    Surface integral of vector field

    Homework Statement find ∫E.dS, where E = (Ar^2, Br (sinθ),C cosρ), over the outside conical surface S, given by 1≤r≤2, θ=\pi/3 (this is an open surface, excluding the end faces).Homework Equations The Attempt at a Solution from the context I believe ρ is the plane polar angle on the x-y...
  15. B

    Area of a cylinder inside a sphere (surface integral)

    Homework Statement Find the area of the cylinder x^2 + y^2 -y = 0 inside the sphere x^2 + y^2 +z^2 =1 Homework Equations dA = sec \gamma dydz where sec \gamma = \frac{|\nabla \phi|}{|\partial \phi/ \partial x|} The Attempt at a Solution The method shown in this section is to...
  16. sa1988

    Surface Integral Confusion: Clarifying the Steps and Methods

    Homework Statement Homework Equations The Attempt at a Solution I've been reading up on surface integrals for several hours now but I can't get my head around this at all. There are just too many things flying around. Do I project dS onto a single plane and work it out as...
  17. Feodalherren

    How Do You Calculate the Mass of a Conical Surface Using Integrals?

    Homework Statement Find the mass of z= \sqrt{x^{2}+y^{2}} when 1 ≤ z ≤ 4. The density function is ρ(x,y,z) = 10 - z Homework Equations The Attempt at a Solution \int\int_{s} ρ dS S = <x, y, \sqrt{x^{2}+y^{2}} > therefore dS = < \frac{-x}{\sqrt{x^{2}+y^{2}}} ...
  18. J

    Is There an Equivalent Expression to ∫ f·dr for Surface Integral?

    The line integral can be expressed, at least, in this three different ways: \int \vec{f} \cdot \hat{t} ds = \int \vec{f} \cdot d\vec{s} = \int \vec{f} \cdot d\vec{r} The surface integral too (except by least expression above): \iint \vec{f} \cdot \hat{n} d^2S = \iint \vec{f} \cdot d^2\vec{S} My...
  19. H

    Surface integral, spherical coordinates, earth

    Homework Statement Find the surface area of the Earth (as a fraction of the total surface of the earth) that lies above 50 degrees latitude North. Homework Equations $$A = \int_R\sqrt{|\det(g)|}d\theta d\phi$$ The Attempt at a Solution Hence I get $$\int_0^{360}...
  20. F

    What is an example of a surface integral using the method of projection?

    I'm a little unsure about an example of a surface integral I've come across, in which the method of projection is used. The example finds the surface area of a hyperbolic paraboloid given by z=(x2-y2)/2R bounded by a cylindrical surface of radius a, such that x2+y2=<2. The first issue I'm...
  21. M

    Evaluating surface integral (solving for unknown variables)

    The solutions have came up with 5 equations, I'm not confused how they got those 5 equations but I don't understand how it was concluded that L = 0 and m = p = 1/√2.
  22. M

    Surface Integral Problem (Answer Already Given)

    Homework Statement The problem is to calculate the flux emanating from the exhaust of a jet engine. The air gas velocity from a jet engine varies linearly from a maximum of 300 m/s at the center of the circular exhaust opening to zero at the edges. If the exhaust diameter is 1.6 m, find...
  23. M

    Calculating surface integral using diverg. thm.

    I believe the book is wrong.. Can some one please check my work. PROBLEM IS IN THE 2nd POST ( SORRY I COULDNT ADD BOTH PICS FOR SOME REASON )
  24. N

    Converting surface integral to line

    Homework Statement I have the following integral \int_{ABC}{\mathbf{v}\cdot \nabla f_id\sigma} where $d\sigma$ is an area element, $\mathbf v$ is a velocity vector and f_i some function. The integral is performed across a triangle ABC and it is assumed that f is linear. In my book this...
  25. J

    Definition for Surface Integral

    Hello! The definition of Line Integral can be this: \int_s\vec{f}\cdot d\vec{r}=\int_s(f_1dx+f_2dy+f_3dz) And the definition of Surface Integral can be this: \int\int_S(f_1dydz+f_2dzdx+f_3dxdy) However, in actually: \\dx=dy\wedge dz \\dy=dz\wedge dx \\dz=dx\wedge dy What do the...
  26. PsychonautQQ

    Scalar Surface Integral over parameterized surface

    Homework Statement Calculate ∫∫ f(x,y,z)dS for the surface G(r,θ) = (rcosθ,rsinθ,θ), 0<r<1, 0<θ<2pi. f(x,y,z) = (x^2+y^2)^(1/2) = r Homework Equations The Attempt at a Solution so the surface is given so I have to find the normal vector... G_r = cos(θ),sinθ,0 G_θ =...
  27. A

