Theorem 23 (Taylor's Theorem in two variables)

Given a function with all its partial derivatives of order less than or equal to continuous on

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Let ( ) lie in , then for every ( ) there exists a between and , and a between and such that

with

,

where and denote the partial derivatives with respect to and respectively.

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The error term is given by

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This Taylor series expansion can be obtained by the MapleV function 'mtaylor()':

> readlib(mtaylor):mtaylor(ff(x,y),[x=x0,y=y0],4);

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Now try to reproduce the following result manually:

> f:=(t,y)->t^2-t+t*sin(y)-y^2;

> readlib(mtaylor):mtaylor(f(t,y),[t=1,y=Pi],5);

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