You've seen Taylor series for functions of 1 variable. For a function satisfying the appropriate conditions, we have

is the * remainder term*:

z is a number between c and x. The Remainder Term gives the error that occurs in approximating with the degree Taylor polynomial.

There is a similar formula for functions of several variables. To
make the notation a little better, I'll define *
higher-order differentials* as follows. Let .

And so on. Here's Taylor's formula for functions of several variables. With more variables, it's more complicated and technical; try to see the resemblance between the formula here and the one for functions of one variable.

* Theorem.* Suppose , where U is an open set in . Suppose f has continuous partial derivatives at all
points of U through order . Let , where and the segment from c to x is
contained in U. Then for some point z on the segment from c to x,

* Example.* Write out the Taylor expansion
through terms of degree 2 for a function of 2 variables .

Let's say we're expanding at a point . Then

* Example.* For a function ,

Write out the Taylor expansion of f at through terms of degree 2.

* Example.* Construct the Taylor series through
the order for at
.

* Example.* Let . Use
a -order Taylor approximation to approximate
.

I'll use a Taylor expansion at , since it's the closest "nice" point to .

The series is

Then

Copyright 2018 by Bruce Ikenaga