Chapter 2 - Preliminaries

Summary

MULTIVARIABLE CALCULUS

Introduction

We plan to briefly introduce the calculus onℝⁿ, namely the concept of totalderivative of multivalued function,. We are indeed familiar with the notionof partial derivatives. In the sequel, we will introduce the importantconcept of total derivative and discuss its connection tothe partial derivatives. We remark that the total derivative (known also asFrechét derivative) can be extended to infinitedimensional normed linear spaces, which is used in the analysis of morecomplicated problems especially arising from optimal control problems,calculus of variations, partial differential equations, and so on.

Motivation: One of the fundamental problems in mathematics (andhence in applications as well) is the following: Let f :ℝⁿ →ℝⁿ. Given y∈ ℝⁿ, solve the system ofequations

f(x) = y (2.1)

and represent the solution as x =g(y) and if possible find goodproperties of g, namely its smoothness. More generally, iff : ℝ^n+m→ ℝⁿ; x ∈ℝⁿ; y ∈ℝ^m, solve the implicit system ofequations

f(x; y) = 0 (2.2)

and represent the solution as x =g(y). Consider the one-dimensionalcase, where f : ℝ → ℝ which isC¹. Suppose that for somea. Then, by the continuity of, we see that in aneighborhood interval I of a.Hence preserves the sign in I, fis monotonic in I andf(I) is an interval. Thus, iff(a) = b, then theabove argument shows that f(x) =y is solvable for all y inf(I), a neighborhood ofb. This is the local solvability that is obtained bythe non-vanishing property of the derivative of f ata. This immediately shows the importance ofunderstanding the derivatives in the solvability of algebraic equations. Weremark that the mere existence of all partial derivatives does not guaranteethe local solvability. We need the stronger concept of total derivative.

Linear Systems: Let us look at the well-known linear system

Ax = y, (2.3)

where A = [a_ij] is a given n × n matrix.That is f(x) = Ax. Thesystem (2.3) can be rewritten as

The system (2.3) or (2.4) is uniquely solvable for x interms of y if and only if det A ≠ 0(global solvability).