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Topic: Closed form calculus

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	Closed and exact differential forms - Wikipedia, the free encyclopedia
		In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms.
		The set of all forms cohomologous to each other form an element of a de Rham cohomology class; the general study of such classes is known as cohomology.
		The implication from 'exact' to 'closed' is then a consequence of the symmetry of second derivatives, with respect to x and y.
	en.wikipedia.org /wiki/Closed_and_exact_differential_forms (442 words)

	Pi-calculus - Wikipedia, the free encyclopedia
		In theoretical computer science, the π-calculus is a process calculus originally developed by Robin Milner, Joachim Parrow and David Walker as a continuation of the body of work on the process calculus CCS (Calculus of Communicating Systems).
		The simplicity of the calculus is due to the fact that names play a dual role as communication channels and variables.
		This polyadic extension can be encoded in the monadic calculus by passing the name of a private channel through which the multiple arguments are then passed in sequence.
	en.wikipedia.org /wiki/Pi_calculus (2290 words)

	C
		The calculus has been developed to treat functions not only of a single variable but also of several variables and is the foundation for the larger branch of mathematics known as Analysis.
		The Leyden jar, a form of capacitor invented at the University of Leiden in the 18th century, consists of a narrow-necked glass jar coated on part of its inner and outer surfaces with conductive metal foil.
		To form the screen display, or image, the electron beam is deflected in the vertical and horizontal directions either by the electrostatic effect of electrodes within the tube or by magnetic fields produced by coils located around the neck of the tube.
	www.hydrocut.com /Terms/C.html (6244 words)

	Menger on the Calculus of Variations
		History does not describe the form of the territory she chose, but if she was a good mathematician she covered the territory in the form of a circle, for today we know: Of all surfaces bounded by curves of a given length, the circle is the one of largest area.
		In differential calculus we deal thus with maxima and minima of so-called functions of points, i.e., of numbers associated with points; in calculus of variations, however, with maxima and minima of so-called functions of curves, that is, of numbers associated with curves or of numbers associated with still more complicated geometric entities, like surfaces.
		While the minimum and maximum problems of calculus of variations correspond to the problem in the ordinary calculus of finding peaks and pits of a surface, the minimax problems correspond to the problem of finding the saddle points of the surface (the passes of a mountain).
	www-history.mcs.st-andrews.ac.uk /Extras/Calculus_of_Variations.html (1947 words)

	[No title]
		A Calculus of Number Based on Spatial Forms by Jeffrey M. James A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Engineering University of Washington 1993 Abstract A calculus for writing and transforming numbers is defined.
		This form creates a repeated multiplication where the arity is determined by the exponent and the base is introduced as the arguments using transfer.
		Because of this dual use, the boundary form of cardinality is considered to be a generalized form of cardinality.
	www.lawsofform.org /docs/jjames-thesis.txt (15548 words)

	PlanetMath: Cartan Calculus
		Equations of this form are called supercommutation relations and are usually written in the form
		Since the Cartan Calculus operators are closed under the Lie superbracket, the vector space spanned by the Cartan Calculus operators has the structure of a Lie superalgebra.
		This is version 1 of Cartan Calculus, born on 2005-11-30.
	planetmath.org /encyclopedia/CartanCalculus.html (284 words)

	Calculus in Context
		The typical student in calculus has not been driven to study calculus in order to come to grips with his or her own scientific questions--as those pioneering students had.
		If calculus is to emerge organically in the minds of the larger student population, a way must be found to involve that population in a spectrum of scientific and mathematical questions.
		Calculus is fundamentally a way of dealing with functional relationships that occur in scientific and mathematical contexts.
	www.math.smith.edu /Local/cicintro/cicintro.html (1628 words)

