E600 Mathematics
           Chapter 3: Multivariate Calculus
              Martin Reinhard
              August 27, 2021
   1. Introduction
   Motivation
    This chapter discusses
            Aformal introduction to multi-dimensional functions
            Key function properties: invertability, convexity (and concavity)
            Multivariate differentiation (main focus)
                   Formal definition and derivation
                   Application
            Multivariate integration: concept and key theorems
              Martin Reinhard                     Ch. 3: Multivariate Calculus                    August 27, 2021      1/56
   1. Introduction
   Motivation
            Thus far: Linear Algebra (linear operations, equation systems)
            Now: analysis of functions, study of (small) variations
            Here: generalizing the derivative to functions f : Rn 7→ Rm
            Why?: Optimization problems with many variables (goods,
            production inputs, statistical parameters)
            Many struggles in the 1st PhD semester were encountered because of
            issues with understanding derivatives...
              Martin Reinhard                     Ch. 3: Multivariate Calculus                    August 27, 2021      2/56
   1. Introduction
   Key Concepts
            Function f : X 7→ Y with domain X, codomain Y and image
            im(f ) = f [X]
                   X ⊆R: univariate function
                             n
                   X ⊆R : multivariate function
                   Y ⊆R: real-valued function
                   Y ⊆Rm: vector-valued function
                   How to call f : R3 7→ R2?
            Examples:
                   Multivariate, real-valued function: x 7→ kxk, x 7→ x′Ax, (x,y) 7→ x · y
                   Multivariate, vector-valued function: x 7→ Ax
            Graph:
                    G(f) = {(x,y) ∈ X ×Y : y = f(x)} = {(x,f(x)) : x ∈ X}
              Martin Reinhard                     Ch. 3: Multivariate Calculus                    August 27, 2021      3/56