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Topic: Inverse function theorem

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Inverse function

Jacobian determinant

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	PlanetMath: inverse function theorem
		The inverse function theorem is a special case of the implicit function theorem where the dimension of each variable is the same.
		Cross-references: variable, dimension, implicit function theorem, strictly monotonic function, monotonic, strictly, neighbourhood, open interval, function, Jacobian, point, open set, mapping, vector-valued function, continuously differentiable
		This is version 6 of inverse function theorem, born on 2002-08-24, modified 2002-12-28.
	planetmath.org /encyclopedia/InverseFunctionTheorem.html (118 words)

	Inverse function theorem - Wikipedia, the free encyclopedia
		In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain.
		That is, an inverse function to F exists in some neighborhood of F(p).
		Using the inverse function theorem, a derivative of a function's inverse indicates the derivative of the original function.
	en.wikipedia.org /wiki/Inverse_function_theorem (350 words)

	Inverse function - Wikipedia, the free encyclopedia
		In mathematics, an inverse function is in simple terms a function which "does the reverse" of a given function.
		If f is a real-valued function, then for f to have a valid inverse, it must pass the horizontal line test, that is a horizontal line y = k placed on the graph of f must pass through f exactly once for all real k.
		is also used for the (set valued) function associating to an element or a subset of the codomain, the inverse image of this subset (or element, seen as a singleton).
	en.wikipedia.org /wiki/Inverse_function (537 words)

	Trigonometry Tutorials and Problems
		The tangent function f(x) = a*tan(bx+c)+d and its properties such as graph, period, phase shift and asymptotes by changing the parameters a, b, c and d are explored interactively using an applet.
		The graph of the inverse trigonometric function arctan and its properties are explored using an applet.
		The graph and the properties of the inverse trigonometric function arcsin are explored using an applet.
	www.analyzemath.com /Trigonometry.html (792 words)

	Learn about Robot Inverse Kinematics
		The inverse kinematics problem (at the position level) for this robot is as follows: Given X
		You may have to use your imagination a bit, but the schematic above is the planar part of the SCARA robot we discuss in the industrial robots section.
		The inverse kinematics solution for Cartesian robots is trivial as all axes are perpendicular by definition and thus there is no coupling of the motions.
	www.learnaboutrobots.com /inverseKinematics.htm (484 words)

	The Implicit Function Theorem
		The foundation for such an study is provided by the implicit function theorem, formulated below.
		This is the content of the inverse function theorem.
		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)

	Inverse functions (Site not responding. Last check: 2007-10-18)
		A continuously differentiable function which has non-zero derivative at a point has an inverse in a neighbourhood of that point (see Figure 6).
		It is clear that a strictly monotone function is one-to-one.
		Show that there is a function g on the range of f so that g(f (x)) = x for all x in the interval a < x < b.
	www.imsc.res.in /~kapil/geometry/prereq1/node12.html (230 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		The latter construction is reduced to the contruction of some analytic functions on the manifold, and this is done by Hilbert space methods.
		The proof is based on the Cauchy-Kowalewski Theorem (a theorem on the existence of analytic solutions to analytic differential equations), and, in particular, no implicit function theorem is used.
		In this paper there is a proof of an analytic inverse function theorem for something called a Banach scale, but this should be more general than one single Banach space.
	www.math.niu.edu /~rusin/known-math/95/embed.analy (714 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		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.
		The Jacobian determinant of this is non-zero at P, and so there is an inverse function psi of this defined near (x_0, 0).
	www.math.niu.edu /~rusin/known-math/99/ift (311 words)

	Inv2_Tang_Derivative.html
		The range of an inverse is the domain the original function.
		So the blue is f, red is the inverse, the lower green is f ', upper green is the derivative of the inverse, and the plums are the tangent lines.
		are excluded in the conclusion of the theorem.
	www.adeptscience.co.uk /maplearticles/f1010.html (1867 words)

	ipedia.com: Inverse function Article (Site not responding. Last check: 2007-10-18)
		In mathematical analysis, an inverse function is in simple terms a function which "does the reverse" of a given function.
		For example, if the function x → 3x + 2 is given, then its inverse function is x → (x - 2) / 3.
		As such, the prefix arc is sometimes used to denote inverse trigonometric functions, eg arcsin x for the inverse of sin x).
	www.ipedia.com /inverse_function.html (444 words)

