Linearization in mathematics and its applications in general refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations. This method is used in fields such as engineering, physics, economics, and ecology. For other meanings of mathematics or math, see mathematics (disambiguation). ... Linear approximation is a method of approximating otherwise difficult to find values of a mathematical function by taking the value on a nearby tangent line instead of the function itself. ... Partial plot of a function f. ... A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... The word stability has a number of technical meanings, all related to the common meaning of the word. ... Look up equilibrium in Wiktionary, the free dictionary. ... System (from the Latin (systēma), and this from the Greek (sustēma)) is an assemblage of entity/objects, real or abstract, comprising a whole with each and every component/element interacting or related to another one. ... To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Engineering is the application of scientific and technical knowledge to solve human problems. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... Economic activity: buyers bargain for good prices while sellers put forth their best front in Chichicastenango Market, Guatemala. ... Ernst Haeckel coined the term oekologie in 1866. ...

With a Function y = f(x)

Linearizations of a function are a lines; ones that are usually used for purposes of calculation. Linearization an efficacious method used to approximate the output of function at any x = a based on the value and slope of a function y = f(x) at x = b, given that f(x) is continuous on [a,b] (or [b,a])and that a is close to b. In, short, linearization approximates the output of a function near x = a. Look up Function in Wiktionary, the free dictionary. ... A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ... Look up Slope in Wiktionary, the free dictionary The slope or the gradient is commonly used to describe the measurement of the steepness, incline or grade of a straight line. ...

For example, you might know that . But without a calculator, what would be a good approximation of ?

For any given function y = f(x), f(x) can be approximated if it is near a known continuous point. The most basic requisite is that, where L_a(x) is the linearization of f(x) at x = a, L(a) = f(a). The point-slope form of an equation forms an equation of a line, given a point (H,K) and slope M. The general form of this equation is: y − K = M(x − H). A linear equation is an equation involving only the sum of constants or products of constants and the first power of a variable. ...

Using the point (x,f(x)), L_a(x) becomes y = f(a) + M(x − a)-. Because functions are locally linear, the best slope to substitute in would be the slope of the line tangent to f(x) at x = a. Local linearity is a property of differentiable functions that says – roughly – that if you zoom in on a point on the graph of the function (with equal scaling horizontally and vertically), the graph will eventually look like a straight line with a slope equal to the derivative of the function... In mathematics, the word tangent has two distinct, but etymologically-related meanings: one in geometry, and one in trigonometry. ...

While the concept of local linearity applies the most to points arbitrarily close to x = a, those relatively close work relatively well for linear approximations. After all, a linearization is only an approximation. The slope M should be, most accurately, the slope of the tangent line at x = a. In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...

An approximation of f(x) at (x, f(x))

Visually, the accompanying diagram shows the tangent line of f(x) at x. At f(x + h), where h is any small positive or negative value, f(x+h) is very nearly the value of the tangent line at the point (x + h,L(x + h)). Download high resolution version (823x586, 6 KB) tangent e= File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (823x586, 6 KB) tangent e= File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

The final equation for the linearization of a function at x = a is:

For x = a, f(a) is f(x) at a. The derivative of f(x) is f'(x), and the slope of f(x) at a is f'(a). In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...

Instead of a linear function, perhaps, would x not be better approximated with polynomial? This can be accomplished via Taylor series. As the degree of the Taylor series rises, it approaches the correct function. ...

Example

To find , we can use the fact that . The linearization of at x = a is , because the function defines the slope of the function at x. Plugging in a = 4, the linearization at 4 is . In this case x = 4.001, so is approximately . The true value is closer to 2.00024998, which is an amazingly accurate.

With Analysis of Linear Functions

The analysis of linear functions is well defined, but most representations of actual systems are nonlinear. Linearization allows us 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 A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ... A linear system is a model of a system based on some kind of linear operator. ... As the degree of the taylor series rises, it approaches the correct function. ...

,

the linearized system can be written as

where is the point of interest and is the Jacobian of evaluated at . In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

In stability analysis, one can use the eigenvalues of the Jacobian matrix evaluated at an equilibrium point to determine the nature of that equilibrium. If all the eigenvalues are positive, the equilibrium is unstable; if they are all negative the equilibrium is stable; and if the values are of mixed signs, the equilibrium is a saddle point. Any complex eigenvalues will appear in complex conjugate pairs and indicate spiral (or circular if the real components are zero) around the equilibrium. The word stability has a number of technical meanings, all related to the common meaning of the word. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... The point is called an equilibrium point for the differential equation if for all . ... A negative number is a number that is less than zero, such as −3. ... Plot of y = x3 with a saddle-point at (0,0). ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In mathematics, a spiral is a curve which turns around some central point or axis, getting progressively closer to or farther from it, depending on which way you follow the curve. ... In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ...