Mathematical Methods for Physics and Engineering - Matematica.NET

PARTIAL DIFFERENTIATIONvaried. However, it is often the case in physical problems that not all the variablesused to describe a situation are in fact independent, i.e. some relationshipbetween the variables must be satisfied. For example, if we walk through a hillylandscape and we are constrained to walk along a path, we will never reachthe highest peak on the landscape unless the path happens to take us to it.Nevertheless, we can still find the highest point that we have reached during ourjourney.We first discuss the case of a function of just two variables. Let us considerfinding the maximum value of the differentiable function f(x, y) subject to theconstraint g(x, y) =c, wherec is a constant. In the above analogy, f(x, y) mightrepresent the height of the land above sea-level in some hilly region, whilstg(x, y) =c is the equation of the path along which we walk.We could, of course, use the constraint g(x, y) =c to substitute for x or y inf(x, y), thereby obtaining a new function of only one variable whose stationarypoints could be found using the methods discussed in subsection 2.1.8. However,such a procedure can involve a lot of algebra and becomes very tedious for functionsof more than two variables. A more direct method for solving such problemsis the method of Lagrange undetermined multipliers, which we now discuss.To maximise f we requiredf = ∂f ∂fdx + dy =0.∂x ∂yIf dx and dy were independent, we could conclude f x =0=f y . However, herethey are not independent, but constrained because g is constant:dg = ∂g ∂gdx + dy =0.∂x ∂yMultiplying dg by an as yet unknown number λ and adding it to df we obtaind(f + λg) =( ∂f∂x + λ ∂g∂x)dx +( ∂f∂y + λ∂g ∂y)dy =0,where λ is called a Lagrange undetermined multiplier. In this equation dx and dyare to be independent and arbitrary; we must therefore choose λ such that∂f∂x + λ ∂g =0, (5.27)∂x∂f∂y + λ∂g =0. (5.28)∂yThese equations, together with the constraint g(x, y) =c, are sufficient to find thethree unknowns, i.e. λ and the values of x and y at the stationary point.168