082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

25.6 Taylor polynomials and Taylor series in two variables 835
6 Given
φ =
(a) Show
√
x 2 +y 2
∂ 2 φ
∂x 2 =y2 (x 2 +y 2 ) −3/2
(b) Verify that φ is asolution of
∂ 2 φ
∂x 2 + ∂2 φ
∂y 2 = (x2 +y 2 ) −1/2
25.6 TAYLORPOLYNOMIALSANDTAYLORSERIES
INTWOVARIABLES
In Chapter 18 we introduced Taylor polynomials and Taylor series for functions of a
single variable. We now extend this to include functions of two variables. Recall the
mainideabehindTaylorpolynomialsandseriesforafunctionofonevariable.Knowing
thevaluesofafunction, f (x),anditsderivativesatx =awecanwritedowntheTaylor
polynomial generated by f aboutx = a. This polynomial approximates to the function
f. The values of the Taylor polynomial and the function are usually in close agreement
for values of x near to x = a. To put it another way, knowing the value of f and its
derivatives atx =aallows us toestimatethe value of f near tox =a.
The same idea holds when f is a function of two variables,xandy. If we know the
valueof f anditspartialderivativesatapointx =a,y =b,thentheTaylorpolynomials
allow ustoestimate f atpoints near to (a,b).
25.6.1 First-orderTaylorpolynomialintwovariables
Suppose f is a function of two independent variables, x and y, and that the values of
f, ∂f ∂f
and areknown atthe pointx =a,y =b, that isweknow
∂x ∂y
f(a,b)
∂f
∂x (a,b)
∂f
∂y (a,b)
The first-order Taylor polynomial, p 1
(x,y), generated by f about (a,b) isgiven by
p 1
(x,y)= f(a,b)+(x−a) ∂f
∂x (a,b)+(y−b)∂f ∂y (a,b)
We note the following properties of a first-order Taylor polynomial:
(1) ThevaluesoftheTaylorpolynomialandthefunctionareidenticalatthepoint (a,b).
(2) The values of the first partial derivatives of the Taylor polynomial and the function
are identical atthe point (a,b).
(3) Thehighestderivative needed tocalculatetheTaylor polynomialisthefirstderivative.
(4) The first-order Taylor polynomial contains only linear terms; that is, there are no
powers ofxoryhigher than 1.
The first-order Taylor polynomial represents a plane which is tangent to the surface
f(x,y)at (a,b).
We can use p 1
(x,y) toestimatethe value of f near to (a,b).