
	Where results make sense

Topic: Jacobian determinant

Related Topics

Jacobian matrix

Inverse function theorem

Determinant

Matrix (mathematics)

Linear transformation

Substitution rule

Jacobian variety

Invertible

Jordan curve

Algebraic geometry

Matrix theory

Karl Gustav Jacobi

2004 in sports

Number

AGN

In the News (Fri 27 Nov 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Jacobian - Wikipedia, the free encyclopedia
		In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.
		In this sense, the Jacobian is akin to a derivative of a multivariate function.
		The absolute value of the Jacobian determinant at p gives us the factor by which the function F expands or shrinks volumes near p; this is why it occurs in the general substitution rule.
	en.wikipedia.org /wiki/Jacobian (685 words)

	Boundaries of Trivariate Solids
		Visualizing these solid objects is difficult, because it requires the determination of the boundary surface of the solid, which is a combination of parametric and implicit surfaces.
		Using an approximation to this determinant, the domain space is adaptively subdivided until a mesh can be determined such that the boundary surface is close to linear in the cells of the mesh.
		Interval approximation techniques are used to approximate the Jacobian determinant and to approximate the Jacobian determinant gradient for use in the adaptive subdivision methods.
	graphics.idav.ucdavis.edu /research/bou_tri_sol (471 words)

	Integration Through the Jacobian
		The area of a parallelagram in the plane is given by the determinant of its spanning vectors.
		The determinant of the jacobian is a local magnification factor on area.
		Multiplying f(g(p)) times the determinant of the jacobian at p is practically the same as multiplying f(q) times the area of the cell based at q, which is the image of the original rectangle.
	www.mathreference.com /ca-int,mvsub.html (903 words)

	The Characterization of the Regularity of the Jacobian Determinant in the framework of Potential spaces - Sickel, ...
		In almost all cases the regularity of the Jacobian determinant is calculated exactly.
		Sickel, W. and Youssfi, A.: The characterization of the regularity of the Jacobian determinant in the framework of potential spaces.
		@misc{ sickel-characterization, author = "W. Sickel and A. Youssfi", title = "The characterization of the regularity of the Jacobian determinant in the framework of potential spaces", text = "Sickel, W. and Youssfi, A.: The characterization of the regularity of the Jacobian determinant in the framework of potential spaces.
	citeseer.ist.psu.edu /96737.html (586 words)

	[No title]
		The Jacobian border is identified by the sweep of the point P to the end position and is identified by the dotted line in Fig.
		The determinant of a square Jacobian is equated to zero and solved for its roots.
		The sweep Jacobian is computed as EMBED Equation (22) Since this is a square Jacobian, the determinant is evaluated and equated to zero as EMBED Equation (23) Solving for the roots yields EMBED Equation and EMBED Equation .
	www.icaen.uiowa.edu /~amalek/sweep/Frmltn.doc (1392 words)

	Jacobian conjecture - Wikipedia, the free encyclopedia
		In mathematics, the Jacobian conjecture is a celebrated problem on polynomials in several variables.
		The Jacobian conjecture has been proved for polynomials of degree 2, and it has also been shown that it follows from the special case where the polynomials are of degree 3.
		The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors.
	en.wikipedia.org /wiki/Jacobian_conjecture (344 words)

	Lecture 14 (Site not responding. Last check: 2007-10-29)
		In the case of the above transformation x=au, y=by, the Jacobian is ab, and the "fudge factor" is ab.
		The "fudge factor" is again the absolute value of a Jacobian determinant: if x,y,z are functions of u,v,w, the corresponding Jacobian is the determinant of the 3 by 3 matrix consisting of the partials of x, y, and z with respect to u, v, and w.
		In example 4 in the book this determinant is computed in the case when u,v,w are spherical coordinates.
	www.math.uiuc.edu /~dikim/m242/lec14.html (1594 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		In particular, since the Jacobian of the inverse is the (matrix) inverse of the Jacobian, the Jacobian determinant of the inverse is the reciprocal of the determinant of the Jacobian.
		When x = f(u) but dim x != dim u the machinery of differentials, Jacobian matrices and determinants, and the IFT can also be applied but in a more circumspect manner.
		From du = f_x dx + f_y dy +f_z dz we observe that the Jacobian f' = [f_x,f_y,f_z] is the transpose of the gradient vector field, grad f.
	new.math.uiuc.edu /math280/notes05feb (598 words)

	On the Dynamics of Genetic Algorithms (and Other Evolving Systems)
		Under appropriate conditions, this determinant is proportional to the product of the values of the fitness function, divided by a positive integer power of the average fitness of the population.
		According to the chain rule, the Jacobian of the two-stage GA is then given by the product of the determinant of this mutation matrix with the determinant of the crossover Jacobian.
		The numerator simply scales the determinant by a power of the geometric mean of the fitness function; and the denominator decreases the determinant whenever the fitness increases, and vice versa.
	journal-ci.csse.monash.edu.au /ci/vol02/gozl2/gozl2.html (2494 words)

