# Jacobian matrix

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## Jacobian matrix

[jə′kō·bē·ən ′mā·triks]
(mathematics)
The matrix used to form the Jacobian.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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and, for the equilibrium [P.sup.+](x, y, z) = (+ [square root of b], + [square root of b], 0), the Jacobian matrix of (6) at points [P.sup.+] is obtained as
Since J = J + O(h), we conclude that the Jacobian matrix J = [[partial derivative][[PHI].sub.n]/[partial derivative]U] in (3.8) is nonsingular for all sufficiently small h and for [x.sub.n] in a small neighbourhood of the exact solution of problem (1.1).
For the equilibrium point [P.sub.1] = (0,0, 0) the Jacobian matrix is given by
The Frechet derivative matrix in (4) is a Jacobian matrix with large dimensionality.
The key point here is that the diagonal elements of the simplified diagonal Jacobian matrix are identical to those of the flux Jacobian matrix in the time-domain solution algorithm.
The Jacobian matrix of model (3) around the disease-free equilibrium, [E.sub.0], is given by
where [mathematical expression not reproducible] is the overall Jacobian matrix of the 2-RPU&2-SPS SPM.
We evaluate the Jacobian matrix (41) at the endemic equilibrium to obtain
where the values of each element in the Jacobian matrix can be expressed as follows:
At the disease-free equilibrium [D.sub.o] the corresponding Jacobian matrix [J.sub.o]([zeta]) of system (3) is computed as follows:

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