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Topic: Implicit function

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Implicit differentiation

Derivative

Inverse function

Product rule

Related Rates

Graph of a function

Exponential function

Implicit surface

Vector calculus

Integral

Calculus

3D computer graphics

Derivative

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	Implicit function theorem - Wikipedia, the free encyclopedia
		In multivariable calculus of mathematics the implicit function theorem says that for a suitable set of equations, some of the variables are defined as functions of the others.
		There are some natural limitations on this use of a mathematical relation to define implicit functions, which may be seen in trying to use the unit circle as the graph of a function.
		The implicit function is only locally defined, and points at which the first-order behavior would be problematic are outside the scope of the result.
	en.wikipedia.org /wiki/Implicit_function_theorem (672 words)

	Implicit function - Wikipedia, the free encyclopedia
		That is, an implicit function can sometimes be defined successfully only by modifying the relation by 'zooming in' to some part of the x-axis, and 'cutting back' unwanted function branches.
		In less technical language, implicit functions exist and can be differentiated, unless the tangent to the supposed graph would be vertical.
		For the important generalisation to functions of several variables, see implicit function theorem.
	en.wikipedia.org /wiki/Implicit_function (479 words)

	PlanetMath: implicit differentiation
		Implicit differentiation is a tool used to analyze functions that cannot be conveniently put into a form
		Most of your derivatives will be functions of one or all the variables, including the implicit function itself.
		This is version 2 of implicit differentiation, born on 2002-02-25, modified 2003-10-30.
	planetmath.org /encyclopedia/ImplicitDifferentiation.html (247 words)

	Derivative - Wikipedia, the free encyclopedia
		If a function is not continuous at x, then there is no tangent line and the function is therefore not differentiable at x; however, even if a function is continuous at x, it may not be differentiable there, as in the case of the function y=
		Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.
		For complex functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument.
	en.wikipedia.org /wiki/Derivative (2298 words)

	Derivative - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-05)
		In mathematics, the derivative of a function is one of the two central concepts of calculus.
		The derivative of a function at a certain point is a measure of the rate at which that function is changing as an argument undergoes change.
		If a function is not continuous at c, then there is no slope and the function is therefore not differentiable at c; however, even if a function is continuous at c, it may not be differentiable.
	encyclopedia.learnthis.info /d/de/derivative.html (1805 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		The implicit surface is defined by means of a function whose arguments are coordinates in the space in which the surface lies.
		Although implicit definitions of this type are a powerful tool for modelling surfaces, they are less suitable for efficient direct rendering of those surfaces, that is for computing the values of the pixels of an image containing such a surface.
		In case Bezier functions are used, the second step 32 decides from the implicit definition how many different surface patches should be used to approximate the implicitly defined surface and the second step computes the coordinates of a number of those control points R for each of those patches.
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=00/02164.000113&ELEMENT_SET=DECL (3137 words)

	PlanetMath: implicit function theorem
		The inverse function theorem is a special case of the implicit function theorem where the dimension of each variable is the same.
		This is version 8 of implicit function theorem, born on 2002-08-24, modified 2004-05-25.
		I assume it is the derivative with resepect to the jth component of the argument, but it would be nice to have it defined or cross-referenced.
	planetmath.org /encyclopedia/ImplicitFunctionTheorem.html (188 words)

	Implicit function: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-05)
		In mathematics, a smooth function is one that is infinitely differentiable, i.e., has derivatives of all finite orders....
		In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant....
		In mathematics, in the field of calculus of several variables, the implicit function theorem says that for a suitable set of equations, some of the variables...
	www.absoluteastronomy.com /encyclopedia/i/im/implicit_function.htm (1283 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-05)
		In order to be a *function, of course, given a value of the independent variable, there must be a unique* value of the dependent variable which makes the equation true.
		This means that for y to be an implicit function of x, if you graph the solution set of the equation, every vertical line x = a should intersect the graph in a unique point.
		One simple case where an implicit function always exists is if the graph is monotone increasing, that is, if (a,c) and (b,d) are points on the graph, and a > b, then c > d.
	mathforum.org /library/drmath/view/51453.html (484 words)

	Implicit Differentiation
		In spite of the fact that the circle cannot be described as the graph of a function, we can describe various parts of the circle as the graphs of functions.
		(this is an implicit function defined by the equation).
		If we compute the derivative at a point through implicit differentiation and it is defined at this point, then the Implicit Function Theorem justifies the calculation we have just done and we need not worry about anything.
	www.ugrad.math.ubc.ca /coursedoc/math100/notes/derivative/implicit.html (822 words)

	Contouring Implicit Surfaces
		This function typically evaluates to the distance of the point from the contour of the volume, giving a negative non-zero result for points inside the volume, a positive non-zero result for points outside the volume, and a zero result for points lying exactly on the contour of the volume.
		Thus evaluating the field function at (0,0,0) would correctly return a result indicating the point is outside the volume, while evaluating the point at (-5,0,0) would correctly return a result indicating the point is inside the volume.
		The Marching Cubes algorithm, published in 1987 [4], laid the foundation for the extraction of a polygonal approximation of an implicit surface.
	www.sandboxie.com /misc/isosurf/isosurfaces.html (5375 words)

	College Mathematics Journal, The: A surface useful for illustrating the implicit function theorem (Site not responding. Last check: 2007-11-05)
		Most of the level sets of the defining function for this surface are smooth curves, but there are two points where the hypotheses of this theorem break down, and the level sets at these points display interesting singularities.
		The Implicit Function Theorem asserts that a level set of a function z = f(x, y) is locally a smooth curve at a point P(a, b) if grad f(a, b) [not =] 0.
		Although the function g(x, y) = x^sup 3^ - y^sup 3^ has grad g = 0 at the origin, the level set through (0, 0) is simply the line y = x.
	www.findarticles.com /p/articles/mi_qa3773/is_200309/ai_n9290176 (511 words)

	The Implicit Function Theorem
		The foundation for such an study is provided by the implicit function theorem, formulated below.
		As a preparation, we discuss consequences of the relationship between the domain dimension n and the target dimension m of the function f.
		This is the content of the implicit function theorem.
	www.ualberta.ca /dept/math/gauss/fcm/calculus/multvrbl/basic/ImplctFnctns/implct_fnctn_thrm.htm (462 words)

	VTK: vtkImplicitFunction Class Reference (Site not responding. Last check: 2007-11-05)
		Implicit functions are real valued functions defined in 3D space, w = F(x,y,z).
		Two primitive operations are required: the ability to evaluate the function, and the function gradient at a given point.
		The implicit function divides space into three regions: on the surface (F(x,y,z)=w), outside of the surface (F(x,y,z)>c), and inside the surface (F(x,y,z)
	noodle.med.yale.edu /vtk/classvtkImplicitFunction.html (722 words)

	Steps for Implicit Differentiation (Site not responding. Last check: 2007-11-05)
		In some cases, it may be possible to solve an implicitly defined function for y as one or more explicitly defined function of x.
		We assume that this implied function is differentiable.
		Whenever you differentiate an expression in y, think of y as a function of x and use the chain rule.
	www.gpc.edu /~jcraig/cal1_ch3/3s6_implicit_diferentiation.htm (294 words)

	The implicit function theorem
		The implicit function theorem is an important example where the linearized system qualitatively determines the behavior of the nonlinear system.
		The implicit function theorem asserts precisely that this is the case.
		A proof of the implicit function theorem can be found in advanced calculus texts.
	www.math.vt.edu /people/renardym/class_home/nova/bifs/node14.html (172 words)

	Warping of Graphic Objects
		It is also possible to apply the deformation map when implicit function is evaluated, and before any representation is used, then it is likely to produce most faithful result, but computation time will be more since deformation function is evaluated every time implicit function is evaluated.
		One of the techniques of local deformation is based on implicit functions modeling, where a scalar valued function is used to determine ISO-potential value to determine inside or outside of an object.
		A scalar valued function with similar properties is used to determine a subset of vertices of an object that are candidates for deforming, and is called the modulating function in this context.
	www.gignews.com /warping.htm (1019 words)

