In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, constraint counting is counting the number of
constraints in order to compare it with the number of
variables,
parameters
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
, etc. that are free to be determined, the idea being that in most cases the number of independent choices that can be made is the excess of the latter over the former.
For example, in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
if the number of constraints (independent equations) in a
system of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three ...
equals the number of unknowns then precisely one solution exists; if there are fewer independent equations than unknowns, an infinite number of solutions exist; and if the number of independent equations exceeds the number of unknowns, then no solutions exist.
In the context of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s, constraint counting is a crude but often useful way of counting the number of ''free functions'' needed to specify a solution to a
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
.
Partial differential equations
Consider a second order partial differential equation in three variables, such as the two-dimensional
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
:
It is often profitable to think of such an equation as a ''rewrite rule'' allowing us to rewrite arbitrary partial derivatives of the function
using fewer partials than would be needed for an arbitrary function. For example, if
satisfies the wave equation, we can rewrite
:
where in the first equality, we appealed to the fact that ''partial derivatives commute''.
Linear equations
To answer this in the important special case of a
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
partial differential equation, Einstein asked: how many of the partial derivatives of a solution can be
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
? It is convenient to record his answer using an
ordinary generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series ...
:
where
is a natural number counting the number of linearly independent partial derivatives (of order k) of an arbitrary function in the solution space of the equation in question.
Whenever a function satisfies some partial differential equation, we can use the corresponding rewrite rule to eliminate some of them, because ''further mixed partials have necessarily become linearly dependent''. Specifically, the power series counting the variety of ''arbitrary'' functions of three variables (no constraints) is
:
but the power series counting those in the solution space of some second order p.d.e. is
:
which records that we can eliminate ''one'' second order partial
, ''three'' third order partials
, and so forth.
More generally, the o.g.f. for an arbitrary function of n variables is
:
where the coefficients of the infinite
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 ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
of the generating function are constructed using an appropriate infinite sequence of
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s, and the power series for a function required to satisfy a linear m-th order equation is
:
Next,
:
which can be interpreted to predict that a solution to a second order linear p.d.e. in ''three'' variables is expressible by two ''freely chosen'' functions of ''two'' variables, one of which is used immediately, and the second, only after taking a ''first derivative'', in order to express the solution.
General solution of initial value problem
To verify this prediction, recall the solution of the
initial value problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or oth ...
:
Applying the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
gives
:
Applying the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
to the two spatial variables gives
: