# Taylor Series for Functions of Several Variables

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

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