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. F

    Proving \int\int_{S} n dS = 0 for Closed Surface S

    Homework Statement Prove that \int\int_{S} n dS = 0 for any closed surface S.Homework Equations The Attempt at a Solution I used divergence theorem. But i thought it is applicable only if there is another vector multiplied to that outward unit vector (n). \int\int_{S} F {\cdot} n dS
  2. J

    Question about surface integrals on E&M

    I've seen on most books and in class that Gauss's law is usually stated like \oint \vec{E} \cdot d\vec{A} = \frac{q_{en}}{\epsilon_0} Shouldn't the integral be a surface integral rather than a line integral? I've seen times in problem resolution where they evaluate the integral as a...
  3. T

    Surface Integrals: Online Guide to Solving 3D Problems

    Can someone either explain here, or link me to an online document, on how to do surface integrals over surfaces in 3d (not simple ones like planes with x, y, or z held constant). I learned this back in Calculus 2 five years ago, and now I need to do it for my Electrodynamics course and I can't...
  4. M

    Surface Integrals: What am I doing wrong?

    Problem: evaluate the double integral of yz over that part of the plane z=y+3 which is inside the cylinder x2+y2=1 I evaluated with respect to z, from z=0 to z=y+3 I got (y3+9y+6y2)/2. Then I integrated this over x2+y2=1. To do that, I switched to polar coordinates, letting x=rcos(theta)...
  5. R

    Surface Integrals: Flux of F across S

    Homework Statement 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) = xy i + yz j + zx k S is the part of the paraboloid z = 3...
  6. D

    Surface Integrals in Polar Coordinates

    Homework Statement Find the area cut from the surface z = 2xy by the cylinder x^2 + y^2 = 6. [Hint: Set up the integral using rectangular coodinates, then switch to polar coordinates.] Homework Equations A = \iint \sqrt{{z_x}^2+{z_y}^2+1}dxdy = \iint...
  7. L

    How Do You Calculate Surface Integrals in Higher Dimensions?

    How would I calcluate a surface integral in dimensions greater than 3. For example, from the definition of a surfrace integral over a vector field: http://en.wikipedia.org/wiki/Surface_integral#Surface_integrals_of_vector_fields To compute the surface integral, I would first need a vector...
  8. C

    Surface integrals and heat flow

    Homework Statement The temperature u in a star of conductivity 6 is inversely proportional to the distance from the center: u = \frac{3}{\sqrt{x^{2} + y^{2} + z^{2}}} If the star is a sphere of radius 3, find the rate of heat flow outward across the surface of the star. Homework...
  9. B

    Surface Integrals: Computing Flux Through a Half Cylinder

    I'm working on an electrostatics problem that I'd appreciate some clarification on. I'm trying to compute the surface integral of a field \lambda(ix +jy) over a surface that is the half cylinder centered on the origin parallel to the x-axis - that is the end caps of the cylinder are located at...
  10. V

    Vector Surface Integrals: Limits and Unit Vectors

    Homework Statement I am given a vector field associated with a square area of a certain side, let's call this side dx, centered on the x-axis at a certain point, say x = x1. The sides of this cross sectional square are parallel to the axis of y and z. I have to compute the flux of this...
  11. I

    Mean Value Theorem in Surface Integrals

    I'm reading Div, Grad Curl, and All That, and in coming up with a formula for the divergence, H.M. Schey starts with a small cube centered at (x,y,z), labels the face parallel to the yz-plane as S1 and calculates \int\int_{S_1}\mathbf{F}\cdot\hat{\mathbf{n}}dS=\int\int_{S_1}F_x(x,y,z)dS...
  12. D

    Surface Integrals: Find Value w/Divergence Theorem

    Homework Statement Find the value of the surface integrals by using the divergence theorem \vec{F} = (y^2z)\vec{i} + (y^3z)\vec{j} + (y^2z^2)\vec{z} S: x^2 + y^2 + z^2 Use spherical coordinates. Homework Equations The Attempt at a Solution I've gotten the integral I think...
  13. I

    Jacobians and Surface integrals

    Why is it that when we evaluate a surface integral of: f(x, y ,z) over dS, where x = x(u, v) y = y(u, v) z = z(u, v) dS is equal to ||ru X rv|| dA Why don't we use the jacobian here when we change coordinate systems?
  14. S

    Surface Area of Cylinder Bounded by x^2 + y^2 = a^2

    Find the surface area of the the portions of the cylinder y^2 + z^2=a^2 bounded by x^2 + y^2 = a^2 not really sure how to go about this. tried to set up a double integral and use polar coordinates but don't know what boundaries to use etc.
  15. P

