
	Where results make sense

Topic: Integral curve

Related Topics

Social conservativism

Monroe Township, Michigan

Winchester, Connecticut

Galapagos Islands

Eduardo Mata

Emergency Rations

Lavaca, Arkansas

	VARIATIONS, CALCULUS OF - Online Information article about VARIATIONS, CALCULUS OF
		According to this method the curve is defined by specifying x and y as one-valued functions of a parameter O. The integral is then of the form f fof(x, y, x, y)do, where the dots denote differentiation with respect to 0, and f is a homogeneous function of x, y of the first degree.
		The integral to be made a minimum is ry(i2+y2)Ido, and the principal equation is d (]) yx a (Z2 — o +.y2)i — of which the first integral is Yx (x2+5,2)1= c, 1+(dx)2': c and the stationary curves are the catenaries y=c cosh((x—a)fc}.
		curve is a stationary curve for this integral.
	encyclopedia.jrank.org /VAN_VIR/VARIATIONS_CALCULUS_OF.html (4356 words)

	Integral
		The area under the curve formed by plotting function f(x) as a function of x can be approximated by drawing rectangles of finite width and height f equal to the value of the function at the center of the interval.
		The idea of the integral is to increase the number of rectangles N toward infinity by taking the limit as the rectangle width approaches zero.
		Integrals are useful for finding the area under curves which can be approximated by geometrical methods.
	hyperphysics.phy-astr.gsu.edu /hbase/integ.html (346 words)

	Springer Online Reference Works
		tangent to the curve (2); curves consisting of an arbitrary segment of the curve (2) and the two straight lines of the family (3) tangent to (2) at each end of this segment.
		The family (3) forms the general solution, while the curve (2), which is the envelope of the family (3), is the singular solution (see [2]).
		Therefore, geometric problems in which it is required to determine a curve in terms of a prescribed property of its tangents (common to all points of the curve) leads to a Clairaut equation.
	eom.springer.de /c/c022350.htm (240 words)

	Sensible Calculus IV.D (Site not responding. Last check: 2007-11-03)
		The instruction now is to draw a curve starting at some point (perhaps based on some initial condition) so that if the curve passes through or near one of the line segments in the field, the tangent to the curve should have a slope close to that of the line segment of the field.
		Integral Curves: Since the curves drawn in the tangent field assemble information from the differential equation and represent solutions to the differential equation, they are called integral curves.
		Since a tangent field and integral curves for a differential equation display only a sampling of the information contained in the differential equation, these tools are most useful to suggest either results about a particular solution to the differential equation or to give more qualitative understanding about the general solution to the differential equation.
	www.humboldt.edu /~mef2/book/ch4/IVD/IV_D.html (1762 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.
		Where the domain is now the curve C and the function is a function that assigns a scalar (number) to each point on the curve.
		Recall that a parametrization of a curve is a representation of the points on the curve with a vector valued function of a scalar parameter t.
	omega.albany.edu:8008 /calc3/line-integrals-dir/define-m2h.html (749 words)

	Arc length - Wikipedia, the free encyclopedia
		Curves with closed-form solution for arc length include the catenary, circle, cycloid, logarithmic spiral, parabola, semicubical parabola and (mathematically, a curve) straight line.
		In the 1600s, the method of exhaustion led to the rectification by geometrical methods of several transcendental curves: the logarithmic spiral by Evangelista Torricelli in 1645 (some sources say John Wallis in the 1650s), the cycloid by Christopher Wren in 1658, and the catenary by Gottfried Leibniz in 1691.
		In 1659 van Heuraet published a construction showing that arc length could be interpreted as the area under a curve - this integral, in effect - and applied it to the parabola.
	en.wikipedia.org /wiki/Arc_length (938 words)

	[No title]
		The bounds on the integrals have to be chosen to describe the region we are integrating over.
		A CURVE INTEGRAL (OF A SCALAR) (which is a single integral since a curve is ONE dimensional) is an integral /
		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)

	Conservative Case
		In Figure 3.6(d) we see that there are two integral curves approaching the saddle point S and two integral curves departing from S. These separatrices ``separate'' the phase plane into two distinct regions.
		Each integral curve inside the separatrices goes around one center, and hence corresponds to an asymmetric periodic oscillation about either the left or the right center, but not both.
		The integral curves outside the separatrices go around all three stationary points and correspond to large-amplitude symmetric periodic orbits (Figure 3.6(d)).
	cnls.lanl.gov /People/nbt/Book/node65.html (438 words)

	Integral
		The integral control rod worth is the total reactivity worth of the rod at that particular degree of withdrawal and is usually defined to be the greatest when the rod is fully withdrawn.
		If the slope of the curve for integral rod worth in Figure 9 is taken, the result is a value for rate of change of control rod worth as a function of control rod position.
		The integral rod worth at a given withdrawal is merely the summation of all the differential rod worths up to that point of withdrawal.
	www.tpub.com /doenuclearphys/nuclearphysics74.htm (870 words)