	Orientable Manifold with Boundary
		the Jacobian Matrix of the transformation between coordinates of two charts has positive determinant (oriented charts), so the smaller Jacobian Matrix with one row and one column deleted (corresponding to the only one coordinate axis that runs away from the boundary) has positive determinant.
		hence the determinant of the whole matrix being positive forces also the determinant of the upper left block to be positive also.
		the determinant is positive iff the determinant of the upper left block is positive.
	www.physicsforums.com /showthread.php?t=81202 (1240 words)

	Jacobians (Site not responding. Last check: 2007-10-29)
		We can find it by taking the determinant of the two by two matrix of partial derivatives.
		Remark: A useful fact is that the Jacobian of the inverse transformation is the reciprocal of the Jacobian of the original transformation.
		This is a consequence of the fact that the determinant of the inverse of a matrix A is the reciprocal of the determinant of A. Idea of the Proof
	ltcconline.net /greenl/courses/202/multipleIntegration/jacobians.htm (409 words)

	The Jacobian in Thermodynamics - Introduction & Background
		The use of the Jacobian in thermodynamics was first introduced by Dr. Kali Mukerjee at Michigan State University.
		The Jacobian is a useful way to find a relation between two different vector spaces.
		For a two-dimensional vector space (let us say x,y), the Jacobian determinant represents the ratio of the elementary areas between another vector space (A,B) and the original vector space (x,y).
	www.egr.msu.edu /classes/msm851/details.html (206 words)

	Carl Gustav Jakob Jacobi Summary
		A determinant is the result of a series of mathematical operations performed on a matrix.
		It was in analytical development that Jacobi’s peculiar power mainly lay, and he made many important contributions of this kind to other departments of mathematics, as a glance at the long list of papers that were published by him in Crelle’s Journal and elsewhere from 1826 onwards will sufficiently indicate.
		Students of vector theory often encounter the Jacobi identity, those studying differential equations often encounter the Jacobian determinant, and those working in number theory and cryptography use the Jacobi symbol.
	www.bookrags.com /Carl_Gustav_Jakob_Jacobi (2337 words)

	SBML ODE Solver Library API: Jacobian Matrix: J = df(x)/dx
		Constructs and returns the determinant of the jacobian matrix.
		Returns NULL if either the jacobian has not been constructed yet, or if i or j are >neq.
		Returns NULL if either the jacobian has not been constructed yet, or if the v1 or vi2 are not ODE variables.
	www.tbi.univie.ac.at /~raim/odeSolver/doc/api/group__jacobian.html (218 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-29)
		Date: 08/06/99 at 16:54:24 From: Doctor Anthony Subject: Re: Determinants The Jacobian ------------- When we make a change of variables in multiple integrals the Jacobian gives us a general technique for making the change.
		We must now find a general method for determining J. We start by reminding you of the formula for the area of triangle ABC with coordinates of the vertices (x1,y1), (x2,y2) and (x3,y3) 1 1 1
		The determinant is positive if the points A, B, C are taken in an counterclockwise direction.
	mathforum.org /library/drmath/view/52092.html (874 words)

	PlanetMath: Jacobi determinant
		is the absolute value of the Jacobi determinant or Jacobian.
		Then by the chain rule and definition of the Jacobi matrix,
		This is version 3 of Jacobi determinant, born on 2002-01-04, modified 2004-08-31.
	planetmath.org /encyclopedia/JacobiDeterminant.html (133 words)

	ACRES: Details About the Code
		attraction_parameter : inv jacobian and radius of adaptation !
		jacobian : jacobian of the coordinate transformation !
		det_jacobian, sqrt_det_inv_jacobian : determinant of jacobian and root of !
	cst-www.nrl.navy.mil /~singh/acres/code.html (6355 words)

	No Title (Site not responding. Last check: 2007-10-29)
		Determine if f(x,y) is continuous at (0,0) where
		Find the jacobian determinant of of x and y with respect to u and v.
		Determine if the following statement is true or false.Give a brief reasoing or quote theorem to supposrt your answer.
	www.math.sunysb.edu /~myonghi/OW/Y2000/m5320/m5320.html (236 words)

	determinant - OneLook Dictionary Search
		Determinant : The Computational Beauty of Nature [home, info]
		Phrases that include determinant: mathematical determinant, cayley menger determinant, determinant expansion, genetic determinant, jacobian determinant, more...
		Words similar to determinant: crucial, deciding, determinantal, determinative, determiner, determining, causal factor, determining factor, more...
	www.onelook.com /?w=determinant (281 words)

	[No title]
		Note the central role of the Jacobian matrix Jacobian[G](X) in the determination of dissipation: for flows, the average sum of the eigenvalues occurs while for maps, the average product of the eigenvalues is used.
		The infinitesimal volume dV[i] enclosed by the simplex of points X, X+V1,..., X+VN is then given by the determinant of the matrix whose columns are the vectors: dV[i] = Det[ {V1,..., VN} ], where the subscript i denotes that this is at the ith iteration of the map G(X).
		This is determined as a kind of inverse of the correlation dimension.
	www.phy.duke.edu /~hsg/213/lectures/10-22-03.txt (9502 words)