    Surface Integrals of Vector Fields question

    Homework Statement F=<0,3,x^2> computer the surface integral over the hemisphere x^2 + y^2 + z^2 = 9 z greater than or equal to 0, outward pointing normal. Homework Equations The Attempt at a Solution I don't know why I keep getting this problem wrong. The general formula for...
  16. C

    Calculating Flux for a Cylinder in the First Octant: A Parametric Approach

    Homework Statement calculate the upward flux of f(x,y,z) = <yz,2x+y,y^2+z> Let S be the portion of the cylinder z=4-y^2 lying in the first octant to the right of the plane y=4. a parametrization into the u v plane is:r(u,v)=(u,v,4-v^2) region is a rectangle in the uv plane with bounds, (0,0) ...
  17. P

    SURFACE INTEGRALS appying to z=f(x,y) x=f(y,z) and y=f(x,z)

    You know when a definition is given in terms of z=f(x,y) like the surface integral and its assmed to apply to y=f(x,z)and x=f(y,z) too ... Why is this? I know theyre just variables ...but since x y and z mean something specifically wrt the coordinate system Would it be trivial...
  18. C

    Surface Integrals: Calculating dxdy, dxdz, dydz

    When calculating surface integrals do you have to calculate double integrals for dxdy, dxdz and dydz and add up or what?
  19. A

    Double Surface Integrals in Polar Coordinates

    Homework Statement Find the surface area of the cone z=3x^2+y^2and above a region in the xy-plane with area 4. Homework Equations double integral sqrt( (dz/dx)^2 + (dz/dy)^2 +1) The Attempt at a Solution I was able to simplify the equation, I just don't know what to do...
  20. rohanprabhu

    Differential Area Element and surface integrals

    [SOLVED] Differential Area Element While doing surface integrals, I am not sure as to which of the following is the correct differential area element to be considered: i] dA = dx dy or ii] A = xy hence, using the product rule: dA = xdy + ydx
  21. R

    : Guass's Law and surface integrals

    I'm in need of help. I need the formulas for total-electric-flux enclosed for a final exam *tommorow. My teacher (nice guy but total slacker) never did any handouts, and I am not the quickest to catch on when he tried to explain in about 20 minutes the concept of surface integrals. I was just...
  22. K

    How to Parametrize the Surface Enclosed by a Curve of Intersection?

    1) (a past exam question) http://www.geocities.com/asdfasdf23135/advcal30.JPG I am stuck with the parametrizations. For part b, to evaluate the resulting surface integral, I think I need to parametrize the surface. How can I parametrize the surface enclosed by the curve of intersection in...
  23. K

    How Do You Compute and Visualize Surface Integrals?

    1) http://www.geocities.com/asdfasdf23135/advcal23.JPG For this question, we have to use the surface integral to compute the area, but I just can't picture what is going on geometrically, so I am stuck at the very begining...can someone please help me? 2)...
  24. H

    What the hell are surface integrals?

    I still can't quite see what I'm doing using surface integrals. I'd like an intuitive definition, something like Reinmann sums are to integrals. Believe me, I've seen enough theory with them and I'd rather have a feel for them before I go over the proofs. Here is what I think: You...
  25. M

    Surface Int. Homework: Compute g = xyz on x^2+y^2+z^2=1 Above z^2=x^2+y^2

    Homework Statement Compute the surface integral: g = xyz on x^2+y^2+z^2 = 1 above z^2=x^2+y^2. Homework Equations The Attempt at a Solution I'm only doubtful about the parameterization. Under normal circumstances, since x^2+y^2+z^2 = 1 is a sphere, we can write: r =...
  26. M

    Surface Area and Surface Integrals

    Homework Statement (Q) Find the area of the surface cut from the paraboloid x^2+y+z^2 = 2 by the plane y=0. Homework Equations The Attempt at a Solution The unit normal vector in this case will be j. Moreover, the gradient vector will be sqrt(4x^2+4z^2+1). And the denominator...
  27. M

    Surface Area and surface Integrals

    Homework Statement (Q) Find the area of the portion of the surface x^2 - 2z = 0 that lies above the triangle bounded by the lines x = sqrt(3), y = 0, and y = x in the xy-plane. Homework Equations The Attempt at a Solution The know how to find the gradient vector. The part...
  28. P

    Alright, so I forgot how to do Surface Integrals

    Homework Statement I have a vector function, and I need to take the surface integral of it over a hemisphere, top half only. I'm "confirming" the divergence theorem by doing a volume integral and surface integral. Already did the volume one so I have something to compare to already. The...
  29. B

    How to solve a surface integral problem involving a helicoid?