    Computing the surface integral of a parabloid

    Homework Statement Homework Equations The Attempt at a Solution I am having difficulty understanding how the author determined the limits of integration of ##R##. The author used ##\theta=\pi/3\quad to\quad \theta=\pi/2## and ##r=1\quad to\quad r=1##. More accurately, I'm...
  28. P

    Surface Integral Homework: Flux Through a Cylinder

    Homework Statement The problem asks to find the flux through a cylinder of radius R and height h. Homework Equations Flux = ∫∫FndS over S F = (ix + jy)*ln(x2+y2) The Attempt at a Solution After finding the unit normal vector (n) to the curved surface of the cylinder, the...
  29. A

    What is the Surface Area of a Cone Between Two Planes in Calculus III?

    Hello, everyone. I've used this forum several times in the past for help with problems or just to see other vantage points on a subject. This is my first post, so please bare with me in trying to format my post correctly. I'm studying for my final exam in Calculus III next week and am working...
  30. J

    Surface level and surface integral

    I have some questions, all associated. So, first, if a curve level is defined as: f(x,y)=k or vectorially as: f(c(t))=k and its curve integral associated as: \bigtriangledown f(c(t))\cdot c'_{t}(t)=k Then, how is the equation of a surface integral associated to surface level: f(x,y,z)=k...
  31. C

    Calculating Mass for a Cylinder in the First Octant

    If the mass per unit area of a surface is given by ρ=xy, find the mass if S is the part of the cylinder x2+z2=25 which is in the first octant and contained within the cylinder x2+y2=16.So here was my attempt. I parametrized the curve. x2+z2=25 r(u, v) = <5cos(u), v, 5sin(u)> I then plugged...
  32. PsychonautQQ

    Surface Integral Homework: Double Integral of (xy+e^z)dS

    Homework Statement Take the double Integral of (xy+e^z)dS where S is the triangle with vertices (0,0,3),(1,0,2),(0,4,1). Homework Equations The Attempt at a Solution So the equation of the plane for the triangle given is z = 3 - x - (1/2)y. We plugged that Z into the z from the...
  33. T

    Computing a discrete surface integral of a scalar function

    Consider a triangulated discrete manifold (a polyhedron) with known vertices (i.e. each vertex is given in terms of its $$(x,y,z)$$ coordinates ). Assign scalar values (some kind of potentials) to each vertex (i.e. at each vertex, a $$k_t(\mathbf{v})$$ is known through its value, no...
  34. W

    Evaluating a Surface Integral for S: How Can the Given Formula be Used?

    Problem: Use the fact that \int_S \vec{v} \cdot d\vec{S}=\int_S \vec{v} \cdot \frac{\nabla f}{\partial f/\partial x} dy\ dz to evaluate the integral for ##S=\{(x,y,z):y=x^2 ; 0 \geq x \geq 2; 0 \geq z \geq 3 \}## and ##\vec{v}=(3z^2, 6, 6xz)##. Attempt at a Solution: I'm having...
  35. D

    MHB Surface Integral Check: \(\mathbf{F}\cdot\hat{\mathbf{n}} \int_0^1\int_0^1dxdy\)

    I haven't done a surface integral in a while so I am asking to get this checked. \(\mathbf{F} = \langle x, y, z\rangle\) and the surface is \(z = xy + 1\) where \(0\leq x\leq 1\) and \(0\leq y\leq 1\). \(\hat{\mathbf{n}} = \nabla f/ \lvert\nabla f\rvert = \frac{1}{\sqrt{3}}\langle 1, 1...
  36. S

    Surface Integral From Div, Grad, Curl and all that

    First, this is my first time actually posting anything so hi PF!Second, I have been working out of Div, Grad, Curl and all that. This problem has me stumped for some reason. My answer never comes out to be the same as the books. If you could help me figure out where I am going wrong I would...
  37. W

    Evaluate the following surface integral

    Homework Statement \int_S 2z+1 dS where S is the surface z = 16-x^2-y^2 \quad z>0 Homework Equations \int_S f dS = \int_S f(S(u,v)) | \frac{\partial S}{\partial u }\times \frac{\partial S}{\partial v } | dudv The Attempt at a Solution let x= u \quad y=v z = 16 - u^2-v^2...
  38. K

