structural:expansion
Uses of linearization

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

the linearized system can be written as

where x 0 {\displaystyle \mathbf {x_{0}} } is the point of interest and D F (x 0, t) {\displaystyle D\mathbf {F} (\mathbf {x_{0}},t)} is the x {\displaystyle \mathbf {x} } - Jacobian of F (x, t) {\displaystyle \mathbf {F} (\mathbf {x},t)} evaluated at x 0 {\displaystyle \mathbf {x_{0}} }.