    Homework Statement Evaluate ∫∫S √(1 + x^2 + y^2) dS where S is the helicoid: r(u, v) = ucos(v)i + usin(v)j + vk, with 0 ≤ u ≤ 4, 0 ≤ v ≤ 3π The Attempt at a Solution What I tried to do was say x=ucos(v) and y=usin(v), then I plugged those into the sqrt(1+x^2+y^2) eq, which I ended...
  30. B

    Help Evaluate ∫∫F∙dS on Surface Integrals

    Please Help! Surface integrals I am wondering if someone can help me with the following? I am asked to evaluate ∫∫F∙dS where F(x,y,z) = z^2xi + (1/3y^3 +tanz)j + (x^2z+y^2)k and S is the top half of the sphere x^2+y^2+z^2 = 1. ∫∫F∙dS = ∫∫∫divFdV. Here, div F = x^2+y^2+z^2. I know that...
  31. B

    Evaluate ∫F∙dr: Surface Integrals Help

    [Can someone help with the following? I am supposed to evaluate the line integral of ∫F∙dr. The curve is oriented counterclockwise as viewed from above. So suppose that F(x,y,z) = (x+y^2)I + (y+z^2)j + (z+x^2)k, and C is the triangle formed by (1,00), (0,1,0), (0,0,1). I know that the...
  32. B

    How to evaluate a surface integral involving a paraboloid and a cylinder?

    I am wondering if someone could help me evaluate the following: I am asked to find the surface integral ∫∫ydS where S is part of the paraboloid y = x^2+z^2 that lies inside the cylinder x^2+z^2 = 4. The double integral could be rewritten as ∫∫y*√(4(x^2+z^2)+1)dS, or...
  33. S

    Solving Surface Integrals: A Challenge

    I'm working from H.M. Schey's Div, grad, curl, and all that, and am trying to figure out surface integration. One of the example problems boils down to the following surface integral over a projection, with z = 1 - x - y \sqrt{3} \int \!\!\! \int_R 1 - y \,dx \,dy I made x and y go from 0 to...
  34. M

    What is the difference between double and surface integrals?

    Hi eveybody, I'm having trouble in surface integrals. I know already what the double integrals measure; a multivariable function ( drawing surface) over "a region of domain".. Now, the surface integrals are for surfaces given by 2-dependent parametrizations over " the surface". my questions...
  35. J

    Surface integrals in spherical coordinates

    Hi, I am studying for finals and I'm having trouble calculating flux over sections of spheres. I can do it using the divergence theorem, but I need to know how to do it without divergence thm also. The problem is, when calculating a vector field such as F(x, y, z) = <z, y, x>, say over...
  36. M

    Surface integrals (without real integration)

    Given F= (ix+jy) Ln(x^2+y^2) and given S, which is a cylinder of radius r, and height h(in the z axis) evaluate \int\int_s F.n \,ds. It says that you shouldn't need to do any work if you think about it enough. I figured I could find the area of the main part to be 2 \pi r h then multiply that...
  37. B

    Solving Part (b) of Question Involving Surface Integrals

    Hi, can someone help me out with the following question parts? a) Let W be a compact region in R^3 bounded by a piecewise smooth closed surface S. Let f:W \to R be a C^1 scalar function. Prove that for all constant vectors \mathop c\limits^ \to , \int\limits_{}^{} {\int\limits_S^{} f...
  38. H

    How to Evaluate Surface Integrals Using Divergence Theorem?

    Please help! I'm soo confused with surface integrals and have several to do by tues for my tutorial. I don't really understand how to approach surface integrals! :cry: Could someone give me an over-view and help me through the question below - hopefully then I can manage the rest myself...
  39. B

    Surface Integrals Explained for Beginners

    At the risk of sounding imbecilic, I'm going to pose this question anyway. If I integral a vector function over a surface {a defined region R on a surface S} then what in fact am I doing? I know it sounds bizarre but I can see the logic of the process to find surface areas..but what does this...
  40. T

    Calculating Surface Integrals with Stokes' Theorem

    I expected Stokes theorem to make my life easier but these problems are even harder than the normal ones I've been doing. Use Stokes' Theorem to evaluate \int\int_ScurlFdS where F(x, y, z) = < x^2*y^3*z, sin(xyz) ,xyz > S: Part of cone y^2 = x^2 + z^2 that lies between the planes y =...
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