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Topic: Line integral

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	PlanetMath: path integral
		The path integral is a generalization of the integral that is very useful in theoretical and applied physics.
		A directed path integral on a closed path is denoted by summa and a circle with an arrow denoting direction.
		This is version 12 of path integral, born on 2002-02-03, modified 2004-09-10.
	planetmath.org /encyclopedia/PathIntegral.html (374 words)

	Area and line integrals (Site not responding. Last check: 2007-10-18)
		A line integral is also used for the general definition of work in mechanics.
		The line integral of electric field around a closed loop is equal to the voltage generated in that loop (Faraday's law):
		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)

	sciforums.com - Line-path
		a line integral is the integral of a vector field projected onto a path.
		a path integral is the integral of a function along a path.
		In my calculus III class, the integral along a path was called a line integral with no reference to fields.
	www.sciforums.com /showthread.php?t=28104 (323 words)

	Introduction to Line Integrals
		The integral above is called a line integral of f(x,y) along C. It is also called a line integral with respect to arc length.
		Line integrals are not restricted to curves in the xy plane.
		If C is a curve in the xy plane and R=0, it might be possible to evaluate the line integral using Green's theorem.
	www.math.oregonstate.edu /home/programs/undergrad/CalculusQuestStudyGuides/vcalc/lint/lint.html (449 words)

	Line Integrals (Site not responding. Last check: 2007-10-18)
		For one variable integration the geometrical figure is a line segment, for double integration the figure is a region, and for triple integration the figure is a solid.
		Fortunately, there is an easier way to find the line integral when the curve is given parametrically or as a vector valued function.
		The main application of line integrals is finding the work done on an object in a force field.
	www.ltcconline.net /greenl/courses/202/vectorIntegration/lineIntegrals.htm (571 words)

	UNL Math 107H, CalculusII, Section 005, Dunbar, 9:30-10:20 M-F
		The vector field is constant along this line as 2 j, the motion is along a straight line, the line integral is 2i \cdot 5 j = 0.
		Estimate the line integral of the vector field F = y i - x j along the unit circle centered at the origin with counterclockwise orientation as being positive, negative, or zero.
		The line integral measures the extent to which a curve in a vector field is going with the vector field or against it.
	www.math.unl.edu /~sdunbar/Teaching/Calculus208/Lessons/LineIntegrals/idealine.shtml (1449 words)

	Definition of a Line Integral
		The integral is then defined as the limit of the sequence of values Sn when the size of the partitions (usually taken to be the size of the largest piece) goes to zero.
		The line integral of a scalar function f along a curve C (also refered to as the path of integration) is defined as above.
		Let us compute, straight from the definition, the value of the line integral of the function f(x,y)=x^2-y along the segment of the y=x line from the origin up to (1,1).
	omega.albany.edu:8008 /calc3/line-integrals-dir/define-m2h.html (749 words)

	11. Line Integrals and Path Independence
		This is a good spot to say again: line integrals depend not only on the integral, but on the path (line) taken between the starting and ending points.
		where the integral is evaluated by using 1 and 2 for the lower and upper limits on x and 0 and 1 for the same on y.
		The former have line integrals that are independent of path.
	scholar.chem.nyu.edu /0652/notes/math/node20.html (1522 words)

	Review for Midterm
		Use Green's theorem to evaluate a line integral on a closed curve of a vector field by calculating a related double integral.
		Evaluate a line integral of a conservative vector field over a given path by finding an easier path for the integration.
		Use the fundamental theorem of line integrals to calculate the line integral of a conservative vector field.
	www-math.cudenver.edu /~billups/courses/ma2423/review_final.html (807 words)

	AMPERE’S LAW & MAGNETIC FIELDS
		If the closed line integral is not zero, you know that there is a net current within the closed path which is generating a magnetic field.
		In the previous figure are four groups of line integral contributions (groups 1 – 4, from left to right), from the four rectangular path sides (a, b, c, d).
		Group 3 (8 line segments of 0.02 meters each) involves the contributions from the path through the solenoid tunnel on axis, where the B field is strong and parallel to the path.
	www.physics.rutgers.edu /ugrad/labs/online/Ampere.html (2591 words)