	Visual Calculus - Maxima and Minima (Site not responding. Last check: 2007-11-01)
		We apply these results to finding maxima and minima of functions having only one critical points and functions which are continuous on a closed interval.
		The maximum and minimum of the function f(x) = x on the closed interval [1, 5] are the endpoints of the interval and are not critical points of f.
		Suppose that the function f is continuous on the closed interval [a, b].
	archives.math.utk.edu /visual.calculus/3/max.2/index.html (344 words)

	Calculus II worksheets for Maple R6
		The easy arguments for graphing calculators are that they are relatively inexpensive (about the cost of a standard calculus textbook), moderately easy to learn to use, and are in the students' hands so that the students have the same set of tools available in class, for homework, and during quizzes and tests.
		Nevertheless, as the students move through the calculus sequence there are topics where a computer algebra system is much more effective than a graphing calculator.
		I find it is best to take the class to the lab for a worksheet, assigning the students to finish it on their own time.
	euler.slu.edu /courseware/Calculus/Calculus2-R95/MapleForCalculus2.html (891 words)

	Numerical Solution, Closed-Form Solution
		Models take many forms, but they generally comprise a set of assumptions that are formalized with mathematical equations.
		A closed-form solution (or closed form expression) is any formula that can be evaluated in a finite number of standard operations.
		Closed form solutions and numerical solutions are similar in that they both can be evaluated with a finite number of standard operations.
	www.riskglossary.com /articles/closed_form_solution.htm (698 words)

	BookRags: Fundamental Theorem of Calculus Summary
		The Fundamental Theorem of Calculus states that the area under the graph of a function over an interval can be calculated by evaluating any antiderivative of the function at the endpoints of the interval.
		Differential calculus is the study of derivatives, which can be thought of as slopes of tangent lines to curves, or as instantaneous rates of change of functions.
		Integral calculus is the study of areas under curves, which are defined in terms of limits of Riemann sums -- the area under a curve is approximated by a collection of rectangles, and the widths of these rectangles are taken to be smaller and smaller to obtain better and better estimates of the actual area.
	www.bookrags.com /research/fundamental-theorem-of-calculus-wom (658 words)

	Karl's Calculus Tutor - Box 3.0c Proof of the Extreme Value Theorem
		This means that the sequence of lower endpoints forms a Cauchy sequence and the sequence of upper endpoints forms a Cauchy sequence.
		is unbounded over the closed interval, then it must be unbounded over at least one half of that closed interval, and that half can be made to be a closed interval as well.
		over a closed interval is nearly identical, except you use the greatest lower bound instead of the least upper bound, and some of the inequalities are turned around the other way.
	www.karlscalculus.org /bndproof.html (2354 words)

	Calculus, Vector and Complex Numbers
		Limits are employed in calculus and its applications to define key number or quantities - saying how to compute a number in the limit via a sequence of approximations defines it.
		The calculus preview provides motivation for the discussion of derivatives - the approximation of what they should be, and then a definition using the limit of approximation (should that limit be defined).
		This Geometric Preview gives a first image of calculus - to explain why derivatives are calculated and how they are used, and to give a context for earlier studies of slopes and rates of change.
	whyslopes.com /Calculus-Introduction (2489 words)

	Math Forum - Ask Dr. Math Archives: College Calculus
		Problem: Find the equation of the line tangent to the ellipse b^2x^2 + a^2y^2 = a^2b^2 in the first quadrant that forms* with the coordinate axes the triangle of smallest possible area (a & b are positive constants).
		The crescent is formed in 2 areas that are not overlapping.
		I learned that dy/dx was a notation that implied 'derivative of y with respect to x.' But in some calculus work, dy and dx seem to have individual meanings and dy/dx is treated as a fraction.
	mathforum.org /library/drmath/sets/college_calculus.html?... (1031 words)

	Calculus - Numericana
		To see whether a local extremum actually occurs, you must examine the second-order behavior of the function close to each of the candidate points (in the rare case where the second-order variations are zero, it's necessary to examine the situation further).
		It's nice to consider a given oriented closed boundary as a projection of a three-dimensional loop whose apparent area is defined as a path integral.
		The fundamental object is the square matrix W formed with the n columns corresponding to the n independent solutions of the homogeneous system.
	home.att.net /~numericana/answer/calculus.htm (5293 words)

