Editing Envelope (mathematics) (section)

=== Partial differential equations ===
Envelopes can be used to construct more complicated solutions of first order [[partial differential equation]]s (PDEs) from simpler ones.<ref>{{Citation | last1=Evans | first1=Lawrence C. | title=Partial differential equations | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-0772-9 | year=1998}}.</ref>  Let ''F''(''x'',''u'',D''u'')&nbsp;=&nbsp;0 be a first order PDE, where ''x'' is a variable with values in an open set Ω&nbsp;⊂&nbsp;'''R'''<sup>''n''</sup>, ''u'' is an unknown real-valued function, D''u'' is the [[gradient]] of ''u'', and ''F'' is a continuously differentiable function that is regular in D''u''.  Suppose that ''u''(''x'';''a'') is an ''m''-parameter family of solutions:  that is, for each fixed ''a''&nbsp;∈&nbsp;''A''&nbsp;⊂&nbsp;'''R'''<sup>''m''</sup>, ''u''(''x'';''a'') is a solution of the differential equation.  A new solution of the differential equation can be constructed by first solving (if possible)
:<math>D_a u(x;a) = 0\,</math>
for ''a''&nbsp;=&nbsp;φ(''x'') as a function of ''x''.  The envelope of the family of functions {''u''(·,''a'')}<sub>''a''∈''A''</sub> is defined by
:<math>v(x) = u(x;\varphi(x)),\quad x\in\Omega,</math>
and also solves the differential equation (provided that it exists as a continuously differentiable function).

Geometrically, the graph of ''v''(''x'') is everywhere tangent to the graph of some member of the family ''u''(''x'';''a''). Since the differential equation is first order, it only puts a condition on the tangent plane to the graph, so that any function everywhere tangent to a solution must also be a solution. The same idea underlies the solution of a first order equation as an integral of the [[Monge cone]].<ref>{{citation |first=Fritz |last=John |authorlink=Fritz John |title=Partial differential equations |publisher=Springer |edition=4th |year=1991 |isbn=978-0-387-90609-6 |url-access=registration |url=https://archive.org/details/partialdifferent00john_0 }}.</ref>  The Monge cone is a cone field in the '''R'''<sup>''n''+1</sup> of the (''x'',''u'') variables cut out by the envelope of the tangent spaces to the first order PDE at each point. A solution of the PDE is then an envelope of the cone field.

In [[Riemannian geometry]], if a smooth family of [[geodesic]]s through a point ''P'' in a [[Riemannian manifold]] has an envelope, then ''P'' has a [[conjugate point]] where any geodesic of the family intersects the envelope. The same is true more generally in the [[calculus of variations]]: if a family of extremals to a functional through a given point ''P'' has an envelope, then a point where an extremal intersects the envelope is a conjugate point to ''P''.