(^482) 16. THE CALCULUS
9 = 2
\j = 3+arctan fcx/ arctan fc
J/ = 4
FIGURE 7. The functional $(y) = J_ 1 (xy'(x)) dx does not as-
sume its minimum value for continuously differentiable functions
y(x) satisfying y(-l) = 2, y{+l) = 4. The limiting position of a
minimizing sequence is the dashed line.
restrictive assumptions about differentiability and taking account of the distinction
between a lower bound and a minimum.^10
An important example in this connection was Riemann's use of Dirichlet's
principle to prove the Riemann mapping theorem, which asserts that any simply
connected region in the plane except the plane itself can be mapped conformally
onto the unit disk Ä = {(÷, y): ÷^2 +y^2 < 1}. That principle required the existence
of a real-valued function u(x, y) that minimizes the integral
Ä
among all functions u(x, y) taking prescribed values on the boundary of the disk.
That function is the unique harmonic function in Ä with the given boundary values.
In 1870 Weierstrass called attention to the integral
Ö(ö)= É+\^2 (ø'(÷))^2 Ü÷,
which when combined with the boundary condition ø(—1) = á, ö(+1) = b, can be
made arbitrarily small by the function
a + b b — a arctan(fcx)
fix) = -ã- + 2 arctan(jfc) '
yet (if á ö b) cannot be zero for any function ø satisfying the boundary conditions
and such that ö' exists at every point.
Weierstrass' example was a case where it was necessary to look outside the
class of smooth functions for a minimum of the functional. The limiting position of
the graphs of the functions for which the integral approximates its minimum value
consists of the two horizontal lines from (—1, o) to (0, a), from (0, b) to (+1, b), and
the section of the y-axis joining them (see Fig. 7).
Weierstrass thought of the smoothness assumptions as necessary evils. He
recognized that they limited the generality of the results, yet he saw that without
them no application of the calculus was possible. The result is a certain vagueness
about the formulation of minimal principles in physics. A certain functional must be
(^10) This distinction was pointed out by Gauss as early as 1799, in his criticism of d'Alembert's
1746 proof of the fundamental theorem of algebra.