Introduction to Taylor Series for Functions of Two Variables

Overview

• Presenter: Alex
• Topic: Introduction to the Taylor series for functions of two variables.
• Purpose: Learn how to expand a function (f(x, y)) using the Taylor series around a point ((a, b)).

Taylor Series Expansion

• Expression in Powers:
  • Expand (f(x, y)) in the powers of (x-a) and (y-b).

Taylor Series Formula

• General Formula: [ f(x, y) = f(a, b) + \frac{1}{1!} [(x-a)f_x(a, b) + (y-b)f_y(a, b)] + \dots ]

Steps to Write the Formula

1. First Term:

   • (f(a, b))
   • Substitute (a) and (b) for (x) and (y) in (f(x, y)).

2. Second Term:

   • ( + \frac{1}{1!} [(x-a)f_x(a, b) + (y-b)f_y(a, b)] )
   • Use partial differentiation of (f) with respect to (x) and (y) at ((a, b)).

3. Third Term:

   • ( + \frac{1}{2!} [(x-a)^2 f_{xx}(a, b) + 2(x-a)(y-b)f_{xy}(a, b) + (y-b)^2 f_{yy}(a, b)] )
   • Involves second partial derivatives.

4. Fourth Term:

   • ( + \frac{1}{3!} [(x-a)^3 f_{xxx}(a, b) + 3(x-a)^2(y-b)f_{xxy}(a, b) + 3(x-a)(y-b)^2 f_{xyy}(a, b) + (y-b)^3 f_{yyy}(a, b)] )
   • Involves third partial derivatives.

5. Continue the Pattern:

   • Continue using the formula for higher-order terms as required._

Key Concepts

• Partial Differentiation: Essential for constructing each term of the series.
• Higher-order Terms: Follow the pattern of expansion using derivatives.

Application

• Use this expansion to approximate functions of two variables around a point.

Conclusion

• The Taylor series provides a method to expand and approximate functions of two variables about a specific point ((a, b)). The series starts from (f(a, b)) and adds terms involving higher-order derivatives.