	PlanetMath: integral curve
		Anyone with an account can edit this entry.
		Cross-references: functions, local coordinates, tangent vector, open interval, curve, point, vector field, smooth, smooth manifold
		This is version 2 of integral curve, born on 2005-05-17, modified 2005-05-18.
	planetmath.org /encyclopedia/IntegralCurve.html (43 words)

	Line integral - Wikipedia, the free encyclopedia
		In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given vector field along a given curve.
		The integral is then the limit of this sum, as the lengths of the subdivision intervals approach zero.
		However, path integrals in the sense of this article are important in quantum mechanics; for example, complex contour integration is often used in evaluating probability amplitudes in quantum scattering theory.
	en.wikipedia.org /wiki/Line_integral (750 words)

	Function Fields
		The function field of the curve is the corresponding field of fractions in the affine case and the homogeneous degree 0 part of this in projective cases.
		Given an element f on a curve C and a place p of C return a series which is the expansion of f at p and the uniformizing element of p.
		The L-polynomial and the zeta function of the curve C over the extension of degree m of the base ring of C, which must be a finite field.
	www.math.lsu.edu /magma/text1176.htm (2592 words)

	Structure Operations
		Given an elliptic curve E, this function returns a sequence consisting of the Weierstrass coefficients of E; this is the sequence [a_1, a_2, a_3, a_4, a_6] such that E is defined by y^2z + a_1xyz + a_3yz^2=x^3 + a_2x^2z + a_4xz^2 + a_6z^3.
		Given an elliptic curve E, defined over Q with integral coefficients, and a prime number p, this function returns the local Tamagawa number of E at p, which is the index in E[Q_p] of the subgroup E^0[Q_p] consisting of points with non-singular reduction modulo p.
		Given an elliptic curve defined over Q with integral coefficients, this function returns generators for the Mordell-Weil group of E, in the form of a sequence of points of E. The i-th element of the sequence corresponds to the i-th generator of the group as returned by the function Mordell-Weil.
	www.math.ufl.edu /help/magma/text474.html (1122 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		The centroid of a curve is just the center of gravity of a piece of wire bent into the shape of the curve.
		Formula for the y-coordinate of the centroid of a 1-D curve: Integral from a to b y ds Total moment of the curve ------------------------- = ------------------------- Integral from a to b ds Length of the curve where ds is the arc-length differential.
		Formula for the y-coordinate of the centroid of a 2-D shape: Integral from a to b y^2/2 dx Total moment of the shape ----------------------------- = ------------------------- Integral from a to b y dx Area of the shape Here is a trivial example: Let y=mx, we will generate a cone of height h.
	www.math.niu.edu /Papers/Rusin/known-math/99/pappus (379 words)

	Calculus in Case You Forgot
		A derivative is the slope of a curve.
		In other words, at x = 0, where the sine curve is at its steepest, the slope is cos(0) = 1, and at 90 degrees, where the sine curve reaches maximum and starts to decrease, the slope is cos(90) = 0.
		The integral of a function is usually written f f(x)dx, where the dx term represents the width of the narrow strips under the curve.
	www.uwgb.edu /dutchs/MATHALGO/calculus_in_case_you_forgot.htm (2198 words)

	Curve fitting made easy - The Industrial Physicist
		With the correct model and calculus, one can determine important characteristics of the data, such as the rate of change anywhere on the curve (first derivative), the local minimum and maximum points of the function (zeros of the first derivative), and the area under the curve (integral).
		To find the values of the model’s parameters that yield the curve closest to the data points, one must define a function that measures the closeness between the data and the model.
		The prediction interval is represented by two curves lying on opposite sides of the fitted curve.
	www.aip.org /tip/INPHFA/vol-9/iss-2/p24.html (2314 words)

	Sample Configurable JCM Applet: IntegralCurves
		The curves will also be stopped automatically if their coordinates become undefined or unreasonably large.
		Curves will also be cleared if a new vector field is drawn.
		An integral curve will be started at each of these points when the example is loaded.
	math.hws.edu /javamath/config_applets/IntegralCurves.html (983 words)

	Creation Functions
		Returns the j-invariant of the elliptic curve E, which equals c_4^3/Delta, in terms of the c-invariants and the discriminant of the curve.
		Given an elliptic curve E, defined over Q with integral coefficients, and a prime number p, this function returns the local Tamgawa number of E at p, which is the index in E[Q_p] of the subgroup E^0[Q_p] consisting of points with non-singular reduction modulo p.
		Given an elliptic curve E defined over Q with integral coeffients, as well as a prime number p this function returns the local information at the prime p as a 5-tuple, consisting of p, its multiplicity in the discriminant, its multiplicity in the conductor, the Tamagawa number at p and the Kodaira symbol.
	www.math.uiuc.edu /Software/magma/text444.html (571 words)