	OBAN tests
		Although the particular prototype methods chosen have the property that the line integral method truncates the amplitude of the input field slightly more than the traditional scheme, the pattern of the field is significantly better using the line integral technique than with the traditional method.
		However, Davies-Jones (1993) demonstrated that all line integral methods are essentially the same when they assume a linear variation of the field between vertices of the triangles typically used for estimating the derivatives.
		By using line integrals to estimate the derivatives, the triangle method is doing something different, but we are not prepared to give a complete analysis of the process in comparison to the traditional approach.
	www.cimms.ou.edu /~doswell/oban_test/oban_test.html (6714 words)

	Path integral of a scalar-valued function* (Site not responding. Last check: 2007-10-18)
		Moreover, we assume each line segment has constant density, so the mass of the line segment is simply its length times its density.
		The density of each line segment is given by the density of the slinky at the point where the upper end of the line segment touches the slinky.
		Since the total length of the line segments approaches the slinky length, the total mass of the line segments approaches the mass of the slinky (notice that mass(dt) on the green slider approaches the red mark labeled "actual mass").
	www.math.umn.edu /~nykamp/m2374/readings/pathint (832 words)

	Line Integrals - Page 3 - Physics Help and Math Help - Physics Forums
		I asked about line integrals with respect to x and y and why it was different from line integrals with respect to arc length.
		I was confusing this definite integral with that of the line integral with respect to x or y or z.
		It seems that line integrals with respect to x,y and z are the result of line integrals with respect to arclength of vector functions ONLY, Thats what they are and where the come from.
	www.physicsforums.com /showthread.php?p=411328 (1896 words)

	14
		A line integral along its path might be able to find the necessary fuel load for the rocket launch.
		To see how a line integral may be used for this, think about a force field F and consider an object moving along a path C in the field.
		This line integral appears in other applications and is the basis for this definition of the line integral of a vector field.
	www.ac.cc.md.us /~donr/CalcIII/unit5/lesson2/u5l2.html (1073 words)

	Line integral of a vector field (Site not responding. Last check: 2007-10-18)
		The line integral of a vector field will play a crucial role in the rest of the class.
		Out of the four fundamental theorems that form the cornerstones of the second half the course, three of the them involve line integrals of vector fields.
		(And the fourth involves surface integrals of vector fields, which are closely related to line integrals of vector fields.) I cannot emphasize too strongly the importance of these integrals.
	www.math.umn.edu /~nykamp/m2374/readings/lineint (564 words)

	PHYS208 Line of charge example (Site not responding. Last check: 2007-10-18)
		The only variable which matters is R, the distance from the line of charge.
		There is no axial dependence because the line of charge is infinitely long, i.e.
		The infinite line of charge is certainly a physically-impossible situation to set up (infinite amount of charge) but one whose consideration is useful in interpreting finite distributions, e.g.
	www.physics.udel.edu /~watson/phys208/lineofcharge/lineofcharge1.html (97 words)

	Why are line integrals interesting? (Site not responding. Last check: 2007-10-18)
		The real problem is in our lack of ability to imagine in our minds what the line integral of a fluctuating quantity might look like.
		If the spectrum is not isotropic, then a line integral measurement still provides an accurate measurement of a part of the fluctuation spectrum, and is therefore useful in diagnosing the plasma and benchmarking computer models.
		Line integrated measurements are inherently less desirable than full 2-D measurements, but a line-integrated measurement with excellent spatial and temporal resolution such as PCI can be a vital complement to other measurements which provide 2-D measurements with less resolution.
	fusion.gat.com /diag/pci/line_integral.html (234 words)

	Line Integral
		The line integral measures the energy involved in tracing the path.
		The line integral through a linear combination of fields is equal to the linear combination of the individual line integrals, traveling through each field separately.
		Also, the line integral across an entire path is the sum of the line integrals along the subpaths, provided the subpaths join together to make the entire path.
	www.mathreference.com /ca-vec,intro.html (395 words)

	Complex Line Integrals I, part 1
		Then the complex line integral of f over C is given by
		Calculate the line integral of the square function, f
		Calculate the integrals of all four functions over each of the two curves and record your results.
	www.math.duke.edu /education/ccp/materials/engin/cint1/cint1.html (225 words)

	Line Integrals
		The total work done is the Integral along the length of the curve of the work on each piece.
		The component of a force which does the work is the one which acts along the curve, (in physics, the tangential component) so we can write this as f.t ds where t is the unit vector tangent to the curve.
		The line integral along r2 contributes nothing, so the line integral around arc+bottomline is the same as the line integral around arc.
	www.ma.iup.edu /projects/CalcDEMma/vecicalc/vecintcalc1.html (379 words)

