Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

5.7 TAYLOR’S THEOREM FOR MANY-VARIABLE FUNCTIONS


Find the Taylor expansion, up to quadratic terms inx− 2 andy− 3 ,off(x, y)=yexpxy
about the pointx=2,y=3.

We first evaluate the required partial derivatives of the function, i.e.


∂f
∂x

=y^2 expxy ,

∂f
∂y

=expxy+xyexpxy ,

∂^2 f
∂x^2

=y^3 expxy ,

∂^2 f
∂y^2

=2xexpxy+x^2 yexpxy ,

∂^2 f
∂x∂y

=2yexpxy+xy^2 expxy.

Using (5.18), the Taylor expansion of a two-variable function, we find


f(x, y)≈e^6

{


3+9(x−2) + 7(y−3)

+(2!)−^1

[


27(x−2)^2 + 48(x−2)(y−3) + 16(y−3)^2

]}


.


It will be noticed that the terms in (5.18) containing first derivatives can be

written as


∂f
∂x

∆x+

∂f
∂y

∆y=

(
∆x


∂x

+∆y


∂y

)
f(x, y),

where both sides of this relation should be evaluated at the point (x 0 ,y 0 ). Similarly


the terms in (5.18) containing second derivatives can be written as


1
2!

[
∂^2 f
∂x^2

(∆x)^2 +2

∂^2 f
∂x∂y

∆x∆y+

∂^2 f
∂y^2

(∆y)^2

]
=

1
2!

(
∆x


∂x

+∆y


∂y

) 2
f(x, y),
(5.19)

where it is understood that the partial derivatives resulting from squaring the


expression in parentheses act only onf(x, y) and its derivatives, and not on ∆x


or ∆y; again both sides of (5.19) should be evaluated at (x 0 ,y 0 ). It can be shown


that the higher-order terms of the Taylor expansion off(x, y) can be written in


an analogous way, and that we may write the full Taylor series as


f(x, y)=

∑∞

n=0

1
n!

[(
∆x


∂x

+∆y


∂y

)n
f(x, y)

]

x 0 ,y 0

where, as indicated, all the terms on the RHS are to be evaluated at (x 0 ,y 0 ).


The most general form of Taylor’s theorem, for a functionf(x 1 ,x 2 ,...,xn)ofn

variables, is a simple extension of the above. Although it is not necessary to do


so, we may think of thexias coordinates inn-dimensional space and write the


function asf(x), wherexis a vector from the origin to (x 1 ,x 2 ,...,xn). Taylor’s

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