Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PARTIAL DIFFERENTIATION

minimum

x

y

− (^2) − (^1123)

2

− 3 −^2

− 0. 2

− 0. 4

0. 2

0. 4

0

0

0

maximum

Figure 5.3 The functionf(x, y)=x^3 exp(−x^2 −y^2 ).

If we define the matrixMto have elements given by

Mij=

∂^2 f

∂xi∂xj

,

then we can rewrite (5.25) as

∆f=^12 ∆xTM∆x, (5.26)

where ∆xis the column vector with the ∆xias its components and ∆xTis its

transpose. SinceMis real and symmetric it hasnreal eigenvaluesλrandn

orthogonal eigenvectorser, which after suitable normalisation satisfy

Mer=λrer, eTres=δrs,

where theKronecker delta, writtenδrs, equals unity forr=sand equals zero

otherwise. These eigenvectors form a basis set for then-dimensional space and

we can therefore expand ∆xin terms of them, obtaining

∆x=

∑

r

arer,