    Surface Integral with Parametrization

    Homework Statement \int\int _{S} \sqrt{1 + x^2 + y^2} dS Given that S is the surface of which \textbf{r}(u,v) = u\cdot cos(v)\textbf{i}+u\cdot sin(v)\textbf{j}+v\textbf{k} is a parametrization. (0 \leq u \leq 1, 0 \leq v \leq \pi) Homework Equations dS = \left| \frac{\partial...
  39. LunaFly

    Surface Integral involving Paraboloid

    Homework Statement Evaluate the surface integral: ∫∫s y dS S is the part of the paraboloid y= x2 + z2 that lies inside the cylinder x2 + z2 =4.Homework Equations ∫∫sf(x,y,z)dS = ∫∫Df(r(u,v))*|ru x rv|dAThe Attempt at a Solution I've drawn the region D in the xz-plane as a circle with...
  40. F

    Solving Surface Integral: Find I Over Half Sphere A

    Homework Statement Find integral I = ∫∫xz^2 dydz + (x^2y − z^3) dzdx + (2xy + y^2z) dxdy (Integrate over A) if A is half a sphere(radius is a). Sphere is given with equation z=(a-x^2-y^2)^1/2 and z=0. Homework Equations The excercise is in 2 parts , find it with just integrating and b)...
  41. B

    Surface Integral Homework: Evaluate ∫∫σ

    Homework Statement Evaluate ∫∫σ where S is a surface with sides S1, S2, and S3. S1 is a portion of the cylinder x2+y2 = 1 whose bottom S2 is the disk x2+y2 ≤ 1 and whose top S3 is the portion of the plane z = 1 + x that lies above S2. Homework Equations Surface integrals, and vector...
  42. B

    Surface Integral Homework: ∫∫σ3x2 + 3y2 + 3z2 dS

    Homework Statement Evaluate ∫∫σ3x2 + 3y2 + 3z2 dS where σ is the part of the cylinder x2 + y2 = 4 between the planes z = 0 , and z = 1, together with the top, and bottom disks. Homework Equations Surface integrals, maybe divergence theorem? The Attempt at a Solution I am having...
  43. U

    Surface Integral (two different answers?)

    Homework Statement Homework Equations The Attempt at a Solution Not sure what's wrong with mine or the provided solution..both seems to be right. My Solution: Provided Solution:
  44. G

    Reducing surface integral to line integral?

    Homework Statement hi I am trying to adjust this general integral to my problem, my problem consists of a semi-infinite rod, i.e. x in [0,∞) the primed variables are the integration variables Homework Equations http://img339.imageshack.us/img339/5038/42247711.jpg The Attempt at a...
  45. J

    How to Set Up a Surface Integral for Finding Average Value on a Unit Sphere?

    Homework Statement Find the average value of the function f(x,y,z)=xyz on the unit sphere in the first octant Homework Equations I know that I need the surface integral of xyz over the sphere and then need to divide by the surface area of the region, but I'm having a hard time setting...
  46. X

    Surface integral of potential*electric field

    could someone please explain why ∫(φE).dS (where φ is the potential and E is the electric field) is equal to zero in a general electrostatic case? Thank you
  47. L

    The Surface Integral of a Vector Field with Restrictions: How Do I Solve It?

    Homework Statement Given is the vector field, \overline{A} = (x2-y2, (x+y)2, (x-y)2). The surface: \overline{B} = (u+v, u-v, uv). The restrictions are the following: -1≤ u, v≤ 1, and the z-component of the normal has to be positive. Calculate I, I = ∫∫\overline{A}\cdot\overline{n}dS...
  48. K

    Surface Integral With Divergence Thm

    Homework Statement Let ##\mathit{F}(x,y,z) = (e^y\cos z, \sqrt{x^3 + 1}\sin z, x^2 + y^2 + 3)## and let ##S## be the graph of ##z = (1-x^2-y^2)e^{(1-x^2-3y^2)}## for ##z \ge 0##, oriented by the upward unit normal. Evaluate ##\int_{S} \mathit{F} \ dS##. (Hint: Close up this surface and use the...
  49. A

    Surface Integral of Vector F over x=xy+1

    Hi to all, Homework Statement Evaluate the surface integral of the vector F=xi+yj+zk over that portion of the surface x=xy+1 which covers the square 0≤x≤1 , 0≤y≤1 in the xy plane Homework Equations ∫∫F.ndσ n=∇g/|∇g| maybe transformation to the volume integral The Attempt at a...
  50. M

    Surface integral - I don't understand it

    Hello, So I am trying to understand surface integrals so I can can more insight to understand Gauss's Law. I am reading a book about it, and the example that is used to explain a surface integral is to have a flat surface that has a mass density that changes as a function of position in the x...
Back
Top