This course provides an introduction to functions of multiple variables with a focus on functions of two- and three-variables.  In the course we extend the definition of the derivative to a multi-dimensional context, find local approximations of functions, identify maxima and minima of a function of multiple variables subject to constraints on the independent variables, and learn to integrate functions of several variables. 

In most areas of inquiry where functions arise, they are functions of multiple variables, so this extension of calculus is important to understanding problems in statistics, economics, biology, engineering, and physics.

The course also includes an introduction to vectors, vector operations, and vector notation.  We use vectors to describe points in two- or three-space, and use vector valued functions to generate mappings from a line to a curve and from a plane to a surface, as well as to describe vector fields.  Vector fields arise in the context of motion of water and of air, as well as in many more abstract contexts.  After defining flux of a vector field through a surface and circulation around a curve, we will study the vector calculus analogs of the Fundamental Theorem of Calculus.  Our course will conclude with a brief introduction to complex numbers.

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