Taylor series in several variables
The Taylor series may also be generalized to functions of more than one variable with
For example, for a function that depends on two variables, x and y, the Taylor series to second order about the point (a, b) is:
where the subscripts denote the respective partial derivatives.
A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as
where is the gradient of evaluated at and is the Hessian matrix. Applying the multi-index notation the Taylor series for several variables becomes
which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, again in full analogy to the single variable case.
Compute a second-order Taylor series expansion around point of a function
Firstly, we compute all partial derivatives we need
The Taylor series is
which in this case becomes
Since log(1 + y) is analytic in |y| < 1, we have
for |y| < 1.