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.
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