Taylor Series for Multi-variable Functions

For a single-output function , around , the Taylor series of this function is

where is the sum over “multi-indices.”

function of two inputs

For a planar function at origin

Taylor Approximation

We can use Taylor series to approximate multi-variable functions as polynomial.

Composition and Substitution

Sometimes it is hard to work with Taylor series of multi-variable functions. Nevertheless, we can simplify the process by using substitution to transform the function into a function of one variable.

Example: Taylor expand about up to terms of order 3

Let , then

And then we can expand the above equation and discard higher order terms

References