	Math Forum - Ask Dr. Math Archives: High School Calculus (Site not responding. Last check: 2007-11-01)
		I'm soon to be a Calculus student in high school.
		I am a grade twelve student taking calculus and was wondering if you could help me with this problem: y=2x(6x+5)exp4 - solve to the second derivative.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/drmath/sets/high_calculus.html?...&num_to_see=40 (858 words)

	08. Differential 2
		Terms 2 and 3 are identical to their previous form, but notice term 1.
		In Graph 1, we see that, as one would expect, at the beginning of the fall when the velocity is close to zero, the acceleration is 9.8 m/s/s, e.g.
		This says that one may integrate the derivative of a function f(x) to reacquire the original function (a restatement of the Fundamental Theorem of Calculus), but a discarded constant term C must be accounted for.
	www.arachnoid.com /calculus/differential2.html (732 words)

	Closed form (Site not responding. Last check: 2007-11-01)
		a finitary expression, rather than one involving (for example) an infinite series - this meaning usually occurs in a phrase like solution in closed form;
		a closed differential form: see closed form (calculus).
		This is a disambiguation page, that is, one that just points to other pages that might otherwise have the same name.
	bopedia.com /en/wikipedia/c/cl/closed_form.html (90 words)

	Calculus Applets at SLU
		When working through the understanding of various kinds of functions it is useful to be able to graph a function with parameters a, b, and c, in the definition of the function, with the parameters controlled by sliders.
		The projection of the curve obtained from the intersection of the two surfaces is then either the graph of f, g, or fg, depending on the plane it is projected onto.
		Trig Functions Applet is a Geometer's Sketchpad applet that connects the 6 standard trig functions with lengths of line segments from a diagram connected to the unit circle.
	www.slu.edu /classes/maymk/MathApplets-SLU.html (1893 words)

	Math Help - Calculus - Derivatives - Technical Tutoring
		Notice that we have an open circle at the point (0, 0) to indicate that the point has been excluded from the graph and a closed circle at (0, 1) to show that this point as been included.
		Let's try the same calculation with the "slope form" of the definition of the derivative to be sure it gives the same result.
		You should practice using both forms until proficient with the calculations, then you can choose the form which best suits the problem you are working.
	www.hyper-ad.com /tutoring/math/calculus/Derivatives.html (1904 words)

	The LISP Programming System (Site not responding. Last check: 2007-11-01)
		When a word is required to form some additional list structure, the first word on the free-storage list is taken and the number in register FREE is changed to become the location of the second word on the free-storage list.
		This process, because it is entirely automatic, is more convenient for the programmer than a system in which he has to keep track of and erase unwanted lists.
		This is because the reclamation process requires several seconds to execute, and therefore must result in the addition of at least several thousand registers to the free-storage list if the program is not to spend most of its time in reclamation.
	www-formal.stanford.edu /jmc/recursive/node4.html (2233 words)

	Earliest Uses of Symbols of Calculus (Site not responding. Last check: 2007-11-01)
		The photo reveals that the column devoted to 1000 on this abacus is inscribed with a symbol quite close in shape to the lemniscate symbol, and which Menninger shows would easily have evolved into the symbol M, the eventual Roman symbol for 1000 [Randy K. Schwartz].
		The footnote indicates that Hamilton had originally intended to use the nabla symbol that is used today but then decided to rotate it to avoid confusion with other uses of the symbol.
		The rotated form appears in Hamilton's magnum opus, the Lectures on Quaternions (1853, p.
	hometown.aol.com /jeff570/calculus.html (2309 words)

