# Functions of Two Variables

Vectors allow us to study curves in the xy-plane and in 3 dimensional space, but we cannot yet use vectors to study surfaces and solids in 3 dimensional space. Thus, in this chapter, we introduce the primary tool in the study of surfaces and solids-functions of 2 variables.  Of particular importance in this introduction to functions of 2 variables will be the concept of a partial derivative, and we will use the partial derivative to study concepts related to functions of 2 variables, such as tangent planes, extrema, and saddle points.

Moreover, we will also see that partial derivatives are important beyond their use in studying the geometry of a surface. Indeed, much of the science of the last century has been presented in the form of partial differential equations. Thus, we will see that partial derivatives are also important in the types of equations and models that have dominated mathematics and science for over a century.

 2.1 Functions of 2 Variables 2.6 The Chain Rule 2.2 Limits and Continuity 2.7 Gradients and Level Curves 2.3 Partial Derivatives 2.8 Optimization 2.4 Partial Differential Equations 2.9 Lagrange Multipliers 2.5 Differentiability and Approximations

Summary and Review