	AIPO from CSDC Inc.: CSDC REMOTE LINKS
		By scaling the normal field by a function which is homogeneous of degree 1 in the components of the original vector, a new vector field can be computed.
		The Jacobian matrix of this vector field has similarity invariants which may be evaluated.
		The trace of the Jacobian matrix gives the mean curvature of the resulting surface, while the trace of the adjoint of the Jacobian matrix (the matrix of cofactors transposed) gives the Gauss curvature.
	www.uh.edu /~rkiehn/ed3/ed3remot.htm (579 words)

	FedoraForum.org - Maths anyone?
		I said the the Jacobian was y/(3x^2) but he got something different.
		The integral is transformed to the u,v space using the Jacobian determinant i.e.
		The Jacobian itself depends on your choice of substitution, so if you're doing a different substitution then you'd get a different Jacobian.
	forums.fedoraforum.org /showthread.php?t=98754 (831 words)

	Jacobian determinant
		Of course I'll have to relearn that; I still have my old text (Anton) and intend to go through a generous helping of problems.
		My question, though, concerns something called the Jacobian determinant (and matrix) that are not discussed in my linear algebra text but are introduced right away in Frankel's book when he discusses sub-manifolds.
		I wonder if someone could give me a brief explanation of the significance of the Jacobian determinant and where it comes from.
	www.lns.cornell.edu /spr/2000-05/msg0024591.html (150 words)

	Re: Jacobian determinant (Site not responding. Last check: 2007-10-29)
		In article <390f02e8.696646731@news.ucalgary.ca>, Ken Muldrew wrote: >I wonder if someone could give me a >brief explanation of the significance of the Jacobian determinant and >where it comes from.
		The Jacobian determinant tells you how much a transformation expands or shrinks the volume of a tiny piece of space.
		It comes from the relation between volume and determinants.
	www.lns.cornell.edu /spr/2000-05/msg0024614.html (67 words)

	Tetrahedral Element Shape Optimization via the Jacobian Determinant and Condition Number (Site not responding. Last check: 2007-10-29)
		Because the element condition number is not defined for tetrahedra with negative volume, these objective functions can be used only when the initial mesh is valid.
		Therefore, we formulate a third objective function using the determinant of the element Jacobian that is suitable for mesh untangling.
		We review the optimization techniques used with each objective function and present experimental results that demonstrate the effectiveness of the mesh improvement and untangling methods.
	www.andrew.cmu.edu /user/sowen/abstracts/Fr697.html (229 words)

	DC MetaData for: The Characterization of the Regularity of the Jacobian Determinant in the framework of Bessel ... (Site not responding. Last check: 2007-10-29)
		DC MetaData for: The Characterization of the Regularity of the Jacobian Determinant in the framework of Bessel Potential Spaces on Domains
		The Characterization of the Regularity of the Jacobian Determinant in the framework of Bessel Potential Spaces on Domains
		We give necessary and sufficient conditions on the parameters $s_1, s_2, \ldots, s_m, p_1, p_2, \ldots, p_m$ such that the Jacobian determinant extends to a bounded operator from ${\cal H}^{s_1}_{p_1}\times {\cal H}^{s_2}_{p_2} \times \cdots \times {\cal H}^{s_m}_{p_m}$ into ${\cal S}'$.
	www.minet.uni-jena.de /Math-Net/reports/shadows/97-09report.html (138 words)

	Information Bridge: DOE Scientific and Technical Information - - Document #11915
		Tetrahedral element shape optimization via the Jacobian determinant and condition number.
		We present a new shape measure for tetrahedral elements that is optimal in the sense that it gives the distance of a tetrahedron from the set of inverted elements.
		Because the element condition number is not defined for tetrahedral with negative volume, these objective functions can be used only when the initial mesh is valid.
	www.osti.gov /bridge/product.biblio.jsp?osti_id=11915 (271 words)

	[No title]
		For example 123 requires 0 exchanges and is even; 132, 213, and 321 require one exchange and are odd; 231 and 312 require two exchanges and are even.
		(I.e., the row index, i, is in the numerator and the column index j is in the denominator.) According to equation [6], after interpreting the determinant as the Jacobian, J, then we can write EMBED Equation.3 .
		But Aji is the cofactor of the term in row j and column i of the Jacobian.
	www.csun.edu /~lcaretto/me692/geometry.doc (1457 words)

	CVGMT: Remarks on the total variation of the Jacobian (Site not responding. Last check: 2007-10-29)
		CVGMT: Remarks on the total variation of the Jacobian
		Remarks on the total variation of the Jacobian
		Abstract: The total variation TV(u) of the Jacobian determinant of non-smooth vector fields u has recently been studied in: FONSECA I., FUSCO N., MARCELLINI P., On the Total Variation of the Jacobian.
	cvgmt.sns.it /papers/muc02 (83 words)

Try your search on: Qwika (all wikis)