Scalar-Valued Functions of Two Variables

Abstract

This chapter is devoted to differential calculus of functions of two variables. In particular we will study geometrical objects such as tangents and tangent planes, maxima and minima, as well as linear and quadratic approximations. The restriction to two variables has been made for simplicity of presentation. All ideas in this and the next chapter can easily be extended (although with slightly more notational effort) to the case of n variables. We begin by studying the graph of a function with the help of vertical cuts and level sets. As a further tool we introduce partial derivatives, which describe the rate of change of the function in the direction of the coordinate axes. Finally the notion of the Fréchet derivative allows us to define the tangent plane to the graph. As for functions of one variable the Taylor formula plays a central role. We use it, e.g., to determine extrema of functions of two variables. In the entire chapter D denotes a subset of $\mathbb R^2$, and

$$ f:D\subset \mathbb R^2 \rightarrow \mathbb R: (x,y) \mapsto z=f(x, y) $$

denotes a scalar-valued function of two variables. Details of vector and matrix algebra used in this chapter can be found in Appendices A and B.