A powerful mathematical tool which is the generalization of the
→ Fourier series for the analysis of non-periodic functions.
The Fourier transform transforms a function defined on physical space into a function
defined on the space of frequencies, whose values quantify the "amount"
of each periodic frequency contained in the original function. The inverse Fourier transform
then reconstructs the original function from its transformed frequency components.
The integral F(α) = ∫ f(u)e^{-iαu}du
is called the Fourier transform of
F(x) = (1/2π)∫ f(α)e^{iαx}dx,
both integrals from -∞ to + ∞.

The method of relating a measurement in one
→ reference frame to another moving with a constant velocity
with respect to the first within the → Newtonian mechanics.
The Galilean transformation between the coordinate systems (x,y,z,t) and
(x',y',z',t') is expressed by the relations: x' = x - vt, y' = y,
z' = z. Galilean transformations break down at high velocities and for
electromagnetic phenomena and is superseded by the
→ Lorentz transformations.

An integral transform of a function obtained by multiplying the given
function f(t) by e^{-pt}, where p is a new variable, and integrating
with respect to t from t = 0 to t = ∞.

A mathematical operation that transforms one function into another.
Two differentiable functions f and g are said to be
Legendre transforms of each other if their first derivatives are
inverse functions of each other:
df(x)/dx = (dg(x)/dx)^{-1}.
The functions f and g are said to be related by a
Legendre transformation.

A set of linear equations that expresses the time and space coordinates of one
→ reference frame in terms of those of another one when one
frame moves at a constant velocity with respect to the other.
In general, the Lorentz transformation allows a change of the origin
of a coordinate system, a rotation around the origin, a reversal of
spatial or temporal direction, and a uniform movement along a spatial axis.
If the system S'(x',y',z',t') moves at the velocity v with respect to
S(x,y,z,t) in the positive direction of the x-axis, the Lorentz
transformations will be:
x' = γ(x - vt), y' = y, z' = z,
t' = γ [t - (vx/c^{2})], where c is the
→ velocity of light and
γ = [1 - (v/c)^{2}]^{-1/2}.
For the special case of velocities much less than c, the
Lorentz transformation reduces to → Galilean transformation.

1) A transformation that preserves angles and changes all distances in the same ratio.
2) A transformation of the form B = X^{-1}AX relating two
→ square matrices A and B.

1) Math.: A mathematical quantity obtained from a given quantity
by an algebraic, geometric, or functional transformation.
The transformation itself.
2) To change in form, appearance, or structure; to change in condition,
nature, or character; convert. Physics: To change into another form of energy.
To increase or decrease (the voltage and current characteristics of an alternating-current
circuit), as by means of a transformer. Math.: To change the form of (a figure, expression, etc.) without
in general changing the value.

1) The act or process of transforming. The state of being transformed.
2) The relationship between description of a physical phenomenon in one
→ reference frame to
another. → Galilean transformation;
→ Lorentz transformation.
3) Math.: The act, process, or result of transforming or mapping.

A device that converts low voltages to higher voltages, or vice versa. A transformer consists
of a primary coil and a secondary coil, both traversed by the same magnetic flux.