21. It extends further to differentiable maps between differentiable manifolds, as the points where the rank of the Jacobian matrix decreases. 22. The earliest approximation was to linearize the nonlinear function and apply the resulting Jacobian matrix to the given mean and covariance. 23. The Jacobian matrix of f is formed by the four partial derivatives of u and v with respect to x and y. 24. The topic of multilinear algebra is applied in some studies of multivariate calculus and manifolds where the Jacobian matrix comes into play. 25. The Jacobian matrix is required by many algorithms that solve the ordinary differential equations numerically, particular when the ODEs are stiff. 26. The Jacobian matrix is determined iteratively based on the "'secant equation "'in the finite difference approximation: 27. If the largest real part of the eigenvalues is zero, the Jacobian matrix does not allow for an evaluation of the stability. 28. Recalling that v is the specific volume, u the flow speed, s the specific entropy, the corresponding jacobian matrix is: 29. Jacobian matrix consisting of the partial derivatives of " f i " with respect to " X j ":30. There is an analogous criterion for a continuously differentiable map with a fixed point, expressed in terms of its Jacobian matrix at,.
