Linear Algebra, Theory And Applications, 2012a

182 SPECTRAL THEORY
and so if the new variables are given by
⎛
⎜
⎝
⎞
− √ 1 √6 1 √3 1
2
2
0 √6 − √ 1 ⎟
3 ⎠
√1
√6 1 √3 1
2
⎛
⎝
x ′
y ′
z ′
⎞ ⎛
⎠ = ⎝
it follows that in terms of the new variables the quadratic form is 2 (x ′ ) 2 − 4(y ′ ) 2 +8(z ′ ) 2 .
You can work other examples the same way.
7.7 Second Derivative Test
Under certain conditions the mixed partial derivatives will always be equal. This astonishing
fact was first observed by Euler around 1734. It is also called Clairaut’s theorem.
Theorem 7.7.1 Suppose f : U ⊆ F 2 → R where U is an open set on which f x ,f y ,f xy and
f yx exist. Then if f xy and f yx are continuous at the point (x, y) ∈ U, itfollows
f xy (x, y) =f yx (x, y) .
Proof: Since U is open, there exists r>0 such that B ((x, y) ,r) ⊆ U. Now let |t| , |s| <
r/2,t,s real numbers and consider
h(t)
Δ(s, t) ≡ 1 { }} { { }} {
st { f (x + t, y + s) − f (x + t, y) − (f (x, y + s) − f (x, y))}. (7.16)
Note that (x + t, y + s) ∈ U because
x
y
z
⎞
⎠ ,
h(0)
|(x + t, y + s) − (x, y)| = |(t, s)| = ( t 2 + s 2) 1/2
≤
( r
2
4 + r2
4
) 1/2
= r √
2