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Topic: Surface integral

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	Surface integral - Wikipedia, the free encyclopedia
		In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral.
		Given a surface, one can integrate over it scalar fields (that is, functions which return numbers as values), and vector fields (that is, functions which return vectors as values).
		Surface integrals have applications in physics, especially in the classical theory of electromagnetism.
	en.wikipedia.org /wiki/Surface_integral (932 words)

	Flux - Wikipedia, the free encyclopedia
		The divergence theorem states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the divergence).
		Stokes theorem states that the flux of the curl of a vector field is the path integral of the vector field over this boundary.
		Except in the case of active transport, net flux is directly proportional to the concentration difference across the membrane, the surface area of the membrane, and the membrane permeability constant.
	en.wikipedia.org /wiki/Flux (1627 words)

	[No title]
		The quick answer here is that the individual surface integrals, when divided by the area of the face, would compute the average value of the z coordinate on the face.
		Therefore, we can compute the surface integrals easily by averaging the z coordinates of the vertices of the face, and multiplying by an area: the surface integral int(zdxdy) over a face spanned by (x0 y0 z0), (x1 y1 z1), and (x2 y2 z2) is (z0+z1+z2)/3.
		Then that face contributes to the surface integral the sum A integral(x) + B integral(y) + C integral(1) where the integral may be taken over the projection of the face to the xy-plane (translation: ignore the z coordinates on the face).
	www.math.niu.edu /~rusin/known-math/95/volume.polyh (911 words)

	Surface (Normal) Integrals
		A scalar field produces a "surface integral", while a vector field produces a "surface normal integral", although the latter is sometimes called a surface integral as well, just like the former.
		The surface integral is the double integral of surface area times f, where f is the density function.
		The area of each tiny patch on the surface is multiplied by the intensity of the field at that point, times the sine of the angle between the field and the surface.
	www.mathreference.com /ca-surf,si.html (871 words)

	[No title]
		In words, Ampère's integral law as given by (1) requires that the line integral (circulation) of the magnetic field intensity H around a closed contour is equal to the net current passing through the surface spanning the contour plus the time rate of change of the net displacement flux density
		Surface S is enclosed by contour C having positive direction determined by the right-hand rule.
		The field tangential to the surface current undergoes a jump that is equal in magnitude to the surface current density.
	web.mit.edu /6.013_book/www/chapter1/1.4.html (1505 words)

	Ch160answers
		Thus the answer is independent of the surface chosen as it must be because the left hand side of Ampere's law is independent of the surface chosen.
		In some texts you may see the closed surface integral indicated by a double integral with a circle over it to remind you that it is a two dimensional integral...
		The second surface is a chef's hat surface through which there is no electric current, but there is a changing electric flux.
	www.lawrence.edu /fast/STONEKIM/courses/00_01/Ch160.html (1543 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Surface area = y x integral of SquareRoot(1 + --- + ---) dx dy over 0
		Surface area = b a x + y int int of --------------- dx dy 0 0 SquareRoot(2xy) now it is easy to do the integrals of SquareRoot(x/(2y)) and SquareRoot(y/(2x)) and we get...
		Similarly (but in 1 dimension less) we can compute AREA as a curve (1D) integral instead of a double integral as would normally be expected, via area(R) = double integral 1 dxdy for (x,y) in R = (1/2) curve integral (x,y).
	www.math.temple.edu /~wds/math127hw7ans (515 words)

	Maxwell’s Electromagnetic Field Equation No. 1 (Site not responding. Last check: 2007-10-22)
		The circle on the integral sign indicates that the integral or summation of area is taken of a closed continuous surface.
		This integral equation states that the amount of electric flux density normal to a surface is caused by a specific amount of charge, q, enclosed by the surface.
		The integral of the divergence of a vector summed throughout the volume is equal to the integral of the product of the vector times its effective area summed over the area.
	www.iit.edu /~smile/guests/gsmxsec1.htm (3180 words)

	lesson 3 (Site not responding. Last check: 2007-10-22)
		The most common line integral in physics is the work done by a force from point a to point b along a path.
		Similar to the discussion of line integral of a scalar function, the author prejudicially choose dxdy, that is why in example 1.6.3 (p.34) he gets teh surface integral of dxdy over a hermishpere eqauls to pi and not 2 pi (surface area of a hemisphere with unit radius.)
		The normal to the surface is gradient of f.
	www2.hawaii.edu /~plam/ph400/lesson3.html (412 words)

