Constructing the Taylor Series in Two Variables: A Step-by-Step Guide

The Taylor series is one of the most powerful tools in mathematics, helping us approximate complex functions with polynomials. In single-variable calculus, this series allows us to approximate a function f(x) around a point by considering its values and derivatives at that point. But what about functions of multiple variables? For functions in two variables, we use a similar concept to construct the Taylor series, but the process incorporates partial derivatives with respect to each variable. 

In this blog post, we’ll dive into the Taylor series in two variables, taking a close look at each step to construct it and discussing why it’s useful for approximating functions in applications like optimization, engineering, and even machine learning.


What is a Taylor Series?

The Taylor series for a function around a point (a, b) is a polynomial that approximates the function f(x, y) by adding up terms involving the function's partial derivatives at the point. The degree of the approximation can vary depending on how many terms we include, which allows us to choose an approximation that’s as simple or as accurate as needed.


For a function f(x, y), the general form of the Taylor series around the point (a,b) is:

where T_n(x, y) is the Taylor polynomial of degree n for f(x, y).


Step 1: Set Up and Understand Partial Derivatives

To construct the Taylor series in two variables, we first need to understand the partial derivatives of the function f(x, y). Partial derivatives tell us how the function changes as we vary one variable while keeping the other constant. For a Taylor series in two variables, we’ll use partial derivatives with respect to both x and y, up to the degree we desire for the approximation.


For example:

The first-order partial derivatives f_x(a, b) and f_y(a, b) describe the slopes of f in the x- and y-directions at the point (a,b). The second-order partial derivatives f_{xx}(a, b), f_{yy}(a, b), and f_{xy}(a, b) describe the curvature of f in various directions at (a, b).


Step 2: Construct the Taylor Polynomial

Let’s start building the Taylor series step-by-step for the cases where n = 1 (linear approximation), n = 2 (quadratic approximation), and n = 3 (cubic approximation).


First-Order Taylor Polynomial (Linear Approximation)

The first-order Taylor polynomial (often called the linear approximation) for f(x, y) around (a, b) is:


Second-Order Taylor Polynomial (Quadratic Approximation)

The second-order Taylor polynomial adds terms involving second-order partial derivatives, which account for the curvature of f around (a, b):

Third-Order Taylor Polynomial (Cubic Approximation)

To improve accuracy further, we can construct the third-order Taylor polynomial by adding terms involving third-order partial derivatives:

Each additional derivative term enhances the polynomial’s ability to match the original function’s shape near (a,b). 

Step 3: Interpret and use the Taylor Polynomial

  1. Each degree of the Taylor polynomial gives a different level of approximation accuracy
  2. Each additional derivative term enhances the polynomial’s ability to match the original function’s shape near (a,b). 
  3. Third order (cubic): Adds even more accuracy by capturing how curvature changes, useful for understanding finer details.

In practical applications, we choose the degree of the polynomial based on the desired accuracy and the complexity we can handle. For example:


In optimization, a second-order Taylor approximation (using derivatives up to the second order) is common to find critical points.


In physics and engineering, where precision is crucial, third-order or higher approximations may be used to understand local behavior in detail.


Example: Constructing a Second-Order Taylor Polynomial

Let's consider a concrete example. Suppose we have a function f(x, y) = exp(x+y) and we want to approximate it around the point (0, 0).


1. Evaluate f and its partial derivatives at (0, 0):

  


2. Construct the polynomial:

Simplifying, we get:

This polynomial provides a quadratic approximation of f(x, y) around (0, 0).

Final Thoughts

The Taylor series in two variables is an indispensable tool for approximating and analyzing functions of multiple variables. With each added degree, we capture more details of the function’s local behavior, which is invaluable in fields like engineering, physics, and machine learning. Whether you need a simple linear approximation or a detailed cubic model, the Taylor series provides a flexible and powerful framework to approximate complex functions with relative ease.

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