	Amazon.com: Calculus Without Limits: Almost: Books: John C. Sparks (Site not responding. Last check: 2007-11-01)
		Lets face it, the subject of calculus is difficult enough; anything that can be done to simplify it, at least initially, is a positive development.
		The book is a unique blend of calculus principles, simplified and understandable illustrations, and motivational descriptions of significant contributions by various world-renowned mathematicians.
		I was intrigued about this book because of its' approach to calculus with minimal emphasis on the limit theorem and because my son is entering high school mathematics.
	www.amazon.com /Calculus-Without-Limits-John-Sparks/dp/1418441244 (1703 words)

	Background (Site not responding. Last check: 2007-11-01)
		Given an ordinary differential equation of the form (5.1), it is natural (or even essential) that the question of whether such a solution, y(t), even exists.
		In other words, there are two directions of pursuit given an equation of the form (5.1).
		Recall the separation of variables technique from calculus for solving such an equation, which involves placing all terms involving v on one side of the equation and all non-v terms on the other.
	www.mathcs.emory.edu /ccs/ccs315/ccs315/node22.html (771 words)

	18 Integration - First Fundamental Theorem of Calculus
		The First Fundamental Theorem of Calculus says that if h(x) is continuous between and at the endpoints x = a and x = b, that is continuous on the closed interval [a,b], then all the rectangle-based approximations approach a single finite limiting value as the width of the base tends to zero.
		For each square with one size on the horizontal axis, the union of it with the squares above it form a rectangle with one end on the horizontal axis and another end on the curve, or very close to the curve.
		Site Message: Calculus and calculus preparation give a first reason and focus for learning and teaching most mathematics in school and college.
	whyslopes.com /etc/CalculusAndBeyond/ch18.html (809 words)

	INFORMS von Neumann Theory Prize Winners
		And, his theoretical papers greatly extend the notion of stochastic dominance and revealed close ties between dominance and moments of probability distributions.
		His selection criterion for reducing fill-in when forming basis factors is the well-known Markowitz criterion and is still used in state-of-the-art codes for both LU and Cholesky factorizations.
		The famous Rao-Blackwell theorem on statistical estimation led to a practical method for improving estimates, now known as "Rao-Blackwellization." An elegant and important form of the renewal theorem is due to Blackwell, as is a beautiful characterization of the information content of an experiment.
	www.informs.org /Prizes/vonNeumannDetails.html (8908 words)

	Karl's Calculus Tutor - 5.4 Applications of Derivatives: Like a Steam Locomotive
		is greater than zero anywhere on the closed interval, it must be greater than zero everywhere on the closed interval.
		is less than zero anywhere on the closed interval, it is less than zero everywhere on the closed interval.
		Because it must do so over every closed interval that is contained in the open interval, and those closed intervals can take you as close to the end points as you like.
	www.karlscalculus.org /calc5_3.html (1840 words)

	Mere Math: "The Bare Facts" Calculus
		Once we decide how close to L we want f (x) to be, it is necessary that f (x) be close to L for all x sufficiently close but not equal to a.
		Suppose that f is continuous on the closed interval [a, b].
		A final resubstitution is necessary to write the answer in terms of the original variable x.
	polaris.umuc.edu /~mjohnso5/Calculus.html (883 words)

	Amazon.com: The Pi-Calculus: A Theory of Mobile Processes: Books: Davide Sangiorgi,David Walker (Site not responding. Last check: 2007-11-01)
		It is one of the many examples of process algebra that have appeared in the last two decades, and has been the subject of intense research.
		On the other hand, readers (such as this reviewer) who are familiar with the lambda calculus or functional programming will find familiar territory in the book, and will more fully appreciate the sixth part of the book, which deals with the interpretation of functions as processes.
		Remembering the same concepts in mathematical logic, particularly in the lambda calculus, one speaks of the occurrence of name in a process as 'bound' if it lies within the scope of a binding of occurrence of the name; it is 'free' if it is not bound.
	www.amazon.com /Pi-Calculus-Theory-Mobile-Processes/dp/0521543274 (1830 words)

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