Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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,
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