126 Functions of Several Variables- If the partial derivatives fxy and fyx exist and are continuous in some
neighborhood of the point A, then fxy(A) = fyx(A); the result is known
as Clairaut' s theorem, after the French mathematician ALEXIS CLAUDE
CLAIRAUT (1713-1765).
EXERCISE 3.1.6.C Let r = r(t) be a vector-valued function, and f, a scalar
field. Define the function g(t) = f(r(t)), where f(r(t)) = f(P) if' r(t) —
OP for some fixed reference point O. Assume that r is differentiable at
the point t = to and the function f is differentiable at the point A, where
0l = r(to). Show that the function g is differentiable at the point t = to
and
g'(t 0 ) = Vf(A)-r'(to). (3.1.9)Try to argue directly by the definitions without using partial derivatives.
Hint: by (3.1.6), g(t) - g(t 0 ) « Vf(A) • (r(t) - r{t 0 )).
If r = r(t) is a vector-valued function of one variable and G is a vec-
tor field, then R(t) = G(r(t)) is a vector-valued function of one variable.
How to find the derivative R'(t) in terms of r and G? In cartesian co-
ordinates, G — G\ i + G23 + G3K, and G\,G2,G^ are scalar fields. By
(3.1.9), we conclude that R'{t) = r'(t)-VGi(r(t))i + r'(t)-VG 2 (r(t))j +
r'(t) • VG3(r(t))/«. To make the construction coordinate-free, we define
the vector (r'(t) • V)G(r(t)) so that, for every fixed unit vector n,
((r'(i) • V)G(r(*))) • n = r'(t) • (V(G • n)(r(t))).With this definition,
~G(r(t)) = (r'(t) • V)G(r(t)). (3.1.10)More generally, for two vector fields F, G, we define the vector field
(F • V)G so that, for every fixed unit vector n,
n-((F-V)G) =F-(V(G-n)). (3.1.11)The expressions of the type (F • V).F often appears in the equations de-
scribing the motion of fluids.
Given a scalar field / and a real number c, the set of points {P : f(P) —
c} is called a level set of /. In two dimensions, that is, for A = M^2 , the
level sets of / are curves. In three dimensions (A = M^3 ), the level sets of