	Volume Line Integral Convolution
		Line Integral Convolution (LIC) is an elegant algorithm for visualizing vector fields.
		This, further, inspired Interrante and Grosch to enhance 3D LIC by using a sparsely opaque input texture and to use LIC to correlate both color and opacity values in the direction of the flow.
		If the stream lines are shaded as described in Section 2.1, the local orientation of the flow can be clearly depicted but streamlines that are separated in depth but which flow in a similar direction cannot effectively be distinguished.
	www.cg.tuwien.ac.at /studentwork/CESCG/CESCG98/TTheussl/node6.html (431 words)

	Line Integral Convolution on Triangulated Surfaces - Teitzel, Grosso, Ertl (ResearchIndex) (Site not responding. Last check: 2007-10-18)
		Line Integral Convolution on Triangulated Surfaces - Teitzel, Grosso, Ertl (ResearchIndex)
		Abstract: Line Integral Convolution (LIC), introduced by Cabral and Leedom in 1993, is a powerful technique for generating striking images of vector data.
		Based on local filtering of an input texture along a curved stream line segment in a vector field, it is possible to depict directional information of the vector field at pixel resolution.
	citeseer.ist.psu.edu /teitzel97line.html (361 words)

	Line integrals (Site not responding. Last check: 2007-10-18)
		Note that the integral depends on the route taken between the initial and final points.
		Suppose that we have a line integral which does not depend on the path of integration.
		It is clear that there are two distinct types of line integral.
	farside.ph.utexas.edu /~rfitzp/teaching/em1/lectures/node13.html (168 words)

	Calculus III (Math 2415) - Line Integrals - Line Integrals - Part I (Site not responding. Last check: 2007-10-18)
		It is no coincidence that we use ds for both of these problems. The ds is the same for both the arc length integral and the notation for the line integral.
		Evaluation of line integrals over piecewise smooth curves is a relatively simple thing to do. All we do is evaluate the line integral over each of the pieces and then add them up. The line integral for some function over the above piecewise curve would be,
		So, we can change the direction of a line integral with respect to arc length and not change the value of the integral.
	tutorial.math.lamar.edu /AllBrowsers/2415/LineIntegralsPtI.asp (1245 words)

	sciforums.com - Line-path
		09-06-03 04:42 PM a line integral is the integral of a vector field projected onto a path.
		09-06-03 04:55 PM In my calculus III class, the integral along a path was called a line integral with no reference to fields.
		09-06-03 05:02 PM "Line integral" and "path integral" are different names for the same thing.
	www.sciforums.com /printthread.php?t=28104 (334 words)

	An Implementation of Line Integral Convolution
		Because of the large numbers of lines needed for this vector field, there are severe aliasing problems and loss of information.
		This LIC image clearly shows the lines of force present in the gravity field.
		Note the dead areas (equilibrium points) both toward the center of the image, and between the two point sources in the upper left.
	www.cs.utah.edu /~wmartin/cs523project (694 words)

	16.4 Line Integral Convolution (LIC) with Texture (Site not responding. Last check: 2007-10-18)
		Line integral convolution is texture based technique for visualizing vector fields and has the advantage of being able to visualize large and detailed vector fields in a reasonable display area.
		Line integral convolution involves selectively blurring a reference image as a function of the vector field to be displayed.
		The goal is to process the image with the vector field, using line integral convolution, so you can visualize it.
	www.opengl.org /resources/tutorials/sig99/advanced99/notes/node318.html (364 words)

	lesson 3 (Site not responding. Last check: 2007-10-18)
		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.)
		Like the case of line integral, we have to parametrize the surface.
	www2.hawaii.edu /~plam/ph400/lesson3.html (412 words)

	Computing Integrals
		In many cases, it is an integral of some function that is of interest in the solution of a PDE problem.
		FlexPDE has an extensive repertoire of integration facilities, including volume integrals, surface integrals on bounding surfaces and line integrals on bounding lines.
		In the case of internal boundaries, there is sometimes a different value of the integral on the two sides of the boundary.
	www.pdesolutions.com /userguide/computingintegrals.htm (192 words)

	Lecture Notes Chapter 1 (Site not responding. Last check: 2007-10-18)
		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 evaluation of the line integral will be as follows:
		and is identical to the line integral of
	teacher.nsrl.rochester.edu /PHY217/Exams/SolutionsMidtermExam1/SolutionsMidtermExam1.html (648 words)

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