PARTIAL DIFFERENTIATION
varied. However, it is often the case in physical problems that not all the vari-
ables used to describe a situation are in fact independent, i.e. some relationship
between the variables must be satisfied. For example, if we walk through a hilly
landscape and we are constrained to walk along a path, we will never reach
the 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 our
journey.
We first discuss the case of a function of just two variables. Let us considerfinding the maximum value of the differentiable functionf(x, y) subject to the
constraintg(x, y)=c,wherecis a constant. In the above analogy,f(x, y) might
represent the height of the land above sea-level in some hilly region, whilst
g(x, y)=cis the equation of the path along which we walk.
We could, of course, use the constraintg(x, y)=cto substitute forxoryinf(x, y), thereby obtaining a new function of only one variable whose stationary
points 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 func-
tions of more than two variables. A more direct method for solving such problems
is themethod of Lagrange undetermined multipliers, which we now discuss.
To maximisefwe requiredf=∂f
∂xdx+∂f
∂ydy=0.Ifdxanddywere independent, we could concludefx=0=fy. However, here
they are not independent, but constrained becausegis constant:
dg=∂g
∂xdx+∂g
∂ydy=0.Multiplyingdgby an as yet unknown numberλand adding it todfwe obtain
d(f+λg)=(
∂f
∂x+λ∂g
∂x)
dx+(
∂f
∂y+λ∂g
∂y)
dy=0,whereλis called aLagrange undetermined multiplier. In this equationdxanddy
are to be independent and arbitrary; we must therefore chooseλsuch that
∂f
∂x+λ∂g
∂x=0, (5.27)∂f
∂y+λ∂g
∂y=0. (5.28)These equations, together with the constraintg(x, y)=c, are sufficient to find the
three unknowns, i.e.λand the values ofxandyat the stationary point.