	Brick Fundraising (Site not responding. Last check: 2007-10-22)
		Hard-burned brick should be used for face work exposed to the weather, and soft brick for filling, foundations, and the like.The standard brick measures approximately 2.25 x 4 x 8 inches, and has a crushing strengthof between 1000 and 3000 lb/in² (7 to 21 million pascals) depending on quality.
		Ahighly impervious and ornamental surface may be laid on brick either by salt glazing, inwhich salt is added during the burning process, or by the use of a "slip," which is a glaze material into which the bricks aredipped.
		Subsequent reheating in the kiln fuzes the slip into a glazed surface integral with the brick base.
	www.relativeaccess.com /File/23027-Brick.Fundraising.Html (566 words)

	Surface integrals* (Site not responding. Last check: 2007-10-22)
		As you probably guessed, there are two types of surface integrals: surface integrals of scalar-valued functions and surface integrals of vector fields.
		This integral is a two-dimensional analog of the
		At these points, the fluid is crossing the surface in the opposite direction than it is at most points on the surface.
	www.math.umn.edu /~nykamp/m2374/readings/surfint (689 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Introduction In the presence of an incident wavefront, the surface properties of an object affect scattering of the wave from the object's surface.
		It is known that a hypersingular integral operator and a first-kind integral operator are ideal preconditioners for each other, in the sense that the composition of the two has the spectral characteristics of a second-kind integral operator.
		of the eigenvalues on a smooth surface stems from the short distance behavior of the kernel, it may be seen that the asymptotic behavior of the various operators on spheres should also obtain for any closed surface that can be obtained by smooth deformation of a sphere.
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=01/98953.011227&ELEMENT_SET=DECL (8098 words)

	Calculus III (Math 2415) - Surface Integrals - Surface Integrals (Site not responding. Last check: 2007-10-22)
		Now, how we evaluate the surface integral will depend upon how the surface is given to us. There are essentially two separate methods here, although as we will see they are really the same.
		In order to evaluate a surface integral we will substitute the equation of the surface in for z in the integrand and then add on the often messy square root. After that the integral is a standard double integral and by this point we should be able to deal with that.
		In order to do this integral we’ll need to note that just like the standard double integral, if the surface is split up into pieces we can also split up the surface integral.
	tutorial.math.lamar.edu /AllBrowsers/2415/SurfaceIntegrals.asp (1003 words)

	[No title]
		The bounds on the integrals have to be chosen to describe the region we are integrating over.
		Sometimes doing some integral is a lot easier in a nice frendly coordinate system than it is in cartesian coordinates.
		Sometimes the surface integral is easier (select a good choice of surface - since all surfaces work, choose an easy one to handle) sometimes the curve integral is easier.
	www.math.temple.edu /~wds/m127revvecint (905 words)

	Active Skim View of: A Boundary Integral Approach in Primitive Variables For Free Surface Flows
		In free surface flows, this is a non linear contribution, because ' + 'the boundary normal velocity component to is strictly dependent on the fluid velocity field.
		In ' + 'free surface flows, this is a non linear contribution, because the boundary normal velocity component to is strictly dependent on the fluid velocity ' + 'field.
		In free surface flows, this is a non linear contribution, ' + ' because the boundary normal velocity component to is strictly dependent on the fluid velocity field.
	www.nap.edu /nap-cgi/skimit.cgi?isbn=0309045754&chap=221-238 (11419 words)

	Lecture Notes Chapter 1 (Site not responding. Last check: 2007-10-22)
		Stoke's theorem states that the line integral of a vector function around a closed loop is equal to the surface integral of the curl of this vector function across a surface enclosed by this loop.
		The direction of the surface vector of the triangle can be determined by calculating the vector product between two vectors laying in the plane of the triangle.
		Here we have used the fact that all charge is located on the surface of the sphere, which is an equipotential surface (surface of constant potential).
	teacher.nsrl.rochester.edu /PHY217/Exams/SolutionsMidtermExam1/SolutionsMidtermExam1.html (648 words)

