
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a linear approximation is an approximation of a general
function using a
linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
(more precisely, an
affine function
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''wikt:affine, affinis'', "connected with") is a geometric transformation that preserves line (geometry), lines and parallel (geometry), parallelism, but not necessarily ...
). They are widely used in the method of
finite differences
A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation.
The difference operator, commonly d ...
to produce first order methods for solving or approximating solutions to equations.
Definition
Given a twice continuously differentiable function
of one
real variable,
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the k-th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation a ...
for the case
states that
where
is the remainder term. The linear approximation is obtained by dropping the remainder:
This is a good approximation when
is close enough to since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the
tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to the graph of
at
. For this reason, this process is also called the tangent line approximation. Linear approximations in this case are further improved when the
second derivative
In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the secon ...
of a,
, is sufficiently small (close to zero) (i.e., at or near an
inflection point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph ...
).
If
is
concave down in the interval between
and
, the approximation will be an overestimate (since the derivative is decreasing in that interval). If
is
concave up, the approximation will be an underestimate.
Linear approximations for
vector
Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Disease vector, an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematics a ...
functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the
Jacobian matrix. For example, given a differentiable function
with real values, one can approximate
for
close to
by the formula
The right-hand side is the equation of the plane tangent to the graph of
at
In the more general case of
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
s, one has
where
is the
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued f ...
of
at
.
Applications
Optics
''Gaussian optics'' is a technique in
geometrical optics
Geometrical optics, or ray optics, is a model of optics that describes light Wave propagation, propagation in terms of ''ray (optics), rays''. The ray in geometrical optics is an abstract object, abstraction useful for approximating the paths along ...
that describes the behaviour of light rays in optical systems by using the
paraxial approximation
In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens).
A paraxial ray is a ray that makes a small angle (''θ'') to the optica ...
, in which only rays which make small angles with the
optical axis
An optical axis is an imaginary line that passes through the geometrical center of an optical system such as a camera lens, microscope or telescopic sight. Lens elements often have rotational symmetry about the axis.
The optical axis defines ...
of the system are considered. In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of a
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification and brightness, in terms of the geometrical shapes and material properties of the constituent elements.
Period of oscillation
The period of swing of a
simple gravity pendulum
A pendulum is a device made of a weight (object), weight suspended from a wikt:pivot, pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, Mechanical equilibrium, equilibrium position, it is subject to a res ...
depends on its
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
, the local
strength of gravity, and to a small extent on the maximum
angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
that the pendulum swings away from vertical, , called the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
.
It is independent of the
mass
Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
of the bob. The true period ''T'' of a simple pendulum, the time taken for a complete cycle of an ideal simple gravity pendulum, can be written in several different forms (see
pendulum
A pendulum is a device made of a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate i ...
), one example being the
infinite series
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
:
where ''L'' is the length of the pendulum and ''g'' is the local
acceleration of gravity.
However, if one takes the linear approximation (i.e. if the amplitude is limited to small swings,
[A "small" swing is one in which the angle θ is small enough that sin(θ) can be approximated by θ when θ is measured in radians] ) the
period is:
In the linear approximation, the period of swing is approximately the same for different size swings: that is, ''the period is independent of amplitude''. This property, called
isochronism
A sequence of events is isochronous if the events occur regularly, or at equal time intervals. The term ''isochronous'' is used in several technical contexts, but usually refers to the primary subject maintaining a constant period or interval ( ...
, is the reason pendulums are so useful for timekeeping. Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.
Electrical resistivity
The electrical resistivity of most materials changes with temperature. If the temperature ''T'' does not vary too much, a linear approximation is typically used:
See also
*
Binomial approximation
*
Euler's method
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explic ...
*
Finite differences
A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation.
The difference operator, commonly d ...
*
Finite difference methods
*
Newton's method
In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a ...
*
Power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
*
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
Notes
References
Further reading
*
*
*
{{Authority control
Differential calculus
Numerical analysis
First order methods