	Inverses and the Implicit Function Theorem
		Lecture 4: The Inverse and the Implicit Function Theorems
		Note: It follows from the formula for the derivative of the inverse that the inverse is also continuously differentiable.
		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)

	Mathematics Tutorials and Problems (with applets)
		Calculus Tutorials and Problems and Questions with answers on topics such as limits, derivatives, integrals, natural logarithm, runge kutta method in differential equations, the mean value theorem and the use of differentiation and integration rules are also included.
		Trigonometry Tutorials and Problems for Self Tests on sine, cosine, tangent, secant functions, trigonometric identities and formulas are also included.
		Inverse of a Function and One to One Functions
	www.analyzemath.com (460 words)

	Course Descriptions
		Functions of several variables were briefly considered in the first year core unit MT1222.
		For example, for functions of several variables, the critical points might be maxima, minima or saddle points (which are minima in one direction and are maxima in another direction).
		They will know the statements of Sard's theorem, The mean value theorem, The inverse and implicit function theorems and Taylor's theorem, and have some familiarity with their proofs.
	www.ma.man.ac.uk /DeptWeb/UGCourses/Syllabus/Level3/MT3502.html (282 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		Classical univalence theorems of Hadamard and of Banach and Mazur deal with analytical and/or topological conditions.
		An algebraic result due to Gale and Nikaido asserts that if the Jacobian matrix of a differentiable function on R^n is a P-matrix at every point, then the function is univalent.
		If time permits, I will talk about a local inverse function theorem for nonsmooth functions which, when specialized, yields the classical inverse function theorem for a C^1-function, Clarke's inverse function theorem for a locally Lipschitzian function, the Pang-Ralph inverse function theorem for a piecewise smooth function, etc. *************************************************************************
	www.mts.jhu.edu /~seminar/seminar/20001019gowda.txt (170 words)

	Section (vi) The Novices' Difficulty with Grounding Intuitive Arguments on Appropriate Theorems
		What they cannot do is support their arguments with formal explanations that go beyond the graphical representations of the functions in question and ground their arguments on theorems that they have been recently taught.
		Thus the theorems are seen into a context of applicability and are embedded into the growing domain of the novice's mathematical knowledge.
		This is exactly what the IFT allows but, as with the IVT, the students use the theorem unconsciously keeping thus its assumptions implicit.
	www.uea.ac.uk /~m011/thesis/chapter7/7vi.htm (888 words)

	mat531 homework
		Bredon gives the Inverse Function Theorem as a special case of the Implicit Function Theorem.
		Apply the proof of the implicit function theorem to g(x,y)=(x-1)^2 +(y-1)^2 -2.
		You may use Bredon's proof of Theorem 5.6 as a model, but remember that there the differentials are D : A_k --> A_(k-1), etc., and that the ``connecting homomorphisms'' are D*: C_k --> A_(k-1).
	www.math.sunysb.edu /~tony/archive/top2/homework.html (1095 words)

	Honors Elementary Calculus II Lecture Notes, 03/22/05 (Site not responding. Last check: 2007-10-18)
		Definition: Let f be a function whose domain is D and range is R. If f satisfies the property of being one-to-one, i.e.
		Theorem (Inverse Function Theorem): Let f be a function that is differentiable on a given interval I, and suppose that f is strictly increasing or strictly decreasing on I. The following statement is true for all x in I and y = f(x):
		Definition: The inverse of the natural logarithm function ln is called the natural exponential function, denoted by exp.
	www.assumption.edu /Alfano/MAT132-SP05/Notes/032205.html (416 words)

	Robert Foote's Abstracts
		As the boundary of the region is traced, a wheel attached to the instrument partially rolls and partially slides, recording a component of its motion on the plane.
		The area of the region is a simple function of the net roll of the wheel.
		Functions continuous except on a set of measure zero are also considered.
	persweb.wabash.edu /facstaff/footer/Abstracts.htm (1977 words)

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