	Calculus III (Math 2415) - Surface Integrals - Stokes' Theorem (Site not responding. Last check: 2007-10-22)
		In this theorem note that the surface S can actually be any surface so long as its boundary curve is given by C. This is something that can be used to our advantage to simplify the surface integral on occasion.
		Now, all we have is the boundary curve for the surface that we’ll need to use in the surface integral. However, as noted above all we need is any surface that has this as its boundary curve. So, let’s use the following plane with upwards orientation for the surface.
		In both of these examples we were able to take an integral that would have been somewhat unpleasant to deal with and by the use of Stokes’ Theorem we were able to convert it into an integral that wasn’t too bad.
	tutorial.math.lamar.edu /AllBrowsers/2415/StokesTheorem.asp (597 words)

	Area and line integrals (Site not responding. Last check: 2007-10-22)
		An area integral of a vector function E can be defined as the integral on a surface of the scalar product of E with area element dA.
		Such an integral is also used for the calculation of voltage difference since voltage is work per unit charge.
		The line integral of a force over a path is equal to the work done by that force on the path.
	hyperphysics.phy-astr.gsu.edu /hbase/intare.html (213 words)

	[No title]
		Thus, the open surface integrals of (1.4.1) become closed, while the contour integral vanishes.
		But now, in view of Gauss' law, the surface integral of the electric displacement can be replaced by the total charge enclosed.
		Thus, the volume integral in (2) gives the net charge -q, while contributions to the surface integral only come from where S cuts through the wire.
	web.mit.edu /6.013_book/www/chapter1/1.5.html (1432 words)

	Surface Integrals
		Suppose that the density per unit area of the surface is given by the function P(x,y,z).
		The notation for a surface integral of a function P(x,y,z) on a surface S is
		Note that if P(x,y,z)=1, then the above surface integral is equal to the surface area of S.
	www.math.oregonstate.edu /home/programs/undergrad/CalculusQuestStudyGuides/vcalc/surfint/surfint.html (430 words)

	The Divergence Theorem (Site not responding. Last check: 2007-10-22)
		The Divergence Theorem relates relates volume integrals to surface integrals of vector fields.
		There are various technical restrictions on the region R and the surface S; see the references for the details.
		The surface integral of F on the entire surface is 128*pi.
	www.math.oregonstate.edu /home/programs/undergrad/CalculusQuestStudyGuides/vcalc/diverg/diverg.html (423 words)

	14
		The rest of this unit will focus on surface integrals and their applications.
		Let S be an oriented surface given by z = g(x,y) and R be the projection of S (shadow) onto the xy plane.
		Ex 4 Set up the integral to find the flux of F through S Where N is the 'upper' unit normal vector to S where F = xi+yj and S is the plane 2x + 3y + z = 6 in the first octant.
	www.ac.cc.md.us /~donr/CalcIII/unit5/lesson6/u5l6.html (1170 words)

	5-5.html (Site not responding. Last check: 2007-10-22)
		of the vector field through the surface is a measure of the rate of change of the volume of the flow through the surface, and it is defined by the surface integral
		Uniformity of the field thus implies that every small subrectangle of the square has the same property, so that the total flux--i.e., the rate of change of the total volume of the flow through the square--is 3 times the area of the square, so that
		This is the same result that we obtain with the flux integral.
	math.etsu.edu /MultiCalc/Chap5/Chap5-5/5-53.html (242 words)

	Diffraction Theory
		However, the Kirchhoff integral is a simple average of the two Rayleigh integrals.
		Before we reduce the surface integral to a line integral, a few more aspects of this restricted problem: Clearly the diffraction pattern produced by this one-dimensionally varying (along x') aperture will not vary with z.
		This integral could to be applied to all six components of the surface fields E(x’) and H(x’).
	www.lci.kent.edu /boslab/projects/diffraction_theory (2108 words)

	Nat' Academies Press, Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics (1994)
		in which the line integral is counterclockwise around the outer edges of the computational region and clockwise around the hull.
		The x-derivative in front of the surface integral was obtained from the symmetry of r with respect to
		The range of integration for the surface integrals is taken to be the finite portion of the mean free-surface level which is to be paneled later.
	www.nap.edu /books/NI000061/html/81.html (874 words)

	The momentum equation in integral form (Site not responding. Last check: 2007-10-22)
		Occasionally, it is useful to have an integral relation specifying the momentum balance for certain regions of the fluid.
		The momentum balance in integral form is useful when all the terms can be written as integrals over the bounding surface
		The remaining volume integral, on the LHS, which normally prevents use of the integral relation, is zero in the case of steady motion.
	astron.berkeley.edu /~jrg/ay202/node65.html (260 words)

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