Lagrange's identity

Lagrange's identity can be proved in a variety of ways.

The second term on the left side of Lagrange's identity can be expanded as:

Explicitly, for complex numbers, Lagrange's identity can be written in the form:

In the present approach, Lagrange's identity is actually derived without assuming it a priori.

Lagrange's identity for complex numbers has been obtained from a straightforward product identity.

In a more compact vector notation, Lagrange's identity is expressed as:

In more than one independent variable, Lagrange's identity is generalized by Green's second identity.

Lagrange's identity exhibits this equality.

Lagrange's identity is fundamental in Sturm-Liouville theory.

Lagrange's identity may refer to:

Lagrange's identity (boundary value problem)

To expand the summation on the right side of Lagrange's identity, first expand the square within the summation:

In algebra, Lagrange's identity, named after Joseph Louis Lagrange, is:

It can be shown that for any u and v for which the various derivatives exist, Lagrange's identity for ordinary differential equations holds:

If Lagrange's identity is integrated over a bounded region, then the divergence theorem can be used to form Green's second identity in the form:

For ordinary differential equations defined in the interval [0, 1], Lagrange's identity can be integrated to obtain an integral form (also known as Green's formula):

Since the Cauchy-Schwarz inequality is a particular case of Lagrange's identity, this proof is yet another way to obtain the CS inequality.

In general terms, Lagrange's identity for any pair of functions u and v in function space C (that is, twice differentiable) in n dimensions is:

It is a special case of another formula, also sometimes called Lagrange's identity, which is the three dimensional case of the Binet-Cauchy identity:

In three dimensions, Lagrange's identity asserts that the square of the area of a parallelogram in space is equal to the sum of the squares of its projections onto the Cartesian coordinate planes.

In the study of ordinary differential equations and their associated boundary value problems, Lagrange's identity, named after Joseph Louis Lagrange, gives the boundary terms arising from integration by parts of a self-adjoint linear differential operator.

The virial theorem can be obtained directly from Lagrange's Identity as applied in classical gravitational dynamics, the original form of which was included in Lagrange's "Essay on the Problem of Three Bodies" published in 1772.