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Relating to the work of the mathematician K. G. J. Jacobi.
‘In order to calculate the inverse Jacobian matrix, we need the following derivatives of the functional response [F.sub.ki] (for all values of k and i).’
‘He showed how to find integrals of a general system of partial differential equations by using sequential complete systems instead of passing to Jacobian systems.’
‘Göpel… finally, after ingenious calculations, obtained the result that the quotients of two theta functions are solutions of the Jacobian problem for p = 2.’
‘Asymptotic stability of the EP was also determined by computing 14 eigenvalues of a 14 × 14 Jacobian matrix derived from the linearization of the nonlinear system around the EP (for more details, see Vinet and Roberge).’
noun
Mathematics
A determinant whose constituents are the derivatives of a number of functions (u, v, w, …) with respect to each of the same number of variables (x, y, z, …).
‘In 1889 Poincaré proved that for the restricted three body problem no integrals exist apart from the Jacobian.’
‘He also worked on determinants and studied the functional determinant now called the Jacobian.’
‘These problems form the basis of a conjecture: every elliptic curve defined over the rational field is a factor of the Jacobian of a modular function field.’
‘Therefore, the determinants of the set of variable transformations (Jacobians) must be calculated for both states involved: the original and the new one, to properly weigh up each new configuration.’
‘Advanced calculus gives us a strong tool for finding the change in the area of a given shape under continuously differentiable transformations-namely, the Jacobian.’