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Linearization makes it possible to use tools for studying linear systems to analyze the behavior of a nonlinear function near a given point. The linearization of a function is the first order term of its Taylor expansion around the point of interest. For a system defined by the equation
For infinitesimal deformations of a continuum body, in which the displacement gradient tensor (2nd order tensor) is small compared to unity, i.e. ‖ ‖, it is possible to perform a geometric linearization of any one of the finite strain tensors used in finite strain theory, e.g. the Lagrangian finite strain tensor, and the Eulerian finite strain tensor.
These refinements of Taylor's theorem are usually proved using the mean value theorem, whence the name.Additionally, notice that this is precisely the mean value theorem when =.
Linear approximation of a nonlinear system: classification of 2D fixed point according to the trace and the determinant of the Jacobian matrix (the linearization of the system near an equilibrium point). The roots of the characteristic polynomial det(A - λI) are the eigenvalues of A.
The theorem states that the behaviour of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its linearization near this equilibrium point, where hyperbolicity means that no eigenvalue of the linearization has real part equal to zero. Therefore, when dealing with such dynamical ...
In numerical analysis, the local linearization (LL) method is a general strategy for designing numerical integrators for differential equations based on a local (piecewise) linearization of the given equation on consecutive time intervals. The numerical integrators are then iteratively defined as the solution of the resulting piecewise linear ...
What happens after an executive order is signed? After a president signs an executive order, the White House sends the document to the Office of the Federal Register, the executive branch's ...
A structure of an equivariant sheaf on an invertible sheaf or a line bundle is also called a linearization. Let X be a complete variety over an algebraically closed field acted by a connected reductive group G and L an invertible sheaf on it. If X is normal, then some tensor power of L is linearizable. [4]