	RAND \| Reports \| Implicit Function Theorems for Optimization Problems and for Systems of Inequalities.
		Implicit function formulas for differentiating the solutions of mathematical programming problems satisfying the conditions of the Kuhn-Tucker theorem are motivated and rigorously demonstrated.
		The special case of a convex objective function with linear constraints is also treated with emphasis on computational details.
		Implicit function formulas for differentiating the unique solution of a system of simultaneous inequalities are also derived.
	www.rand.org /pubs/reports/R1036 (253 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Let F be a function of two real variables, and consider the equation F(x,y) = 0.
		Then there is an open neighbourhood V of x_0 in R such that there is a unique continuous function f with the property that f(x_0) = y_0 and F(x,f(x)) = 0 for all x in V. In addition f has a continuous derivative.
		Take m continuously differentiable functions F_1,..., F_m of n + m variables x_1,...,x_n,y_1,...,y_m and replace the non-vanishing of the partial derivative by the non-vanishing of an appropriate determinant of partial derivatives.
	www.math.niu.edu /~rusin/known-math/99/ift (311 words)

	Amazon.com: The Implicit Function Theorem : History, Theory, and Applications: Books: Steven G. Krantz,Harold R. Parks (Site not responding. Last check: 2007-11-05)
		The implicit function theorem is part of the bedrock of mathematical analysis and geometry.
		There are many different forms of the implicit function theorem, including (i) the classical formulation for C^k functions, (ii) formulations in other function spaces, (iii) formulations for non-smooth functions, (iv) formulations for functions with degenerate Jacobian.
		The history of the implicit function theorem is a lively and complex story, and is intimately bound up with the development of fundamental ideas in analysis and geometry.
	www.amazon.com /exec/obidos/tg/detail/-/0817642854?v=glance (805 words)

	Homework Assignments: 08/23 – 08/26
		When we differentiate the function as an implicit function, we don’t have to decide which derivative to use.
		Solution: The overall function is a natural log function; therefore, we take the derivative of the natural log function first, then the sine function, then the power function.
		The derivative of the tangent function is 1 divided by the cosine function squared. Therefore,
	www.math.utep.edu /classes/calculus/help/implicitDifferentiation.htm (931 words)

	Documentation for vtkImplicitFunction
		Implicit functions are of the form F(x,y,z) = 0.
		It is possible to represent almost any type of geometry with implicit functions, especially if you use boolean combinations implicit functions (see vtkImplicitBoolean).
		The transformation matrix transforms a point into the space of the implicit function (i.e., the model space).
	www.geologie.uni-freiburg.de /root/manuals/vtkman/vtkImplicitFunction.html (219 words)

	Shape Transformation (Site not responding. Last check: 2007-11-05)
		Traditionally, shape transformation using implicit functions is performed in two distinct steps: 1) creating two implicit functions, and 2) interpolating between these two functions.
		Zero-valued constraints specify the locations of shape boundaries and positive-valued constraints are placed along the normal direction in towards the center of the shape.
		We then invoke a variational interpolation technique (the 3D generalization of thin-plate interpolation), and this yields a single implicit function in 3D.
	www.gvu.gatech.edu /people/faculty/greg.turk/morph/morph.html (248 words)

	Calculus I Notes, Section 3-6 (Site not responding. Last check: 2007-11-05)
		When an equation cannot be solved for y, we call it an implicit function.
		Because y is not native to what are differentiating with respect to, we need to regard it as a composite function.
		We can see in a plot of the implicit function that the slope of the tangent line at the point (3,3) does appear to be -1.
	www.blc.edu /fac/rbuelow/calc/nt3-6.html (564 words)

	Implicit functions
		Let f (x, y) be a continuous function of two variables so that it is continuous, and differentiable with respect to y when x is kept fixed; in particular, we have expression
		We want to find the implicit function h(x) defined by f = 0 (see 7).
		This function h is the required implicit function.
	www.imsc.res.in /~kapil/geometry/prereq1/node13.html (450 words)

	Inverses and the Implicit Function Theorem
		Lecture 4: The Inverse and the Implicit Function Theorems
		Replacing the rectangle with a smaller one, we can assume the same is true when f is restricted to the closure of the rectangle.
		Exercise 2: Show that the Inverse Function Theorem is a Corollary of the Implicit Function Theorem.
	www.msc.uky.edu /ken/ma570/lectures/lecture4/html/inverse.htm (248 words)

	Citebase - Implicit function theorem over free groups
		We prove the results that can be described as Implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ systems.
		We will also show that the Implicit function theorem is true only for these varieties.
		We also prove a weak version of the Implicit function theorem for NTQ systems which is one of the key results in the solution of the Tarski's problems about the elementary theory of a free group.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0312509 (610 words)

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