OurBigBook.comSite Source code

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 13C / a / Solution

Let

u \mapsto u + ε η

and

v \mapsto v + ε ξ

, where the variations vanish on the boundary. The first variation is

δ L = \iint_{Ω} (f_{u} η + f_{v} ξ + f_{u_{x}} η_{x} + f_{u_{y}} η_{y} + f_{v_{x}} ξ_{x} + f_{v_{y}} ξ_{y}) d x d y .

Applying integration by parts to the four derivative terms and discarding the boundary contributions gives

δ L = \iint_{Ω} (f_{u} - \frac{\partial f _{u_{x}}}{\partial x} - \frac{\partial f _{u_{y}}}{\partial y}) η d x d y

+ \iint_{Ω} (f_{v} - \frac{\partial f _{v_{x}}}{\partial x} - \frac{\partial f _{v_{y}}}{\partial y}) ξ d x d y .

The variations

η

and

ξ

are independent and arbitrary in the interior. The fundamental lemma of the calculus of variations therefore gives the two Euler-Lagrange equations for two fields

f_{u} - \partial_{x} f_{u_{x}} - \partial_{y} f_{u_{y}} = 0,

f_{v} - \partial_{x} f_{v_{x}} - \partial_{y} f_{v_{y}} = 0 .

Solved by gpt-5.6-sol high.

 Ancestors (11)