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 bewritten 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=01
n![(
∆x∂
∂x+∆y∂
∂y)n
f(x, y)]x 0 ,y 0where, 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)ofnvariables, 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