Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

PARTIAL DIFFERENTIATION


Thus, from (5.17), we may write



∂x

=cosφ


∂ρ


sinφ
ρ


∂φ

,



∂y

=sinφ


∂ρ

+


cosφ
ρ


∂φ

.


Now it is only a matter of writing


∂^2 f
∂x^2

=



∂x

(


∂f
∂x

)


=



∂x

(



∂x

)


f

=


(


cosφ


∂ρ


sinφ
ρ


∂φ

)(


cosφ


∂ρ


sinφ
ρ


∂φ

)


g

=


(


cosφ


∂ρ


sinφ
ρ


∂φ

)(


cosφ

∂g
∂ρ


sinφ
ρ

∂g
∂φ

)


=cos^2 φ

∂^2 g
∂ρ^2

+


2cosφsinφ
ρ^2

∂g
∂φ


2cosφsinφ
ρ

∂^2 g
∂φ∂ρ

+

sin^2 φ
ρ

∂g
∂ρ

+


sin^2 φ
ρ^2

∂^2 g
∂φ^2

and a similar expression for∂^2 f/∂y^2 ,


∂^2 f
∂y^2

=


(


sinφ


∂ρ

+


cosφ
ρ


∂φ

)(


sinφ


∂ρ

+


cosφ
ρ


∂φ

)


g

=sin^2 φ

∂^2 g
∂ρ^2


2cosφsinφ
ρ^2

∂g
∂φ

+


2cosφsinφ
ρ

∂^2 g
∂φ∂ρ

+

cos^2 φ
ρ

∂g
∂ρ

+


cos^2 φ
ρ^2

∂^2 g
∂φ^2

.


When these two expressions are added together the change of variables is complete and
we obtain


∂^2 f
∂x^2

+


∂^2 f
∂y^2

=


∂^2 g
∂ρ^2

+


1


ρ

∂g
∂ρ

+


1


ρ^2

∂^2 g
∂φ^2

.


5.7 Taylor’s theorem for many-variable functions

We have already introduced Taylor’s theorem for a functionf(x) of one variable,


in section 4.6. In an analogous way, the Taylor expansion of a functionf(x, y)of


two variables is given by


f(x, y)=f(x 0 ,y 0 )+

∂f
∂x

∆x+

∂f
∂y

∆y

+

1
2!

[
∂^2 f
∂x^2

(∆x)^2 +2

∂^2 f
∂x∂y

∆x∆y+

∂^2 f
∂y^2

(∆y)^2

]
+···, (5.18)

where ∆x=x−x 0 and ∆y=y−y 0 , and all the derivatives are to be evaluated


at (x 0 ,y 0 ).

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