	Manual do Maxima: 63. plotdf
		Integral curves can be obtained by clicking on the plot, or with the option
		If a pair of coordinates are entered in the field "Trajectory at" in the "Config" dialog menu, and the "enter" key is pressed, a new integral curve will be shown, in addition to the ones already shown.
		When "Replot" is selected, only the last integral curve entered will be shown.
	maxima.sourceforge.net /docs/manual/pt/maxima_63.html (1134 words)

	Springer Online Reference Works
		(the integral on the right is an integral over a real interval), and is called a line integral of the first kind, or a line integral with respect to arc length.
		It is the limit of suitable integral sums, which can be described in terms related to the curve.
		The relationship between line integrals and integrals of other types is established by the Green formulas and the Stokes formula.
	eom.springer.de /c/c027410.htm (450 words)

	Curves over the Rationals
		Given an elliptic curve E defined over Q and a prime number p, this function returns the local Tamagawa number of E at p, which is the index in E(Q_p) of the subgroup E^0(Q_p) of points with nonsingular reduction modulo p.
		Four-descent is done on a two-cover, that is a hyperelliptic curve defined by a polynomial of degree four, which the assumption that the quartic of this descendent does not have a rational root.
		The isogenous curves of the minimal twist corresponding to these roots are then computed, and then these curves are twisted back to get the isogenous curves of E. In all cases, if the conductor is squarefree, some small values of the Frobenius traces are checked mod p to ensure the feasibility of a p-isogeny.
	www.math.lsu.edu /magma/text1220.htm (5136 words)

	Calculus III (Math 2415) - Surface Integrals - Stokes' Theorem (Site not responding. Last check: 2007-11-03)
		C. To get the positive orientation of C think of yourself as walking along the curve. While you are walking along the curve if your head is pointing in the same direction as the unit normal vectors while the surface is on the left then you are walking in the positive direction on C.
		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)

	An integral curve
		A trajectory represents an integral curve (or solution trajectory) of the differential equations with initial condition
		9 (Integral Curve for a Constant Velocity Field) The simplest case is a constant vector field.
		The integral curve is generally found by determining the eigenvalues of the matrix
	planning.cs.uiuc.edu /node382.html (228 words)

	Sensible Calculus V.A (Site not responding. Last check: 2007-11-03)
		The problem of solving differential equations was investigated using graphical tools (tangent fields and integral curves), numerical tools (Euler's method for estimating the values of solutions with initial conditions- a numerical integration) and, for a more limited family of differential equations, symbolically (indefinite integrals).
		As the number of summands increases the estimating sums approach a single number which can be interpreted in the two distinct models using the function P to describe either the velocity of a moving object or the graph of a function bounding of a planar region.
		Give an interpretation with a related visualization for the following integrals a) as an area of a region in the plane b) as the net change in position of a moving object, and c) as the net change in the second coordinates of two points and an integral curve for a differential equation.
	www.humboldt.edu /~mef2/book/ch5/VA/CH5A.html (3475 words)

	SECTION 2. MATHEMATICS, STATISTICS AND SAMPLING
		The differential and integral calculus are important mathematical tools in the analysis of the dynamics of population and other systems.
		This is often useful when fitting a theoretical, and possibly complicated curve to observed data; the fitting of a straight line, either graphically by eye, or by regression techniques is relatively simple.
		is negative the slope of the curve is decreasing and the curve is concave downward.
	www.fao.org /docrep/X5685E/x5685e02.htm (5889 words)

	Sample Problems from Chapter 13 (Site not responding. Last check: 2007-11-03)
		Count the squares under the curve from 0 to 1, by approximating (carefully).
		Find the ratio of the no. of squares under the curve from 0 to 1, to the no. of squares in th 1x1 square.
		The natural logarithm is the area under a curve and an infinite series.
	www.shout.net /~mathman/html/prob13.html (615 words)

	SpinWorks: Integrate
		Also, only the relative integral size is meaningful; two integrals with sizes of 1 and 4 tell us the same thing as two integrals with sizes of 0.2 and 0.8.
		Expect deviations in your integrals to be as large as 10-15% (routine) and even larger.
		For example, you might measure integral sizes of 0.9 and 3.3 for two peaks; these measurements are compatible with a molecular formula containing 1H and 3H (0.9:3.3 approximately equals 1:3) or a formula containing 1H and 4H (0.9:3.3 approximately equals 1:4).
	academic.reed.edu /chemistry/alan/201_202/SpinWorks/integrate.htm (903 words)

Try your search on: Qwika (all wikis)